Modeling Subjectivity and Interpretations in Games: A Hypergame Theoretic Approach

Doctoral Dissertation Presented to Department of Value and Decision Science Graduate School of Decision Science and Technology Tokyo Institute of Technology

Submitted by

Yasuo SASAKI Thesis Adviser: Professor Kyoichi Kijima March 2013

Abstract The present thesis studies modeling subjectivity and interpretations in games. That is, we study in a formal way roles of subjectivity and interpretations of agents (decision makers) in interactive decision making. There are mainly three issues we address. The first one is “modeling subjective formulations of games.” We, throughout the thesis, adopt hypergame theory (Bennett (1977)), a game theoretical framework which allows agents to perceive games subjectively. We first argue that several topics on theoretical foundations of hypergames remain a matter of further study. In particular, we discuss two topics in the thesis. One is on rational behaviors of agents in hierarchical hypergames, a type of hypergames that allows an agent to have a “hierarchy of perceptions” such as an agent’s view about another agent’s view and so on. In Chapter 3, we propose a new solution concept called subjective rationalizability and examine its properties. An action of an agent is said to be subjectively rationalizable when the agent thinks it can be a best response to the other’s choices, each of which the agent thinks each agent thinks is a best response to the other’s choices, and so on. Although the concept would appear impractical as it is because we need an infinite sequence of best responses to calculate it, we, in order to make the concept applicable for hypergame analysis, then prove that it is equivalent to rationalizability in the standard game theory under a condition called inside common knowledge. The other topic is on a relationship between hypergames and another model of games with incomplete information, Bayesian games (Harsanyi (1967)). In Chapter 4, we compare the two independently developed models. We first show that any hypergames can naturally be reformulated in terms of Bayesian games in an unified way called Bayesian representation of hypergames, and then prove that some equilibrium concepts defined for hypergames provide us with the same implications as equilibria for Bayesian games. For instance, a hyper Nash equilibrium in a simple hypergame always consists of such actions that the actual type of each agent, types of the agents to which the objective probability 1 is assigned, take in a Bayesian Nash equilibrium in the corresponding Bayesian game, and vice versa. Based on the results, we discuss carefully how each model should be used according to the analyzer’s purpose. Next, the second issue is “modeling revisions of subjective views.” When our interest is in situations in which a game is played repeatedly, not only analyzing what outcome would be obtained but also studying how an outcome of the game may encourage an agent to learn the situation would be important for understanding stationary states in such situations. In a repeated hypergame, a new framework we propose in Chapter 5, agents interact repetitively, and may update each subjective view according to an outcome of the game at each period, while their choices are supposed to be subject to each subjective view at the current period. A game at each period is assumed to be described as a simple hypergame. The framework will be applied to a certain class of hypergames called systems of holding back, where the agents believe they themselves do the best from each agent’s point of view but misperceptions prevent achievements of Pareto-optimal stationary state from an objective point of view. We examine how differences of behavioral ways of agents may work in such situations, and show that apparently irrational behavior can lead to the betterment of such situations while if the agents always choose rational actions, i.e., Nash actions, in light of their own perceptions, i

the situation never be improved. The idea is inspired by a recently developed systemic concept called systems intelligence (H¨am¨al¨ainen and Saarinen (2004)), and our study can be considered as a game theoretic characterization of it. Finally, the third main issue is “modeling interpretations of actions.” By interpretation, we here mean how, in an extensive form game, an agent formulates a subgame she faces, having observed another agent’s choice. For the purpose, we, in Chapter 6, first propose a new framework called extensive hypergames, which is an extension of hypergames to extensive forms. We restrict our attention to a simple class of it called Stackelberg hypergames where an agent may misperceive the opponent’s utility function in Stackelberg games, and examine their properties. Then we explicitly formulate interpretations in the sense above by introducing interpretation functions. An interpretation function is a mapping from actions to what we call intentions. For example, if some two actions, such as to raise a hand and to say “Hello.” are indifferent in the context of a game, they might be interpreted as an intention, to greet someone. We assume that, when an agent formulates her subjective game with intentions, she may choose any actions to achieve an intention which she considers she chooses. Then it is shown that whether agents formulate the situation at the level of actions or intentions can be critical to determine stationary states. That is, the set of equilibria may be different in those cases. Therefore, when a game is considered as a mental model of an agent, how she identifies the set of alternatives is very crucial.

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Acknowledgement I would like to express my greatest gratitude to my thesis advisor, Professor Kyoichi Kijima. Ever since I joined his research group, his valuable advice not only for my research scheme but also for my entire student life has helped me greatly. The systems perspective I learnt from him has surely changed my way of thinking, and hence my life. The excellent research environment of Tokyo Institute of Technology is indispensable for my working out the dissertation. Above all I appreciate Professor Takehiro Inohara and Assistant Professor Norimasa Kobayashi teaching me about game theory and hypergame theory, the theoretical bases of the dissertation, in their lectures as well as in informal discussions. Then careful review of the dissertation and valuable comments by Professor Shigeo Muto, Professor Akifumi Tokosumi and Associate Professor Mayuko Nakamaru are gratefully acknowledged. I also thank all the members in Kijima and Inohara laboratory for intellectual conversations, kind supports and so on. In addition, I feel so grateful to Professor Raimo P. H¨am¨al¨ainen for accepting me as a visiting researcher in Systems Analysis Laboratory, Helsinki University of Technology (AprDec in 2009). His research program of systems intelligence has inspired my ideas so much. Since 2011, I have worked for Value Management Institute, Inc., where I have a lot of chances to apply what I learnt in Tokyo Tech to real social issues and policy making. I really appreciate that members in the company have supported me to write the dissertation. The acknowledgement is also directed to the financial supports by the 21st century COE program “Creation of Agent-based Social Systems Sciences” and Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows No. 21-9482. Last but not least, I wish to thank my family and my wife Miou for their patience and dedicated supports. March, 2013 Yasuo Sasaki

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Contents Abstract

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Acknowledgement

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1 Introduction 1.1 Motivations and Backgrounds . . . . . . . . . . . . . 1.1.1 Modeling Subjective Formulations of Decision 1.1.2 Modeling Revisions of Subjective Views . . . 1.1.3 Modeling Interpretations of Actions . . . . . 1.2 Aims and Methodologies . . . . . . . . . . . . . . . . 1.3 Key Notions . . . . . . . . . . . . . . . . . . . . . . . 1.4 Structure of the Thesis . . . . . . . . . . . . . . . . .

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2 Theoretical Backgrounds and Basic Concepts 2.1 Rational Choice Model . . . . . . . . . . . . . . 2.1.1 Rationality and Utility Maximization . 2.1.2 Bounded Rationality . . . . . . . . . . . 2.2 Games and Equilibria . . . . . . . . . . . . . . 2.2.1 Normal Form Games . . . . . . . . . . . 2.2.2 Nash Equilibria . . . . . . . . . . . . . . 2.2.3 Extensive Form Games . . . . . . . . . 2.3 Games with Incomplete Information . . . . . . 2.3.1 Bayesian Games . . . . . . . . . . . . . 2.3.2 Limits of Bayesian Decision Theory . . 2.4 Epistemology: State-space Model . . . . . . . . 2.4.1 Modeling Knowledge . . . . . . . . . . . 2.4.2 Modeling Unawareness . . . . . . . . . . 2.4.3 Subjective State-spaces . . . . . . . . . 2.5 Hypergames . . . . . . . . . . . . . . . . . . . . 2.5.1 Simple Hypergames . . . . . . . . . . . 2.5.2 Hierarchical Hypergames . . . . . . . .

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3 Subjective Rationalizability in Hypergames 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Subjective Rationalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Equivalence of Subjective Rationalizability and Rationalizability Under Inside Common Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Inside Common Knowledge . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Equivalence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Hypergames and Bayesian Games: A Theoretical Comparison els of Games with Incomplete Information 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Simple Hypergames and Bayesian Games . . . . . . . . . . . . . 4.2.1 Equilibria in Simple Hypergames . . . . . . . . . . . . . . 4.2.2 Bayesian Representation of Simple Hypergames . . . . . . 4.2.3 Relationships of Equilibrium Concepts . . . . . . . . . . . 4.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Hierarchical Hypergames and Bayesian Games . . . . . . . . . . 4.3.1 Equilibria in Hierarchical Hypergames . . . . . . . . . . . 4.3.2 Bayesian Representation of Hierarchical Hypergames . . . 4.3.3 Relationships of Equilibrium Concepts . . . . . . . . . . . 4.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Decision Making and Perceptual Views in Repeated Hypergames: Theoretical Characterization of Systems Intelligence 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Systems Intelligence and Systems of Holding Back . . . . . . . . . . 5.3 Repeated Hypergame Model . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Preparations for Modeling . . . . . . . . . . . . . . . . . . . . 5.3.2 Rules for Behaviors and View Revisions . . . . . . . . . . . . 5.3.3 Reachability of hypergames . . . . . . . . . . . . . . . . . . . 5.3.4 Two Types of Agents . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Systems of Holding Back as Hypergames . . . . . . . . . . . . . . . . 5.4.1 Definition and Results . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Modeling Interpretations of Actions in Extensive 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 6.2 Extensive Hypergames . . . . . . . . . . . . . . . . 6.3 Stackelberg Hypergames . . . . . . . . . . . . . . . 6.3.1 The Framework . . . . . . . . . . . . . . . . 6.3.2 Equilibria . . . . . . . . . . . . . . . . . . . 6.3.3 Effects of Misperceptions . . . . . . . . . . 6.3.4 Application . . . . . . . . . . . . . . . . . . 6.4 Interpretation of Actions . . . . . . . . . . . . . . . 6.4.1 Concepts . . . . . . . . . . . . . . . . . . . 6.4.2 Equilibria . . . . . . . . . . . . . . . . . . . 6.4.3 Application . . . . . . . . . . . . . . . . . . 6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . .

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7 Concluding Remarks 7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix A. List of mathematical symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. List of symbols used in the thesis . . . . . . . . . . . . . . . . . . . . . . . . . .

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Bibliography

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Hypergames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A Game

Chapter 1

Introduction “The limits of my language mean the limits of my world.” – Ludwig Wittgenstein (1922)

1.1

Motivations and Backgrounds

We start with an introduction of our motivations and backgrounds of the research, while formal definitions of the concepts as well as detailed review of theoretical backgrounds are presented in Chapter 2.

1.1.1

Modeling Subjective Formulations of Decision Problems

In decision making, how to solve a given decision problem, needless to say, is important. But how to formulate the decision problem is as essential as it, or more1 . Human beings, however, are not omniscience. Information we have as well as our cognitive capability are limited, and hence we, when formulating a decision problem, cannot model something of which we are not aware 2 . People, before 911, were not able to even imagine that terrorists may use civilian aircraft to attack the city. If we are unaware of something that is actually relevant to a certain decision problem, then our formulation of it would inevitably become imperfect. As a result, our decision is subject to such an imperfect model, whether being aware of the fact itself. As Savage (1954) has argued, in many decision problems we face in real life, it would be impossible to identify anything that is indeed relevant (see 2.3.2). Throughout the thesis, we call the “true” decision problem which identifies everything relevant to it objective, while such an imperfect model is referred to as subjective in contrast (see 1.3). Decision theory has tackled with modeling subjective formulations of decision problems for some time. Most of the results fall into Bayesian decision theory, which describes subjectivity of an agent (decision maker) as a probability distribution on a set of possible states of the world. As will be discussed in detail in Chapter 2, it requires an agent to identify the complete set of states of the world (relevant to the decision problem). Therefore, according to the theory, we are supposed to include the possibility of 911 in the set of the possible states and assign zero or an extremely small probability, but this would not be consistent with our mind before 911: we, in the first place, did not have such a possibility in mind. Then, not surprisingly, not a few researchers have pointed out the insufficiency of Bayesian 1

For example, as Drucker (1971) says, “The most serious mistakes are not being made as a result of wrong answers. The truly dangerous thing is asking the wrong questions.” 2 In the literature of decision theory, to be unaware of something is said to require at least that one does not know that the one does not know it (Dekel, Lipman and Rustichini (1998)). We shall discuss it formally in 2.4.

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decision theory and the importance of developing a new way to deal with human subjectivity in a more realistic way3 . Our interest is in multi-agent interactive decision problems rather than in single-agent ones, hence we adopt game theoretical frameworks. Game theory has developed models of games with incomplete information and played the leading role in information economics. But the complete-state-space assumption also goes for Bayesian games (Harsanyi (1967)), which are acknowledged as the standard model in this line. Alternatively we adopt hypergame theory proposed by Bennett (1977) for modeling games in which each agent formulates the decision problem subjectively. It does not suppose agents have the common set of possible states as is assumed in Bayesian games, but rather directly formulates how they perceive the situation in question as subjective games (see 2.5). In the present thesis, we shall study the theoretical foundation of hypergames in a more rigorous way, and in addition, develop the model in order to deal with some new kinds of problems which will be mentioned next, to which conventional hypergame models have not been applied.

1.1.2

Modeling Revisions of Subjective Views

We are often subject to imperfect models formulated subjectively. At the same time, however, we, in the course of decisions, may have opportunities to learn how to formulate the decision problem. We now know the possibility of terrorism such as 911. Then, what provides such opportunities for learning? Of course, one might be able to get aware of it by pure reasoning: “Terrorists may use civilian aircraft to attack a city.” is a logically possible statement. But we consider experiences, direct or indirect, as the primary factors for those. For almost all of us, it would be not our reasoning but having observed 911 itself that got us aware of the new way of terrorism. In terms of game theory, an experience corresponds to an outcome of a game. After a game is played, if an agent observes something unexpected, then she would revise her view about the decision problem in order for the outcome to be explicable4 . We consider to study revisions of views in the sense above is essential for understanding of stationary states of games, or our society. This is because, though conceptual and hypothetical, our experience and our views about the world are in a mutual feedback as shown in Figure 1.1, and a stationary state can be regarded as a result of such a feedback loop5 . That is, our experience is subject to how we formulate the decision problem, while how to formulate it depends on our experience. Most decision theory has dealt only with the one way of the loop, from “views” to “experiences.” For example, game theoretical models typically analyze how the agents act or should act in a given game6 . Hence we need to develop such a 3

See e.g. Dekel, Lipman and Rustichini (1998); Binmore (2009); and Gilboa (2010) In social psychology, the agent at such a state is said to feel cognitive dissonance (Festinger (1957)). According to his theory, people have a motivational drive to reduce dissonance by altering existing cognitions, adding new ones to create a consistent belief system, or alternatively by reducing the importance of any one of the dissonant elements. 5 This kind of idea is not new in philosophy or sociology. For example, one can find out it in Wittgenstein’s (1953) language game. Also Hayek’s (1955) notion of external order and internal order which characterize his prominent idea of spontaneous order can be related to the idea. The two kinds of order correspond to stability of behaviors and perceptions of agents, respectively. Furthermore, in philosophy of science, Popper’s (1959) idea of falsificationism is highly relevant to the view. That is, revision of a view in our sense corresponds to a falsification of a theory. Just as a theory need not be the “truth” in Popper’s argument, we suppose a subjective view in a stationary state might not be correct: it is supposed to be such a view that is not inconsistent with the agent’s experiences. 6 In this sense, Kaneko and Matsui (1999) argue that most game theoretic models are deductive. See also Matsui (2002, pp. 204-206). Hargreaves and Varoufakis (1995, Ch. 1) discuss similar issues, too. Incidentally, one may think evolutionary game theory addresses issues described in Figure 1.1. We, however, note that it assumes that agents do not (and need not) have any knowledge about games, and hence deals with different problems with ours. For fundamental assumptions of evolutionary game theory, see e.g. Selten (1991). 4

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framework that can deal with the other way, namely revisions of subjective views according to outcomes in games.

Figure 1.1: Mutual feedback between views and experiences Although update of belief has been studied in Bayesian decision theory, i.e. Bayesian updating, it means update of an agent’s subjective probability distribution on a given set of possible states depending on outcomes. Thus this kind of update is conducted within a fixed model. In contrast, we would like to deal with revisions of subjective views per se7 . Hence, although there are no hypergame literature that addresses the issue, we shall explore the possibility of developing hypergame models for the purpose8 .

1.1.3

Modeling Interpretations of Actions

As another issue in association with modeling formulation of subjective views, we shall study interpretations of actions. In particular, by an interpretation of an action, we mean how, in an extensive form game (see 2.2.3), an agent, having observed another agent’s choice of an action, formulates her decision problem, that is, how she believes which subgame she plays then. Too see the problem, consider an extensive form game of Figure 1.2, where agent 1 (she) first acts, then, having observed her choice, agent 2 (he) makes a decision.

Figure 1.2: Interpretations of actions in an extensive form game 7

For single-agent decision problems, case-based decision theory proposed by Gilboa and Schemeidler (1995, 2001, 2003) is notable for the purpose, where an agent is supposed to assess alternatives based on her own accumulated past decisions (called cases) and similarity between those and the current decision problem. Similarly, inductive game theory deals with how an agent creates a subjective view based on her experiences, i.e., outcomes in the past games (Kaneko and Matsui (1999); Kaneko and Kline (2008); and Matsui (2008)). 8 Kijima’s (1996) I-PALM studies dynamic changes of subjective views in hypergames but the mechanism of such changes is not presented explicitly. Inohara (2000) studies exchanges of information among agents in hypergames and those effects on agents’ subjective views, while we deal with view revisions based on outcomes of games. For a specific type of view revision, drama theory deals with preference changes of agents caused by “dilemmas” induced from the game structure (Howard et al. (1993); and Bennett and Howard (1996)).

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We now consider the possibility that they both share the game structure but interpretations of agent 1’s action might be different between them. Suppose agent 1 chooses u. She considers the choice leads agent 2 to play subgame A. However, if he interprets her choice of u as d for some reason, then the subgame he thinks he plays is not subgame A but subgame B. Therefore such interpretations of actions can largely affect what outcome is obtained. As is often the case in real life, people may interpret someone’s action in different ways. For example, some two countries, say A and B, play a game like above, and they both have the same opinion that A first chooses between “arms reduction” and “military expansion,” then B reacts. But A’s choice of arms reduction might not be interpreted as it is by B. For instance, if A declares peaceful use of nuclear technology with the intention to reduce arms, B may interpret it as A’s being prepared for military expansion. As the example shows, even if they share the views about the game, how an action is implemented might become crucial. This is a quite new issue in formal decision theory9 . We consider hypergames can deal with the problem, and try to develop them so as to enable us to analyze interpretations of actions in games. Furthermore, the discussion on view revisions mentioned above can also be applied to interpretations in this sense. That is, an agent may revise how to interpret a certain action depending on an outcome of the game. We shall study such issues as well.

1.2

Aims and Methodologies

Our aims of the study are summarized as below according to the three topics mentioned above. The first aim is to model subjective formulations of games. As mentioned above, we adopt hypergame models for the purpose, but we shall argue that several topics on theoretical foundations of hypergames remain a matter of further study. In particular, we discuss two issues in the thesis. One is on rational behaviors of agents when they are allowed to have “hierarchies of perceptions” about the game, such as agent i’s view about agent j’s view about agent k’s view, and so on. Such a situation is captured by hierarchical hypergames10 . Pointing out problems of existing solution concepts for hierarchical hypergames, we alternatively propose a new solution concept called subjective rationalizability and examine its properties. The other one is on a relationship between hypergames and Bayesian games. Since Bayesian games are often referred to as general enough as a model of games with incomplete information, one may think we do not need hypergame modeling. But in fact the relationship between the two models has not been studied rigorously enough as they have been established and developed independently. We tackle with the issue in the thesis by comparing the models as well as their equilibrium concepts, and then carefully review how each model should be used according to the analyzer’s purpose. Then, the second aim is to model revisions of subjective views. For the purpose we propose a new framework called repeated hypergames. In a repeated hypergame, agents interact repetitively, and may update each subjective view according to an outcome of the game at each period. An agent’s choice is supposed to be subject to the agent’s subjective game at the current period. We, by using the model, aim to describe the mutual feedback between subjective views and experiences as mentioned above. In addition, we discuss how people can make better decisions on the premise of such a structure, and show that apparently irrational behavior can lead to the betterment of the situation in some cases. Conceptually, such a behavior is inspired by a recently developed systemic concept called systems intelligence (H¨am¨al¨ainen and Saarinen (2004)). The third aim is to model interpretations of actions. Since it is a matter in extensive form games, we first extend hypergame models to extensive forms, and examine their properties. 9 10

The significance of studying it has been pointed out by Kobayashi (2012) For hierarchical hypergames, see e.g. Wang et al. (1988, 1989) and Inohara (2000, 2002).

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Then we explicitly formulate interpretations of actions in the sense described above and discuss how they can affect decision making of agents as well as the outcome of the game.

1.3

Key Notions

We here explain the usage of some key notions used throughout the thesis. Objects and Subjects The distinction between objects and subjects is very important in the thesis. An objective viewpoint, in our context, is synonymous with our, i.e. an analyzer’s, point of view, while a subjective viewpoint means a particular agent’s viewpoint. In the standard game theory, we usually need not distinct objects and subjects in this sense because the agents are supposed to know about the game as much as an analyzer unless otherwise stated (Myerson, 1991). On the other hand, we deal with agents whose views might be restricted in some sense compared to that of an analyzer, and hence need to define such views as subjective ones and set an analyzer’s view separately as objective one in order to analyze how these kinds of cognitive limits may affect the interactive situation in question. Knowledge and Belief In philosophy, knowledge has often been defined as “justified true belief.”11 The definition, however, is useless for us because our framework cannot discuss anything about “justification” of belief. Instead, we simply say that an agent believes something when it is assumed to be possibly misperceived within the framework, whether the belief is correct or not. For example, in hypergames, we may say, “An agent believes the game she plays is this particular game, and she believes the opponent has this utility function,” and so on, where such her belief might be incorrect. On the other hand, when we say an agent knows something, the knowledge is always true, that is, the something is what we assume she never misperceives within the model. We may also say that an agent believes that she knows something. Note that this is slightly different with usual usage in the standard game theory, where belief is often defined as a probability distribution on a set of possible states. Common knowledge is an important concept in the standard game theory as well as in our study. It is commonly said that something is called common knowledge if everyone knows it, everyone knows everyone knows it, everyone knows everyone knows everyone knows it, and so on12 . In our study, we deal with agents who believe something is common knowledge13 . Misperceptions To misperceive something means to recognize it incorrectly in some sense. In our usage, for example, when we say that agent i misperceives another agent j’s preference, it implies that agent i’s view about it is different from that perceived by agent j himself. In real life, people often understand someone such as his preference better than himself. Thus one may be inclined to deal with the possibility that it may not agent i but agent j himself who misperceives agent j’s preference. But we do not discuss such issues in the thesis, while we do not assume an agent always understands her preference “correctly,” either14 . We, for convenience, simply refer to misperception only in the way above. Equilibria The notion of equilibrium plays a primary role in economics and game theory. For example, Nash equilibrium has been central in game theoretic analysis, however in what sense it is called an equilibrium has often caused controversy (see 2.2). 11

The definition dates back to Plato. See e.g. Ayer (1956). The concept was first introduced by Lewis (1969) in philosophy, and then mathematically defined by Aumann (1976). 13 For example, Inohara (2000) and Kaneko (2002) discuss such situations. 14 Matsumura and Kobayashi (2005) study such issues. 12

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We adopt several equilibrium concepts in the thesis and try to give unambiguous meanings to each of them. There are basically two types of equilibria we use. One type is considered as a consequence of strategic inferences held by rational agents in light of their perceptions in one-shot games. Therefore, when we discuss this type of equilibrium concepts, our interest is in prediction of an outcome likely to happen when the agents interact and make decisions only one time. We sometimes refer to solution concepts as meaning equilibrium concepts used according to this interpretation. The other type is referred to as a stationary state when the game is played repetitively. As discussed above, we consider a stationary state in a game as such a state in which no agent has motives to change behavior as well as perception about the game15 . Hence when we discuss this type of equilibrium concepts, our interest is in a steady state in this sense in a game.

1.4

Structure of the Thesis

The structure of the thesis is depicted in Figure 1.3. Following the introduction, Chapter 2 introduces theoretical backgrounds of our study, such as the rational choice model, games and hypergames. Formal definitions of basic concepts we use throughout the thesis are also introduced here. Chapter 3 to 6 is the main part of the thesis where we present our own frameworks as well as some findings. In Chapter 3, we propose a new solution concept for hierarchical hypergames called subjective rationalizability. We argue that an agent’s choice according to subjective rationalizability is compatible with the agent’s belief in common knowledge of rationality, and examine its properties. Then, in Chapter 4, we compare two models of games with incomplete information, hypergames and Bayesian games. Based on the results, we discuss how each model should be used according to the analyzer’s purpose. Since the concept of subjective rationalizability becomes a key, the chapter should be read after Chapter 3. On the other hand, Chapter 5 and 6 provide independent topics. In Chapter 5, we propose the repeated hypergame framework as a model of the mutual feedback between subjective views and experiences as mentioned above. We also discuss its relation to the concept of systems intelligence. Chapter 6 provides the framework of extensive hypergames, an extension of hypergame models into extensive forms. By using the model, we study how interpretations of actions can influence games. Finally, Chapter 7 gives concluding remarks. A list of mathematical symbols is provided at the end.

15 Some equilibrium concepts in the standard game theory such as sequential equilibrium and perfect Bayesian equilibrium explicitly require each agent’s belief to be consistent. We, however, note that such a belief refers to the agent’s subjective probability distribution on some set in a given game, while we study stability of views in the sense of the agent’s subjective formulation of the game per se.

6

Figure 1.3: Structure of the thesis

7

Chapter 2

Theoretical Backgrounds and Basic Concepts This chapter introduces the theoretical backgrounds of our study and provides some basic concepts which we use in the later chapters as well as those definitions. Following the introduction of rational choice model, we explain about game models, particularly focusing on how interactive decision making among agents whose information are incomplete can be captured in terms of formal decision theoretical models.

2.1

Rational Choice Model

Rational choice model is the foundation of the frameworks we employ in the study. Rationality in the context is synonymous with utility maximization. In this section, we first clarify the meanings of these concepts, and then introduce the idea of bounded rationality by which agents of our concern are characterized.

2.1.1

Rationality and Utility Maximization

What is utility? What does its maximization mean? We here explain these based on the theory of revealed preference, which is “the orthodoxy” (Mas-Colell et al. (1995)) in economics and game theory. As opposed to the traditional utilitarian’s view, it assumes nothing psychological but instead defines utility functions based only on observables, that is, people’s actual choices, which are directly associated with their preferences1 . Consider a single-agent decision problem from a finite set of alternatives X. A binary relation on it, %⊂ X × X, is interpreted as “preferred at least as.” That is, for any two alternatives x, y ∈ X, (x, y) ∈%, or x % y, means “x is preferred at least as y.” In terms of revealed preference, the binary relation is supposed to be deduced directly from actual choices: “one chooses x rather than y hence prefers x to y.” Let us say a function u : X → ℜ represents % if, for any x, y ∈ X, x % y iff u(x) ≥ u(y). When does there exist such a real-valued function u that represents %? Consider the following two axioms: A1 (Completeness): ∀ x, y ∈ X, x % y or y % x. 1 The term of utility was first adopted by Jeremy Bentham, known as the founder of utilitarianism, in the 18th century. He considered it as a single measure of any kind of the pleasure or pain a person feels as a result of a decision. In his view, one’s brain is like a device generating something called utility, and thus “one chooses x rather than y because x generates higher utility than y does.” Economists after Bentham, however, became increasingly uncomfortable with the view, which demanded that economics be based on such psychological foundations, and established the theory of revealed preference in order to define utilities in terms only on observables (Ramsey (1931); and Samuelson (1947)). The movement is said to be as a descendant of logical positivism (Gilboa (2009, p. 60)).

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A2 (Transitivity): ∀ x, y, z ∈ X, x % y and y % z implies x % z. Completeness states that any two alternatives are comparable for the agent: for any two alternatives, she can tell that one is preferable to another or that they are indifferent2 . Transitivity keeps her preference from being cyclic. A complete reflexive transitive binary relation is called a preference relation 3 . Then the following proposition is well known. Proposition 2.1: Let X be a finite set. A binary relation on it, %⊂ X × X, satisfies A1 and A2 iff there exists a function u : X → ℜ that represents %. The proposition states that for any preference relation on a finite set, there exists a real-valued function that represents it. Such a function is called an utility function. Although Proposition 2.1 is sufficient for finite alternatives, its generalization to any sets X (including uncountable ones4 ) requires an additional axiom called separability5 , and then the next proposition has given by Cantor (1915)6 . A3 (Separability): There exists a countable set Z ⊂ X such that for any x, y ∈ X \ Z, if x ≻ y, then ∃ z ∈ Z, x % z % y. Proposition 2.1a: Let X be an arbitrary set. A binary relation on it, %⊂ X × X, satisfies A1-A3 iff there exists a function u : X → ℜ that represents %. Note that the utility function whose existence is assured by Proposition 2.1 and 2.1a is ordinal. That is, only ranking of its values has any significance. Indeed any ordinal utility functions that are results of any strictly increasing transformations of an ordinal utility function represent the same preference relation. For example, suppose X = {x, y, z} and the preference relation is given as x ≻ y ≻ z, where ≻ is the asymmetric part of %, which is interpreted as “strictly preferred to.” Then one can construct an utility function u that represents it, for which (u(x), u(y), u(z)) may be (90, 10, 3) or (10, 5, 0) or (2, 0, −1). In the thesis, we deal only with ordinal utility functions and do not use cardinal utilities, or von Neumann and Morgenstern utilities, mainly due to two reasons. First, it is simply because of mathematical convenience. Second, our main interest is not in issues such as risk and uncertainty, for which cardinal utility functions have been proved to be powerful. Henceforth by utility functions we mean ordinal ones unless otherwise stated. Being different from our colloquial usage, the notion of rationality described in the rational choice model is conventionally synonymous with utility maximization (Osborne and Rubinstein, 1994). That is, a rational agent, in any decision problem with an alternative set X, always chooses arg maxx∈X u(x). Furthermore, as we have seen, her being rational is equivalent to her choice being consistent in the sense that it satisfies the axioms above. Therefore, we say, with the behavioral consistency, that she is rational and acts as if she maximizes her utility function, even if she herself might not notice the fact. Note that rationality is thus different with selfishness, though those are often mistakenly mixed up by non-experts. The latter refers to how one’s preference is, while, in the former, 2 Note that “or” in logic is not exclusive (unless otherwise stated), that is, “A or B” means ”A or B or both.” 3 %⊂ X × X is called reflexive if and only if ∀ x ∈ X, x % x. Completeness implies reflexivity. 4 A set X is countable if it is possible to “count” the elements of it. Formally, it is countable if and only if there is a function with domain {1, 2, 3 . . .} and range X which is onto to X. Finite sets are of course always countable. The set of integers is an example of a infinite countable set, while the set of real numbers ℜ is uncountable. 5 Separability assures that any element in X can be ranked relatively to some countable set Z ⊂ X. The axiom is provided for mathematical necessity: it is often extremely difficult or impossible to exemplify it empirically. 6 See e.g. Kreps (1988); and Gilboa (2009). If one is interested in continuity of utility functions, Debreu’s (1959) theorem that deals with it is required.

9

only consistency of choices matters. One may argue that Mother Teresa is altruistic, whereas Adolf Hitler is selfish. However, if only their choices are consistent, we say they both are rational.

2.1.2

Bounded Rationality

In our study, we deal with boundedly rational agents in a sense. Let us clarify in what sense their rationality is bounded. The term of bounded rationality was first coined by Simon (1957) as an alternative basis for perfect rationality adopted in economics and related fields. He claimed that in decision making, our rationality is limited by, say, the amount of information, the calculation capability, the cognitive abilities, the amount of time to make decisions, and so on, and thus models of human behavior should take into account such bounds. The argument won the sympathy of many at that time but it was not until 1990s that it became a fashionable and concrete research agenda. Nowadays there seem to be mainly two approaches for modeling bounded rationality. One is to employ heuristics as ways of decision making of agents (e.g. Gigerenzer and Selten (2002)). The other is, as proposed by Rubinstein (1998), to specify human decision making procedures explicitly in a formal way. Our study takes up the standpoint of the latter approach. Among those limitations Simon referred to, of our concern is particularly the limitation of the cognitive aspects. We assume in our framework that agents may not be able to formulate decision problems correctly (from an analyzer’s point of view) but are rational in other senses and behave so as to maximize their utilities in the subjectively-perceived decision problems. To capture the idea, Simon’s (1978) distinguishing two kinds of rationality would be useful: substantive rationality and procedural rationality. In Simon’s words, the former is “the extent to which appropriate courses of action are chosen,” while the latter is “the effectiveness, in light of human cognitive powers and limitations, of the procedures used to choose actions” (ibid.). That is, substantive rationality is evaluated by what an agent has done, whereas procedural rationality refers to how she has done it. Our study drops the assumption of substantive rationality but retains that of procedural rationality.

2.2

Games and Equilibria

A game is a formal model of interactive decision making involving two or more agents. A trivial but substantial difference of it from single-agent decision problems is that, in games, utilities of an agent depend not only on her choice (and the state of nature) but also on choices of the other agents, and hence she has to take into account the others’ minds in order to make a rational choice. Since the seminal work of von Neumann and Morgenstern (1944), game theory has been developed explosively and applied to a huge variety of social phenomena. Here we introduce the basic game theoretical models and those equilibria concepts, which are the bases of hypergame models we will study in the later chapters.

2.2.1

Normal Form Games

In a normal form game, each agent makes a decision simultaneously and independently, and once and for all. It consists of three components: a set of agents, sets of actions available to them, and utility functions for each that associate real values (utilities) with outcomes7 . Formally: 7

Since we only consider ordinal utility functions, we do not deal with mixed extension of games throughout the thesis.

10

Definition 2.1 (Normal form games): G = (I, A, u) is a normal form game, where: • I is the finite set of agents. • A = ×i∈I Ai , where Ai is the finite set of agent i’s actions. a ∈ A is called an outcome. • u = (ui )i∈I , where ui : A → ℜ is agent i’s utility function. A two-agent normal form game can be represented as shown in Table 2.1. Let us call the row agent and the column agent 1 and 2, respectively. Then I = {1, 2}, A1 = {t, b} and A2 = {l, r}. The two numbers in each entry express utilities they obtain at each outcome: the first number is the utility for 1, and the second for 2. For instance, when an outcome {t, r} ∈ A obtains, agent 1 and 2 get utility 1 and 4, respectively. 1\2 t b

l 3, 3 4, 1

r 1, 4 2, 2

Table 2.1: A two-agent normal form game

2.2.2

Nash Equilibria

The main concern of game theorists to study games is of course to analyze, in a given game, what outcome obtains. Nash equilibrium (Nash (1951)) has played the central role for the purpose. Let us see the definition first, and then its implications. Definition 2.2 (Nash equilibrium): Let G = (I, A, u) be a normal form game. a∗ = (a∗i , a∗−i ) ∈ A is a Nash equilibrium of G iff ∀ i ∈ I, ∀ ai ∈ Ai , ui (a∗ ) ≥ ui (ai , a∗−i ). We say that, in a game, a′i ∈ Ai is a best response of agent i to some the others’ choices, a−i ∈ A−i , iff ui (a′i , a−i ) ≥ ui (ai , a−i ) for any ai ∈ Ai . In a Nash equilibrium, each agent chooses each own best response to the choices of the others. Thus nobody has an incentive to change the action as long as the others do not change their choices. Let us denote the set of Nash equilibria of a normal form game G by N (G). In general, N (G) may be empty (unless we do not deal with mixed extension of games), or have two or more elements. Furthermore we call an agent’s action that constitutes some Nash equilibrium her Nash action, that is, a∗i ∈ Ai is called agent i’s Nash action in G iff there exists a−i ∈ A−i such that (a∗i , a−i ) ∈ N (G). Let us denote the set of agent i’s Nash actions in G by Ni (G). In the game of Table 2.1, {b, r} is the unique Nash equilibrium. Thus the sets of Nash actions of agent 1 and 2 are {b} and {r}, respectively. Furthermore, we use the concept of Pareto Nash equilibria in order to evaluate the relative desirability among multiple Nash equilibria in Chapter 5. It is defined as a Nash equilibrium that is not Pareto-dominated by any other Nash equilibria even in a weak sense. Definition 2.3 (Pareto Nash equilibrium): Let G = (I, A, u) be a normal form game. a∗ ∈ A is a Pareto Nash equilibrium of G iff a∗ ∈ N (G) and there does not exist a ∈ N (G) such that ∀ i ∈ I, ui (a) ≥ ui (a∗ ) and ∃ j ∈ I, uj (a) > uj (a∗ ). Let us denote the set of Pareto Nash equilibria of G by P N (G). By definition, in any normal form game G, P N (G) ⊆ N (G), and in particular, if the game has the unique Nash equilibrium, P N (G) = N (G). For example, the game of Table 2.2 has two Nash equilibria, {t, l} and {b, r}. Since the former Pareto dominates the latter, only the former is a Pareto Nash equilibrium. We call an agent’s action that constitutes some Pareto Nash equilibrium her Pareto Nash action, that is, a∗i ∈ Ai is called agent i’s Pareto Nash 11

action in G iff there exists a−i ∈ A−i such that (a∗i , a−i ) ∈ P N (G). Let us denote the set of agent i’s Pareto Nash actions in G by P Ni (G). Note that a Pareto Nash equilibrium is not always a Pareto-optimal outcome in a game. For example, in the game of Table 2.1, {b, r} is a Pareto Nash equilibrium, but it is not Pareto-optimal in the game as it is Pareto-dominated by {t, l}. 1\2 t b

l 4, 4 3, 1

r 1, 3 2, 2

Table 2.2: Pareto Nash equilibrium In what sense is Nash equilibrium called an “equilibrium”? In fact the fundamental question frequently arises in game theory and it is still controversial8 . Until 1980s, not a few game theorists had interpreted Nash equilibrium (as well as other kinds of equilibrium concepts) as a consequence of strategic inferences held by rational agents. Hence equilibrium selection had been one of the main issues of game theory until that time as a game in general may have multiple equilibria (e.g. Harsanyi and Selten (1988)). Such a view, however, turned out to be faulty in effect when Bernheim (1984) and Pearce (1984) showed that the precise implication of common knowledge of rationality (and of all the components of the game) is rationalizablity, a weaker concept than Nash equilibrium9 . To put it the other way around, for the agents to play a Nash equilibrium in an one-shot game, they need some “exogenous and obvious” reason to do so (such as culture, social norm, correlated devices, a direction of a coordinator, etc.)10 . Another more satisfying interpretation of Nash equilibrium is to regard it as a stationary state. In this view, a Nash equilibrium achieved in a game is seen as a result of an adaptive process in which agents try to get adjusted to the other’s behavior. Hence this approach, implicitly or explicitly, assume the game is played repeatedly. But Nash equilibrium itself tells us nothing about what the process is like or under what condition the process converges11 . Indeed the interpretation of Nash equilibrium is still controversial, and hence we cannot ignore the problem when we discuss on solution concepts in our frameworks developed in the later chapters based on hypergames. We sometimes adopt Nash equilibrium as a solution concept of an one-shot game, or a candidate of stationary states, but whenever taking about solution concepts, we carefully explain the assumptions as well as the implications of them.

2.2.3

Extensive Form Games

An extensive form games is a model of dynamic situations, in which decisions made by the agents might not be simultaneous. In Chapter 6, we study extensive form games. Although the general definition of an extensive form game requires a somewhat complicated structure, we, in the thesis, restrict our attention to its simpler class, namely, extensive form games with perfect information in which there are no randomizing devices (often called nature or chance). A game is called of perfect information iff each agent, when making any decision, knows all the events that have previously occurred, or more precisely, any information set of any agent consists of one decision nodes. An extensive form game of such a class is formally defined as follows12 : 8

About interpretations of Nash equilibria, for example, see Binmore (1987); Kaneko (1987); and Mas-Corell (1995, pp. 248-249). 9 We discuss rationalizablity in Chapter 3. 10 Schelling (1960) provided the earliest suggestion for the topic. For this respect, Aumann’s (1987) notion of correlated equilibria is notable. See also Gintis (2009). 11 See, e.g. Fudenberg and Levine (1998). 12 The definition is based on that given by Osborne and Rubinstein (1994, Ch. 6).

12

Definition 2.4 (Extensive form games): Ge = (I, W, P, u) is an extensive form game with perfect information (without nature’s move), where: • I is the finite set of agents. • W is the set of finite sequences each of which is called a history, which satisfies the following properties: – Each component of a history h = (a1 , · · · , an ) ∈ W is called an action taken by an agent. – If (a1 , · · · , an ) ∈ W and m < n, then (a1 , · · · , am ) ∈ W . – h0 ∈ W , where h0 is the empty sequence, which is called the initial history. – A history (a1 , · · · , an ) ∈ W is called a terminal history iff there is no an+1 such that (a1 , · · · , an+1 ) ∈ W . Let us denote the set of terminal histories by Z. • P : W \ Z → I is the agent assignment function. • u = (ui )i∈I , where ui : Z → ℜ is agent i’s utility function. Throughout the thesis, we simply refer to extensive form games of the class as “extensive form games” unless otherwise stated. We interpret them as follows. After each nonterminal history h, agent P (h) chooses an action. Let us denote the set of actions available to the agent at the point by A(h). That is, A(h) = {a|(h, a) ∈ W }, where (h, a) is such a history in which a is taken after h (let (h, a) = a if h = h0 ). Note that, by definition, if h is nonterminal, A(h) ̸= φ. In our study, we particularly focus on a specific class of extensive form games called Stackelberg games. A Stackelberg game is a two-agent extensive form game an agent called a “leader” first chooses an action and then another agent called a “follower,” informed of the leader’s choice, chooses an action. As usual, the set of action available to the follower is assumed to be identical regardless of the leader’s choice. Formally it can be defined as follows: Definition 2.5 (Stackelberg games): An extensive form game Ge = (I, W, P, u) is called a Stackelberg game when it satisfies all of the following conditions: • I = {1, 2}, where 1 and 2 are supposed to be the leader and the follower, respectively. Hence P (h0 ) = 1, and for any a1 ∈ A(h0 ), P (a1 ) = 2. • For any a1 a′1 ∈ A(h0 ), A(a1 ) = A(a′1 ). • For any (a1 , a2 ) ∈ W , there is no a3 such that (a1 , a2 , a3 ) ∈ W . When we refer to Stackelberg games, for simplicity, we denote the action set of the leader by A1 and that of the follower lead by the leader’s choice by A2 . Hence the set of terminal histories Z corresponds to A1 × A2 . Let us denote it by A, then each agent i’s utility function is given as: ui : A → ℜ. We may refer to a ∈ A as an outcome. The solution concept usually applied to such games is called Stackelberg equilibrium. Definition 2.6 (Stackelberg equilibrium): Let an extensive form game Ge = (I, W, P, u) be a Stackelberg game. Let A′ = {(a1 , a2 ) ∈ A|∀ a′2 ∈ A2 , u2 (a1 , a2 ) ≥ u2 (a1 , a′2 )}. a∗ ∈ A is called a Stackelberg equilibrium iff a∗ ∈ A′ and ∀

a ∈ A′ , u1 (a∗ ) ≥ u1 (a).

Let us denote the set of Stackelberg equilibria of Ge by S(Ge ). In a Stackelberg equilibrium, the leader first chooses such an action that maximizes her own utility function given that whatever she chooses, the follower maximizes his utility function after the choice, while the follower indeed maximizes his utility function, informed of the leader’s choice. In the definition, we restrict our attention only to outcomes in which 13

the follower takes a best response to the leader’s choice by introducing the subset of A, A′ . It is known that when the game is finite, there always exist Stackelberg equilibria, and, in particular, it is unique when neither of the both agent is indifferent between any two outcomes13 . A Stackelberg equilibrium is also characterized as a backward-induction solution, and it is known that common knowledge of the game structure and rationality implies that the solution of the game is given as a backward-induction solution (Aumann (1995)). Figure 2.1 illustrates an example of Stackelberg game. In the game, I = {1, 2}, where 1 is the leader while 2 is the follower: a number on each node indicates the agent who makes decision at the point. They both have two actions, i.e., A1 = {u, d} and A2 = {u, d} (recall that agent 2 has the same action set regardless of agent 1’s choice). The game has four outcomes, that is, A = A1 × A2 = {(u, u), (u, d), (d, u), (d, d)}, and each pair of numbers described in the right-hand side represents utilities each agent obtains when the outcome is obtained: in each pair, the left number is the utility of agent 1 and the right one is that of agent 2. For instance, when agent 1 chooses d and then agent 2 chooses d, each one get utility as much as 2 and 4, respectively. It is easily shown that this is the Stackelberg equilibrium of the game, namely S(Ge ) = {(d, d)}.

Figure 2.1: A Stackelberg game

2.3

Games with Incomplete Information

We say a game is of complete information if the agents know all the components of the game (and the fact is common knowledge among them). On the other hand, a game is called of incomplete information if some or all of them lack full information about the game. Clearly our own interest of the study is in the latter. In this section, we overview how game theory has dealt with games with incomplete information, focusing particularly on the standard model for those called Bayesian games. Then we discuss it critically. In Chapter 4, we study Bayesian games.

2.3.1

Bayesian Games

Since the early development of game theory, questions about how the analysis would change if some agents do not actually know some components of the game, that is, if the game is of incomplete information, have arisen14 . Harsanyi’s (1967) way of modeling it was the breakthrough, and in fact still remains standard in this field. He argues eloquently that uncertainties as well as perceptual differences 13

A Stackelberg equilibrium is also derived from a subgame perfect equilibrium, which is more popular in game theory. The existence and uniqueness of Stackelberg equilibrium mentioned here generally hold for subgame perfect equilibria in finite extensive form games with perfect information (Osborne and Rubinstein (1994, Ch. 6)). 14 The term of incomplete information was already used by von Neumann and Morgenstern (1944) to refer to those situations, though they were not eager to study such incomplete models. An example of the earliest formalism of it is games with misperception proposed by Luce and Raiffa (1957, Ch. 12), which now can be regarded as a simple case of hypergames. For historical developments of games with incomplete information, see e.g. Myerson (2004).

14

of agents about anything are captured without loss of generality by subjective probability distributions for each agent over the set of possible states (or types), and moreover the state set as well as the probability distributions on it (for each type of each agent) must be common knowledge15 . With those settings, any games with incomplete information can be transformed into Bayesian games, which are defined as follows: Definition 2.7 (Bayesian games): Gb = (I, A, T, p, u) is a Bayesian game, where • • • •

I is the finite set of agents. A = ×i∈I Ai , where Ai is the finite set of agent i’s actions. T = ×i∈I Ti , where Ti is the finite set of agent i’s types. p = (pi )i∈I , where pi is agent i’s subjective prior, which is a joint probability distribution on T−i for each ti ∈ Ti . • u = (ui )i∈I , where ui : A × T → ℜ is agent i’s utility function. A type of an agent is characterized by subjective prior and utility function. A subjective prior describes the type’s perception about the game: each type is assumed to have a probability distribution on the types of the other agents. Unlike normal form games, an agent’s utility is determined not only by actions but also by types of the agents. The transformed game is now of complete (but imperfect) information16 . That is, although an agent might not know the actual type of another agent, the type set of the agent as well as each type’s subjective prior is now common knowledge: everyone knows all the components of the transformed game. We can analyze Bayesian games with the following concept of Bayesian Nash equilibrium, a natural generalization of Nash equilibrium. To define it, we need to introduce “action plans” for each type of each agent, which we call her strategy. A strategy of agent i, si , is a mapping from her types to her actions, namely, si : Ti → Ai . Let us denote the set of agent i’s strategies by Si and let S = ×i∈I Si . We may write s−i (t−i ) as meaning (sj (tj ))j∈I\{i} with sj ∈ Sj and tj ∈ Tj . Then Bayesian Nash equilibrium is defined as follows: Definition 2.8 (Bayesian Nash equilibrium): Let Gb = (I, A, T, p, u) be a Bayesian game. s∗ = (s∗i , s∗−i ) ∈ S is a Bayesian Nash equilibrium of Gb iff ∀ i ∈ I, ∀ ti ∈ Ti , ∀ si ∈ Si , ∑

ui ((s∗i (ti ), s∗−i (t−i )), (ti , t−i ))pi (t−i |ti ) ≥

t−i ∈T−i



ui ((si (ti ), s∗−i (t−i )), (ti , t−i ))pi (t−i |ti ).

t−i ∈T−i

Let us denote the set of Bayesian Nash equilibria of a Bayesian game Gb by BN (Gb ). In a Bayesian Nash equilibrium, each type of each agent maximizes her expected utility given her belief (subjective prior)17 . For a Bayesian game, we can consider a joint probability distribution po on the type set T , which describes probabilities for which a particular combination of types for each agent is chosen actually. We call it the objective prior of the game. In particular, we say subjective priors are consistent in a Bayesian game iff each agent’s subjective probability distribution 15

We discuss the Harsanyi’s claim in Chapter 4. Furthermore, Mertens and Zamir (1985) have showed a general way to construct the universal belief space in order to capture agents’ uncertainties about each other’s belief, each other’s belief about each other’s belief, and so on, which allows us to formulate such kind of uncertainty in terms of Bayesian games. 16 Notice the difference between complete information and perfect information. An extensive form games is called of perfect information if each agent knows all the previous moves at each decision nodes, while of imperfect information otherwise. A Bayesian game is of complete information since the agents are assumed to know the whole structure but is of imperfect information because each agent never knows the others’ actual types. 17 Note that calculations of expected utilities generally need the utility functions to be cardinal.

15

given the agent’s type is just the conditional probability distribution computed from the objective prior by Bayes formula18 . When the objective prior is common knowledge and the agents have consistent subjective priors, it is called a common prior, which is typically assumed in the literature. Although we can analyze Bayesian games with inconsistent priors19 , Harsanyi (1967, Part III) claims that any reasonable game should have a common prior so that any differences among agents can be explained by differences in information they have got. The argument is known as Harsanyi doctrine and the common prior assumption (CPA) has become “virtually universal in economic analysis” (Myerson (2004)). Whether an objective prior is a common prior, we can also formulate a Bayesian game in terms of it (instead of subjective priors) as Gb = (I, A, T, po , u), and define Nash equilibrium of it as follows20 : Definition 2.9 (Nash equilibrium of Bayesian games): Let Gb = (I, A, T, po , u) be a Bayesian game (with objective prior). s∗ = (s∗i , s∗−i ) ∈ S is a Nash equilibrium of Gb iff ∀ i ∈ I,∀ s ∈ S , i i ∑ ∑ ui ((s∗i (ti ), s∗−i (t−i )), t)po (t) ≥ ui ((si (ti ), s∗−i (t−i )), t)po (t). t∈T

t∈T

Let us denote the set of Nash equilibria of Gb by N (Gb ). In a Nash equilibrium of a Bayesian game, each type of each agent maximizes her expected utility given an objective prior. Harsanyi (1967) proved that in any Bayesian games with consistent priors, the set of Bayesian Nash equilibria coincides with that of Nash equilibria. We refer to the both formulation, one with subjective priors and the other with objective priors, as Bayesian games and write them as Gb unless it might lead to any confusion (when we write BN (Gb ) and N (Gb ), we suppose (pi )i∈I and po , respectively). Furthermore we may say that po is the objective prior of Gb = (I, A, T, p, u). Since most literatures adopt CPA, the distinction between the two definitions of Bayesian games as well as that between the two equilibrium concepts make practically no difference in analyses. On the other hand, we discuss Bayesian games without CPA in Chapter 4.

2.3.2

Limits of Bayesian Decision Theory

CPA, however, is not perfectly intuitive, and indeed is controversial21 . But, as we have seen, it is possible to formulate and analyze Bayesian games without it. Rather, it is a more fundamental assumption of the model that we here would like to shed light on, that is, common knowledge of the set of possible states. Moreover, if we take a step further, then we would face a more serious question: how, in the first place, can an agent know all the relevant states? This is a problem of Bayesian decision theory itself rather than Bayesian games22 . To consider about it, Savage’s (1954, p. 16) conceptual distinction between small worlds and large worlds would be useful. According to Savage, small worlds are those in which it is 18 That is, (pi )i∈I are consistent iff there exists a probability distribution po on T such that ∀ i ∈ I,∀ t ∈ T, pi (t−i |ti ) = po (t)/Σt−i ∈T−i po (ti , t−i ). 19 Recall the definitions of Bayesian games and Bayesian Nash equilibrium do not require the consistency. 20 We have introduced two formulations of Bayesian games. Harsanyi (1967) originally named the latter (with objective prior) “Bayesian games,” while the former (Definition 2.8) “standard form of games with incomplete information,” though nowadays the former is also referred to as Bayesian games in textbooks of game theory. The two notions of equilibria we introduced, Bayesian Nash equilibrium and Nash equilibrium of Bayesian games, are sometimes also mixed up. However, if we adopt CPA, those distinctions make no difference in analyses as will be mentioned. 21 For rigorous attempts to characterize common priors, see e.g. Morris (1995); and Halpern (2002). 22 For critical reviews of Bayesian decision theory, see e.g. Kelsey and Quiggin (1992); Binmore (2009, Ch. 7); and Gilboa (2010).

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always possible to “look before you leap,” that is, every possibility can be taken into account in advance. Then every relevant aspect of a certain decision problem can be captured in a set of possible states. For example, a game such as coin tossing might be seen as a small world. It would be reasonable to consider that the only uncertainty is the face of the coin and each side of it will appear with probability 0.5. It is ironical that Savage, normally acknowledged as the founder of Bayesian decision theory, himself held a view that applications of the theory should be restricted to small worlds (ibid.). He considered that there is another type of worlds in which there is no other way but to “cross the bridge when you come to it,” that is, it is extremely difficult or impossible to identify and assess all the relevant possibilities. They are called large worlds. Then he argued that the idea to apply his theory to any worlds as “utterly ridiculous” simply because it is utterly beyond human ability. To sum up, Bayesian decision theory (including Bayesian games) seems to be legitimate only in small worlds but no longer so in large worlds23 . Furthermore, to take the issue more seriously, what can be indeed identified as a small world is not obvious. For example, Savage (1954) argues that even planning a picnic would not be a decision problem in a small world because it is in principle impossible to make a complete list of all the uncertainties relevant to it. Moreover, consider a game whose “rule” is explicitly specified, such as chess. One may argue that chess is no doubt a game in a small world. But notice that common knowledge of the rule of chess itself does not imply that of the “game” that the two players truly play. For example, a gentle father might intentionally be defeated by his son to delight him, who never knows his father’s intention. Or, a player who hates to lose, facing the crisis of being defeated, might pull over the chess table, which might be totally unexpected by the other player. In those cases, playing chess is an event in large worlds: at least one of them might not possess all the possibilities relevant to the situation in mind. It can be said that our aim of the study is to model agents who face decision problems in large worlds. The next section introduces a formal model of knowledge in order to see such a critique to Bayesian decision theory in a formal way, with which we continue the discussion.

2.4

Epistemology: State-space Model

This section introduces a formal model that deals with epistemology of agents such as knowledge and belief. We do not elaborate the model itself but it would be useful to clarify the epistemic foundations of decision theoretic models for understanding the subsequent discussions. We first overview the standard model which underlies most decision theories including Bayesian decision theory, and then introduce a somewhat unorthodox model that deals with unawareness of agents.

2.4.1

Modeling Knowledge

We present the set theoretical model of knowledge (also known as the semantic formalism) called state-space model. The foundation of it was established by Hintikka (1962) and applications to modeling knowledge of agents in games was initiated by Aumann (1976). The primitive is a pair (Ω, P ), where: • Ω is a set of possible states of the world, or simply states. • P : Ω → 2Ω \ {φ} is a possibility correspondence 24 . 23

See also Binmore (2009); and Gintis (2010). In the literature, a possibility correspondence may be called information function, or information partition (if P partitions Ω). 24

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Ω is supposed to include all the possible states relevant to a certain decision problem25 , and each ω ∈ Ω is exclusive with one another, thus only one element in Ω can be the “true” state26 . On the other hand, P is a mapping from Ω to all nonempty subsets of Ω. If the image of P partitions Ω and ω ∈ P (ω) for all ω ∈ Ω, then we say P is partitional 27 . Otherwise it is nonpartitional. P (ω) is interpreted as the set of all those states that she cannot distinguish from ω. Thus when the true state is ω, she considers any state in P (ω) as possible but any state outside as impossible. For example, let Ω = {a, b, c} and suppose the decision maker can distinguish only whether or not it is c. Then her possibility correspondence is given as P (a) = P (b) = {a, b} and P (c) = {c}. E ⊆ Ω is called an event. For example, {a, b} is an event which means “it is a or b.” An agent is said to know an event E given P if and only if P (ω) ⊆ E, because if this is the case, she considers some states in E surely obtains. Then her knowledge operator K is defined by K(E) = {ω ∈ Ω|P (ω) ⊆ E}. K(E) is the set of all states in which she knows E. Note that K(E) itself is an event where she knows E. Consider the following five properties28 , where ¬E is the complement of E: K1: K2: K3: K4: K5:

KΩ = Ω; K(E ∩ F ) = K(E) ∩ K(F ); K(E) ⊆ E; K(E) ⊆ KK(E); ¬K(E) ⊆ K¬K(E).

K1 and K2 are basic property of knowledge in the sense that a knowledge operator defined from a possibility correspondence (like above) always satisfies these two. The following proposition is due to Bacharach (1985). Proposition 2.2: For (Ω, P ), let K be the knowledge operator derived from P . K satisfies K1-K5 if and only if P is partitional. Standard modeling in economics and game theory assumes P is partitional29 , hence, implicitly or explicitly, accepts all K1-K5. It is sometimes called the model of Bayesian epistemology as with partitions, one can reasonably use Bayes’ rule to update her prior probability to a posterior probability in the face of a specific information, and indeed it underlies Bayesian decision theory. Those properties apparently look innocent and allow us to have powerful model. It, however, misses some of the fundamental questions on human knowledge that have been argued among philosophers. Since to clarify the missing things is important for us, let us overview K1-K5 critically30 . 25

A set of states, in Savage’s (1954) definition, is “a description of the world, leaving no relevant aspect undescribed.” In particular, some uncertainty is called relevant when it can affect the agent’s utility. However, in many decision problems, it is unclear what is indeed relevant. Hence what should be identified as states is not obvious (see also 2.3.2), and in fact various definitions of states are defined in various literatures. An extreme interpretation is that they include not only uncertainty about nature or the others but also even the agent’s own actions (e.g. Aumann and Brandenburger (1995)). 26 In the standard model, the true state is assumed to be always included in the set of states. 27 Let the set of theSimages of P be Q. We say Q partitions Ω if and only if (i) A ∩ B = φ for all A, B ∈ Q with A ̸= B and (ii) A∈Q A = Ω. 28 It can be easily shown that with K3, K4 and K5 hold with the equality. 29 In game theory, one can find that, for instance, information sets in extensive form games typically reflect the idea. For more on the standard information partition model, see e.g. Osborne and Rubinstein (1994, Ch. 5); Fagin et al. (1995); and Aumann (1999). 30 For more on critical reviews on the standard partitional model, see e.g. Binmore (1990); Samuelson (2004); and Binmore (2009).

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K1, known as completeness or omniscience, states that, at all states, the agent knows the set of all the possible states, or equivalently, she always knows that the true state is in Ω. However, it would sound strange to claim that we have already conceived of all the possibilities about anything of any time. To accept K1 unlimitedly implies, say, that Newton considered Einstein’s theory as possible. But indeed the idea that one never excludes the true state is the core of Bayesian decision theory. K2 says that knowing both E and F is equivalent to knowing its intersection (or conjunction). This appears innocuous but let us see its implication31 , K2a: E ⊆ F ⇒ K(E) ⊆ K(F ). K2a, known as monotonicity or logical omniscience, states that if E implies F , then knowing E implies knowing F . This implies that once one knows something, then she knows anything that follows it. Hence once she learns some axioms of mathematics, then she must know all the theorems that can be derived from them. K3, known as axiom of knowledge or nondelusion, means that what is known must be true, or equivalently, what is not true never be known. But do we really know only things that are true? It never allows any kinds of misperceptions or misunderstandings that are everyday affairs around us. K4, known as axiom of transparency or positive introspection, states that whenever one knows something, she knows that she knows it. It seems to us less harmful than the other properties. K5, known as axiom of wisdom or negative introspection, means that whenever one does not know something, she must know that she does not know it. It is obviously not intuitive to consider that our knowledge always satisfies the axiom. “There are things that we do not know we do not know.,” said Donald Ramsfeld (2002), the former U.S. Secretary of General, looking back on the September 11th terrorists attack. Almost all of us did not know that terrorists might use civilian aircraft as a weapon. In addition, and more importantly, we did not know that we did not know it. All of these critiques more or less refer to limitations of Bayesian epistemology. Again, Savage (1954) was clear in these respects: as mentioned in 2.3.2, they would be irrelevant if the decision problem is in small worlds but might become drastically critical if it is in large worlds. In particular, given our motivation of the study, it seems to us that at least K1, K3 and K5 need to be relaxed. For the purpose, we next introduce a recently developed field aiming at modeling human unawareness.

2.4.2

Modeling Unawareness

The standard model with K1-K5 has obviously too strong implications as a model of human beings. Nevertheless it still underlies most economic and game theoretical models. Meanwhile, it is also true that recently considerable efforts have been made to develop less restrictive models by researchers who are not satisfied with the standard modeling. They have sought to model human unawareness, which the standard model entirely excludes32 .

To derive from K2 to K2a, assume K2 and E ⊆ F . Then K(E) = K(E ∩ F ) = K(E) ∩ K(F ). Hence K(E) ⊆ K(F ). 32 For a survey of this research agenda, see Dekel et al. (1998). It seems to us that dealing with unawareness is still far from orthodox. For example, Aumann (2005, pp. 89-90) acknowledges its significance but says, “Though there has been work that addresses the problem of awareness, I know of none that has caught on, that is truly satisfactory.” We also note that the importance of unawareness as well as taking into account unforeseen contingencies in decision theoretic models had already pointed out by some literatures as early as late 1980s (e.g. Fagin and Halpern (1988); and Kreps (1988)), though they do not discuss its relation with possibility correspondences. 31

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Unawareness, needless to say, is ubiquitous in our life, and inevitably affects our ways of formulating decision problems and hence those results. Unfortunately we do not have any unified definition of it. But a consensus is that when we say one is unaware of something, it should be required at least that she does not know it, and she does not know that she does not know it (like the Ramsfeld’s statement). For example, it is not only that Newton did not know Einstein’s relativity theory but also that he did not know he did not know it: Newton was unaware of it. To see the difference between unawareness and just not knowing, imagine an undergraduate student whose major is not physics. When he is asked to explain Einstein’s theory, he would not be able to answer because he does not know it, but in this case he knows that he does not know: he is aware of his ignorance, that is, K5, axiom of wisdom, holds here. Unawareness, by the requirement stated above, needs to violate K5, which is a necessary condition for a possibility correspondence to be partitional. Therefore the first major escape from the standard model was naturally to study nonpartitional models, that is, possibility correspondences that may not satisfy K3-K5. They were first studied rigorously by Geanakoplos (1989) with the following example story33 : Sherlok Holmes and Watson are investigating a crime in which a horse was stolen and the keeper was killed. Holmes has a conversation with Gregory, a local detective: Gregory “Is there any other point to which you would wish to draw my attention?” Holmes “To the curious incident of the dog in the night-time.” Gregory “The dog did nothing in the night-time.” Holmes “That was the curious incident.” Then Holmes concludes that there was no intruder in the stable that night (because had somebody intruded, the dog would have barked). On the other hand, although Watson also knows the fact that the dog did not bark, somehow he does not come up with the inference that there was no intruder. Now our interest is in Watson’s mode of thinking34 . Any reader of the story would say that he is unaware of the fact that there was no intruder. Geanakoplos argues, by allowing a possibility correspondence to be non-partitional, we can model unawareness like Watson’s: let the set of states Ω = {a, b}, where a is a state in which there was an intruder while b a state in which there was not, then Watson’s mind can be expressed as P (a) = {a} and P (b) = {a, b}. The simple formulation says that Watson can immediately notice the true state is a when it is true, while in state b, he cannot deduce which one out of the two states is true. The interpretation of the modeling is that, in state b, he should exclude the possibility that state a is true but he is not aware of it. Note that the his possibility correspondence does not satisfy K5. Nonpartitional models have provided us with some interesting results that standard partitional models could not have accommodated35 . But the modeling is in fact quite puzzling in several aspects if we take it seriously. Briefly, in state b above, Watson is supposed to not recognize the possibility that there was an intruder (in the original informal story), yet he considers b possible there. That is, he considers something that he does not recognize possible36 . 33

Due to Doyle (1892). On the other hand, from the decision theoretic point of view, Holmes’s way of thinking is perfectly “Bayesian” in the sense that by using newly acquired information about the dog, he nicely updates his prior belief about the existence of an intruder to the posterior belief. 35 See e.g. Geanakoplos (1989) and Samet (1990). 36 For detail of the critique, see Dekel, Lipman and Rustichini (1998). Furthermore, it might be easily attacked by asking, “Watson knows that if the true state is a, he would know it. Then why does he still conclude that a is possible when the true state is b?” 34

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In an attempt to avoid such puzzling formulations, then, in addition to knowledge operators, unawareness operators that directly deal with an agent’s unawareness have been introduced by Modica and Rustichini (1994)37 . However, it has been shown by Dekel, Lipman and Rustichini (1998, henceforth DLR) that any models based on the state-space model, regardless of the possibility correspondence being partitional or nonpartitional, cannot capture unawareness in an appropriate manner. DLR introduce a function U : 2Ω → 2Ω , called an unawareness operator, with the interpretation that U (E) is the event where the agent is unaware of the possibility that an event E occurs. They do not construct it from K, but instead present four axioms that it should satisfy: U1 U2 U3 U4

(Plausibility): U (E) ⊆ ¬K(E) ∩ ¬K¬K(E); (KU introspection): KU (E) = φ; (AU introspection): U (E) ⊆ U U (E); (Weak necessitation): ¬U (E) ⊆ K(Ω);

Plausibility is the translation of the requirement of unawareness described above: when the agent is unaware of something, then she does not know it, and she dose not know that she does not know it. KU introspection states that whenever she is unaware of something, she cannot know the fact. This should be interpreted carefully. We can be aware of the possibility that there might be something of which we are unaware. But U2 says that the agent cannot identify the thing of which she is unaware even in such cases. Then, AU introspection simply says that if the agent is unaware of something, then she must be unaware of the fact. Finally, weak necessitation states that when she is aware of something, she knows the true state is in Ω. Note that this is a weakening of K1. Then we have the following result proved by DLR. Proposition 2.3: For Ω with the knowledge operator K and the unawareness operator U , if they satisfy U1-U4, then ∀ E, F, G ∈ 2Ω , U (E) = U (F ) ⊆ ¬K(G). In words, if the agent is unaware of anything, then she is unaware of everything and knows nothing. It implies that standard state-space models can describe only either full awareness or full unawareness, that is, they cannot model any nontrivial human unawareness. DLR argue that the result is because of the primary assumption of the model that does not distinguish the agent’s point of view from that of us (an analyzer). She is assumed to be able to list all of the uncertainties relevant to decision problems she faces. But if she is unaware of some possibility, states recognized by herself should be “less complete” than those formulated by the analyzer. Hence, any models without this kind of distinction will fail to capture nontrivial unawareness.

2.4.3

Subjective State-spaces

The implication of DLR’s proposition is that if we want to model nontrivial human unawareness, we need to distinguish between subjective and objective modeling. In particular, the former needs to be less complete than the latter. In fact, several novel models have been recently proposed in this line (e.g. Heifetz et al. (2008); and Li (2009)). They explicitly model subjective state spaces that are supposed to be those agents subjectively conceive of and restricted compared to “full” state spaces that are in hands of analyzers. We, however, do not aim to contribute to the epistemological research itself in the thesis but rather try to argue with the idea of subjective state spaces that our hypergame framework introduced and developed in the later chapters can deal with nontrivial unawareness. For the purpose, reconsider the Watson’s example. Let us interpret his ignorance in Modica and Rustichini define an unawareness operator U as U (E) = ¬K(E) ∩ ¬K¬K(E), which is just one of axioms in Dekel, Lipman and Rustichini (1998). See U1 below. 37

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two ways: assigning (subjective) probability zero and unawareness38 . First, suppose that he was aware of the possibility itself that there was no intruder and, for some reason, conclude consciously that it cannot happen by noticing the fact that the dog did not bark. That is, state b was included in Ω and probability zero was assigned to it. In this case, the inference simply reflects, in terms of Bayesian updating, his (incorrect) conditional probability that describes the relation between the dog’s bark and existence of an intruder. Then it would be captured well by subjective Bayesianism and hence we do not need any novel models. But this is neither compatible with Watson’s actual mind nor the feature that we would like to formulate. Second, suppose, had someone asked Watson, “Could there have been an intruder that night?,” he would have answered, “Of course not, the dog did not bark!” At this time, he clearly knows that the dog’s not barking implies non-existence of any intruder. Then how can we explain his ignorance in the original story? We, as well as the literatures above, interpret it by considering that state b was not included in his subjective state space: he was unaware of it39 . Let us denote Watson’s subjective state space by ΩW while by Ω mean the full state space that are those from objective point of view. Then ΩW ⊂ Ω and, in particular, b ∈ Ω but b ∈ / ΩW . Therefore, it, in the first place, is impossible for him to assign any probability to state b. The idea about the distinction between probability zero and unawareness has been discussed in terms of the set theoretical epistemology introduced in this section but not applied rigorously to games40 . As discussed above, Bayesian games require the agents to have full state space Ω, namely the type set. On the other hand, hypergame models that directly formulate each agent’s subjective view do not require to do so. In the next section, we introduce hypergame models.

2.5

Hypergames

Our study is based on another model of game of incomplete information called hypergames. Hypergame theory, proposed by Bennett (1977), deals with interactive situations where agents may misperceive some components of a game. The basic idea is that each agent is assumed to have her own subjective view of it, which is given as a normal form game and called the agent‘s subjective game, based on which she makes decisions. Therefore each agent may perceive the game in different ways41 . In this respect, hypergame theory discards the basic assumption of standard game theory that every agent sees the “same” game42 . Among several types of hypergame models that have been proposed so far, we introduce two models which we shall use in the thesis: simple hypergames and hierarchical hypergames. The former simply assumes that each agent believes that each subjective game is common knowledge (Bennett and Dando (1979); and Bennett et al. (1989)), while the latter takes into account hierarchy of an agent’s perceptions, that is, an agent’s perception about another agent’s perception, and so on (Bennett (1980); Wang et al. (1988); Wang et al. (1989); Inohara (2000); and Inohara (2002)). In the thesis, we may refer to these models simply as “hypergames” unless it leads to any confusion. 38

For the topic, see also Li (2008). The story is inspired by Li (2009). Li shows that this kind of unawareness satisfies DLR’s axioms, U1-U4, under appropriate redefinition of those. 40 An exception we know is Feinberg’s (2010) model of games with unawareness. 41 There exist some other frameworks that allow agents to perceive a situation in different ways in some sense. Particularly since 1990s, these kinds of models that deal with agents whose cognitions are limited have been studied by not a few researchers. Among those are the theory of self-confirming equilibrium by Fudenberg and Levine (1993, 1998), inductive game theory by Kaneko and Matsui (1999), Feinberg’s (2010) games with unawareness and so on. 42 Note that the assumption holds for Bayesian games. 39

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In this section, we only introduce the basic models, while we discuss their solution concepts in later chapters.

2.5.1

Simple Hypergames

Simple hypergames assume that each agent has each own perception about the game she faces in the form of a normal form game. Such a perception is called her subjective game and a simple hypergame is defined as a collection of subjective games for all the agents43 . Formally: Definition 2.10 (Simple hypergames): H = (I, (Gi )i∈I ) is a simple hypergame, where I is the finite set of agents and Gi = (I i , Ai , ui ) is a normal form game called agent i’s subjective game, where: • I i is the finite set of agents perceived by agent i (I i ⊆ I). • Ai = ×j∈I i Aij , where Aij is the finite set of agent j’s actions perceived by agent i. • ui = (uij )j∈I i , where uij : Ai → ℜ is agent j’s utility function perceived by agent i. Simple hypergames assume, implicitly or explicitly, that each agent believes that each own subjective game is common knowledge among all the agents (who the agent thinks participates the game)44 . For example, agent i believes not only that the situation is Gi but also that all the others see it as well. An agent never knows another agent’s subjective game and hence the whole structure of the hypergame, which are described only from an analyzer’s point of view. Let us denote ×i∈I Aii by Ao . Since, in a simple hypergame, agent i chooses an action from i Ai , Ao is interpreted as the set of all the “realizable” outcomes from an objective viewpoint. Note that each agent i believes the outcome space is not Ao but Ai , and utilities of some agent may not be defined on some elements in Ao . We say that agent i misperceives the set of agents iff I i ̸= I, some agent j’s action set iff Aij ̸= Ajj , and some agent j’s utility function iff uij ̸= ujj , with j ̸= i. As an example, consider a two-agent hypergame of agents 1 (she) and 2 (he), namely H = (I, Gi )i∈I with I = {1, 2}. Table 2.3a illustrates 1’s subjective game, G1 , where A11 = {t, b}, A12 = {l, r} and each entry expresses utilities for the both agents which describe the utility functions perceived by agent 1, u11 and u12 . Similarly, Table 2.3b represents 2’s subjective game, G2 . In the hypergame, they both perceive the set of agents and actions correctly but misperceive each other’s utility function. Since each subjective game is a normal form game, we can define Nash equilibrium of it as usual. In this case, N (G1 ) = {(t, l), (b, r)} while N (G2 ) = {(b, r)}. 1\2 t b

l 4, 4 3, 1

r 1, 3 2, 2

Table 2.3a: 1’s subjective game, G1 1\2 t b

l 3, 4 4, 1

r 1, 3 2, 2

Table 2.3b: 2’s subjective game, G2 43

The literature of hypergames as well as the present thesis typically considers only pure strategies (actions). Mixed extension of simple hypergames is discussed by Sasaki et al. (2007). 44 Kaneko (2002) discusses individual belief of common knowledge in terms of epistemic logic.

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2.5.2

Hierarchical Hypergames

Simple hypergames might be extreme in the sense that the agents may misperceive a interactive decision situation yet they all believe each own perception is common knowledge. Then, once an agent admits the possibility that the others may perceive the game in different ways, she would take into account it when formulating her decision problem. Furthermore, she might notice that the other agent may also notice such possibilities of misperceptions, that is, she may consider another agent may think other agents may misperceive the game. Then she needs to consider the agent’s view about some other agent, and so on. Hierarchical hypergames explicitly model such “hierarchies of perceptions” of agents by introducing the concept of viewpoints. Each perception of each viewpoint is given as a normal form game and called the viewpoint’s subjective game. Definition 2.11 (Hierarchical hypergames): H h = (I, (Gσ )σ∈Σ ) is a hierarchical hypergame, where I is the finite set of agents involved in the situation and Σ is the set of viewpoints relevant in it. For any σ ∈ Σ, Gσ = (I σ , Aσ , uσ ) is a normal form game called viewpoint σ’s subjective game, where: • I σ is the finite set of agents perceived by viewpoint σ. • Aσ = ×i∈I σ Aσi , where Aσi is the set of agent i’s actions perceived by viewpoint σ. • uσ = (uσi )i∈I σ , where uσi : Aσ → ℜ is agent i’s utility function perceived by viewpoint σ. ∪ ik ···in \ {i } for any k(= 2, ..., n)}. Then let Σ = I ∪ ∞ k n=2 {i1 · · · in |in ∈ I, ik−1 ∈ I A viewpoint indicates a specific hierarchy of perception. For example, viewpoint i means agent i’s view, i.e. subjective game, and viewpoint ji is agent j’s view perceived by agent i. In the example of a simple hypergame above, if agent 1 thinks not that G1 is common knowledge but that the opponent sees the game shown as G2 , such a view is described as agent 2’s subjective game perceived by agent 1, namely G21 . In general, viewpoint i1 i2 · · · in is interpreted as agent i1 ’s view perceived by agent i2 . . . perceived by agent in . As with simple hypergames, let us denote ×i∈I Aii by Ao . Ao is interpreted as the set of all the “realizable” outcomes from an objective viewpoint. We specify the set of viewpoints relevant in a hypergame by Σ. A viewpoint is said to be relevant when it is actually taken into account in some agent’s decision making. We assume that, in a hypergame, any viewpoint σ, when formulating the decision situation, considers views of all the agents who σ thinks are participating in the game and does not consider views of anybody else. For example, when agent i is in I, i ∈ Σ by definition, and if another agent j is in I i , viewpoint ji must be in Σ, and otherwise, it is not included in Σ. Furthermore, we suppose that a viewpoint does not contain any successive agents because we identify an agent’s view with the agents’ view perceived by herself. For example, since agent i’s view perceived by agent i is essentially same as agent i’s view, we consider viewpoint i but do not consider viewpoint ii, that is, i ∈ Σ but ii ∈ / Σ, and similarly, neither jii nor iij is included in Σ. In the subsequent discussion, when we refer to viewpoints, we only indicate viewpoints relevant in this sense. We may deal with concatenations of viewpoints. For example, by σ ′ σ with σ = i1 · · · in and σ ′ = j1 · · · jm (with jm ̸= i1 ) we mean viewpoint j1 · · · jm i1 · · · in . When σ = i1 · · · in with n ≥ 2, any viewpoint im · · · in with n ≥ m ≥ 2 is said to be higher than σ. On the other hand, any viewpoint τ σ with τ = j1 · · · jl and jl ̸= i1 is said to be lower than σ. For example, for viewpoint ji, viewpoint i is higher than ji while viewpoint kji is lower than it45 . Furthermore, when σ = i1 · · · in , let us denote i1 by σ1 . We say σ1 is the lowest agent 45 Clearly a binary relation %⊂ Σ × Σ, which is interpreted as “higher than or equal to,” is a partial order. Viewpoints i, j, ... are the highest viewpoints, that is, viewpoints which do not have any higher viewpoints than themselves. On the other hand, there are no lowest viewpoints because any viewpoints have some lower viewpoints.

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in viewpoint σ. For any σ ∈ Σ, let Σσ = σ ∪ {τ σ|τ = j1 · · · jl , jl ̸= σ1 , and τ σ ∈ Σ}. Σσ is the union of σ itself and the set of viewpoints lower than σ. σ Moreover, we assume that for any σ ∈ Σ, ∀ i ∈ I σ , Aiσ i ⊆ Ai . Intuitively, this means that if agent i thinks another agent j is aware that an action is available, then i never excludes the action from j’s action set in i’s subjective game.

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Chapter 3

Subjective Rationalizability in Hypergames In this chapter, we discuss solution concepts of hierarchical hyeprgames introduced in Chapter 2. Although several solution concepts have been proposed for hypergames, we carefully review them and point out some problems, and then propose a new solution concept called subjective rationalizability1 . An action of an agent is called subjectively rationalizable when the agent thinks it can be a best response to the other’s choices, each of which the agent thinks each agent thinks is a best response to the other’s choices, and so on. In order to make the concept applicable for hypergame analysis, we then prove that it is equivalent to rationalizability in the standard game theory under a condition called inside common knowledge. Keywords: hierarchical hypergames, rationalizability, subjective rationalizability, inside common knowledge.

3.1

Introduction

We discuss rational behaviors of agents in light of their perceptions in hierarchical hypergames in this chapter. In order to predict an agent’s choice in hierarchical hypergames, several solution concepts have been proposed (Wang et al. (1988, 1989)). They are typically based on the idea that an analyzer first fixes the level of hierarchy of perceptions (practically three or four at most), calculates an “equilibrium” (e.g. Nash) in the game of the lowest level of the hierarchy, and then supposes that each agent in each level takes a best response to choices of agents in the one step lower level. For example, consider a three-level hypergame played by two agents, i and j. According to the idea, in order to analyze agent i’s choice, we first need to know an equilibrium in agent i’s subjective game perceived by agent j perceived by agent i, namely subjective game of viewpoint iji. Suppose agent i’s action, ai , constitutes the equilibrium there. Then agent i expects agent j would take a best response to ai in viewpoint ji’s subjective game, and agent i chooses a best response to it. We point out two problems of the idea. First, it is not clear what assures that an agent takes an action that constitutes an equilibrium. In fact, in game theory, how a Nash equilibrium can be played is still controversial as discussed in 2.2. There is a leap of logic between that ai constitutes a Nash equilibrium and that agent i takes it. Second, there seems to be no substantial reason that we can fix the finite level of a hypergame. In the previous example, agent i’s choice in viewpoint iji’s subjective game seems to depend on agent i’s expectation about agent j’s choice, which clearly depends on how agent i thinks how agent j perceives the game. It seems that we need to consider viewpoint jiji’s subjective game, and so on. In this study, we propose a new solution concept called subjective rationalizability so as 1

The chapter is based on Sasaki (2011).

26

to avoid these problems. We do not fix level of perceptions and instead consider infinite hierarchy, and suppose that an agent acts according to the following principle: an agent takes a best response to actions of the others, each of which the agent thinks is a best response to actions of the others, each of which the agent thinks the other agent thinks is a best response to actions of the others, and so on. If an agent’s action satisfies this principle, it is called subjectively rationalizable2 . Subjective rationalizability is defined not for agents but for viewpoints, so, for instance, agent i can think of agent j’ choice as viewpoint ji’s subjectively rationalizable action. According to the idea, agent i’s choice can be predicted as viewpoint i’s subjectively rationalizable action. We also say it is agent i’s subjectively rationalizable action3 . The concept, however, is apparently impractical because we need to know each agent’s choice in infinite hierarchy of perception in order to calculate subjectively rationalizable action. For example, to see if an action of agent i is subjectively rationalizable, we need to check if it is a best response (in viewpoint i’s subjective game) to a particular combination of actions of the other agents who agent i thinks are participating in the game, and if each of the actions is a best response (in each viewpoint’s subjective game) to a particular combination of actions of the other agents who agent i thinks the agent thinks are participating in the game, and so on. This process continues infinitely. To make subjective rationalizability a more applicable concept avoiding the problem, we prove that, under a condition, subjectively rationalizability becomes equivalent to the concept of rationalizability commonly used in the standard game theory (Bernheim (1984); and Pearce (1984)). The condition is called inside common knowledge (ICK) (Inohara (2000, 2002)). A particular viewpoint is said to have ICK if and only if the viewpoint considers that every agent sees the same game, every agent thinks every agent sees the same game, and so on, that is, the game structure is common knowledge among every agent. Furthermore, we shall show that, by using the result, it becomes easier to calculate an agent’s subjectively rationalizable action when the hypergame contains ICK. (We note that throughout the thesis we distinguish clearly between “subjective rationalizability” and “rationalizability.”)

3.2

Subjective Rationalizability

We here study hierarchical hypergames introduced in the previous chapter and use the same notations. We newly define the concept of subjective rationalizability for a viewpoint4 . It is considered as such an action that an agent would choose in light of its own perception. Definition 3.1 (Subjective rationalizability): Let H h = (I, (Gσ )σ∈Σ ) be a hierarchical hypergame. a∗σ ∈ Aσσ1 is called subjectively rationalizable for viewpoint σ ∈ Σ if and only if there exists (a∗τ )τ ∈Σσ such that, for all τ ∈ Σσ , a∗τ ∈ Aττ1 and ∀

aτ ∈ Aττ1 , uττ1 (a∗τ , a∗−τ ) ≥ uττ1 (aτ , a∗−τ ),

where a∗−τ = (a∗iτ )i∈I τ \{τ1 } . Then such (a∗τ )τ ∈Σσ is called a best response hierarchy of Σσ . An action of the lowest agent in a viewpoint is called subjectively rationalizable for the viewpoint when it is a best response in the viewpoint’s subjective game to some actions of the other agents, each of which is a best response in the subjective game of one step lower viewpoint to some actions of the other agents, and so on. The concept of subjective 2

Although not a hypergame model, games with unawareness have provided a solution concept called extended rationalizability based on the same motivation (Feinberg (2010)). 3 Hence the concept assumes that an agent believes any agents are rational in any levels in the hierarchy of the perception. On the other hand, level-k models weaken the assumption, and suppose an agent believes another agent’s rationality holds in some fixed levels at most. (see e.g. Crawford (2003)). 4 Note that, in Definition 3.1, a∗σ is a particular action of σ1 , the lowest agent of viewpoint σ.

27

rationalizability can be understood based on the following idea: the lowest agent in a viewpoint would take a best response to actions which she thinks the other agents would choose. When expecting the choices of the others, she considers that each of the other agents takes a best response to actions which she thinks the agent thinks the other agents would choose, and such her inference goes on for even lower viewpoints. When agent i makes decision in this way, her choice can be predicted as a subjectively rationalizable action of viewpoint i. Thus we may also say that it is agent i’s subjectively rationalizable action. Let us see Figure 3.1 and consider how agent i decides her choice according to the idea. In the figure, a circled character indicates an agent, while an action in a square is the predicted choice of the agent, and an arrow describes each agent’s perception. Suppose that agent i considers the other agents, 1, · · · , m, are going to choose a∗1i , · · · , and a∗mi , respectively. She expects so because each of the other agents takes a best response to some particular actions that agent i thinks each agent considers the other agents are going to choose. For example, agent i expects agent 1 to take a∗1i , which is a best response in viewpoint 1i’s subjective game to (a∗21i , · · · , a∗n1i ), actions of the other agents who agent i thinks agent 1 thinks are in the game, namely, 2, · · · , n. Furthermore, agent i considers agent 1 thinks the others are going to choose (a∗21i , · · · , a∗n1i ) because agent i thinks agent 1 thinks each of the other agents takes a best response to some particular actions that agent i thinks agent 1 thinks each agent expects the other agents to choose, and so on. Then she takes a∗i which is a best response in her own subjective game to (a∗1i , · · · , a∗mi ). a∗i is her subjectively rartionalizable action5 .

Figure 3.1: Subjective rationalizability As a result, the actions depicted in the squares in the figure constitute a best response hierarchy: each specified action is a best response in the subjective game of the viewpoint to the actions described in one step lower layer. Therefore, by definition, each action is subjectively rationalizable for each viewpoint. For example, a∗1i is viewpoint 1i’s subjectively rationalizable action. An outcome likely to obtain as a result of each agent’s decision making in this way is thus given as a combination of each agent’s subjectively rationalizable action. Note that, given that a viewpoint’s view may contain some misperceptions, a subjectively rationalizable action may not be always a best response to the others’ actual choices from an objective (an analyzer’s) point of view, and hence it is called “subjectively” rationalizable. (See also 3.4.) We note that, according to the definition, an agent or a viewpoint may have no subjectively rationalizable action. In an extreme example, if there are no common actions in Aij and Aji j , one cannot construct a best response hierarchy of Σi , and thus any actions of agent i cannot be subjectively rationalizable. But the assumption above would be unnatural because if 5

mi i i Note that we here assume that (a∗1i , · · · , a∗mi ) ∈ A1i 1 × · · · × Am is also in A1 × · · · × Am

28

agent i believes another agent j considers a particular action is available, then agent i should include the action in agent j’s action set. That is, Aij should contain all the elements in Aji j . σ for any viewpoint σ ∈ Σ and any k ∈ I σ \ {σ }, It is easily shown that if we assume Akσ ⊆ A 1 k k one can always have a best response hierarchy, and hence there exists at least one subjectively rationalizable action. The concept of subjective rationalizability is based on the consistent idea mentioned above in theory but it entails a bothersome problem in practice. As the definition shows, calculation of a subjectively rationalizable action requires us to identify a best response hierarchy. It, however, contains infinite elements, and therefore it appears practically impossible to calculate it. In the next section, in order to avoid the problem, we shall prove that under some condition a subjectively rationalizable action can be replaced by a rationalizable action, a commonly used solution concept in the standard game theory6 . In a normal form game, rationalizable action is defined as such an action that survives iterative elimination of actions that cannot be best responses to any combinations of the others’ actions (Bernheim (1984); and Pearce (1984)). Definition 3.2 (Rationalizability): Let G = (I, A, u) be a normal form game. a∗i ∈ Ai is called a rationalizable action of agent i iff a∗i



∞ ∩

Hi (t),

t=1

where for any j ∈ I, Hj (t) = {a′j ∈ Aj |∃ a−j ∈ ×k∈I\{j} Hk (t − 1),∀ aj ∈ Aj , uj (a′j , a−j ) ≥ uj (aj , a−j )} with t = 1, 2, . . ., and Hj (0) = Aj . Let us denote the set of rationalizable actions of agent i in G by Ri (G). It is known that, in a normal form game, an agent always has at least one rationalizable action, and may have two or more. As a simple example, consider the game shown in Table 3.1. The game has no Nash equilibrium, but the both actions for the both agents are rationalizable. It is easy to see it because both actions can be a best response to some action of the opponent. 1\2 t b

l 1, 0 0, 1

r 0, 1 1, 0

Table 3.1: Rationalizability in a normal form game

3.3 3.3.1

Equivalence of Subjective Rationalizability and Rationalizability Under Inside Common Knowledge Inside Common Knowledge

We introduce the concept of inside common knowledge (ICK) of viewpoints. A viewpoint σ is said to have ICK when every viewpoint lower than σ has the same subjective game as that of σ (Inohara (2000)). 6

As stated in 2.2, it is known that when the game structure as well as rationality of agents are common knowledge, an agent’s choice is given not as an action that constitutes a Nash equilibrium but as a rationalizable action

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Definition 3.3 (Inside common knowledge, ICK): In a hierarchical hypergame H h = (I, (Gσ )σ∈Σ ), σ ∈ Σ is said to have inside common knowledge if and only if ∀ τ ∈ Σσ , Gτ = Gσ . ICK can be regarded as common knowledge perceived subjectively. For example, viewpoint ji’s having ICK means that agent i thinks agent j considers the game is common knowledge among the agents. In this case, since viewpoint ji’s subjective game may be different with that of agent i, the situation may not be truly common knowledge from an objective point of view. Since it is quite unlikely that, in reality, people consider many different games in infinite hierarchy of perception, it would be natural to assume ICK at some particular level.

3.3.2

Equivalence Results

The next lemma assures that a subjectively rationalizable action of a viewpoint is identical to a rationalizable action of the lowest agent of the viewpoint in its subjective game. Lemma 3.1: In a hierarchical hypergame H h = (I, (Gσ )σ∈Σ ), suppose σ ∈ Σ has ICK. Then, a∗σ ∈ Aσσ1 is subjectively rationalizable if and only if a∗σ ∈ Rσ1 (Gσ ). Proof : Let H h = (I, (Gσ )σ∈Σ ) be a hierarchical hypergame and suppose σ ∈ Σ has ICK. (proof of if part) Suppose a∗σ ∈ Aσσ1 is rationalizable in Gσ , that is, a∗σ ∈

∞ ∩

Hσ1 (t),

t=1

where ∀

i ∈ I σ , Hi (t) = {a′i ∈ Aσi |∃ a−i ∈ ×j∈I σ \{i} Hj (t − 1),∀ ai ∈ Aσi , uσi (a′i , a−i ) ≥ uσi (ai , a−i )}

with t = 1, 2, . . ., and Hi (0) = Aσi . For τ ∈ Σσ and i ∈ I τ \ {τ1 }, let Siτ = {ai ∈ Aτi |∃ (a′τ1 , a−τ1 ,i ) ∈ Sτ × (×j∈I τ \{τ1 ,i} Aτj ),∀ aτ1 ∈ Aττ1 , uττ1 (a′τ1 , ai , a−τ1 ,i ) ≥ uττ1 (aτ1 , ai , a−τ1 ,i )} with Sσ = {a∗σ }. Siτ is interpreted as the set of agent i’s actions that can constitute action profile of all the agents except for τ1 to which some action of τ1 in Sτ is a best response in Gτ . Due to σ’s ICK, for any τ ∈ Σσ and i ∈ I τ \ {τ1 }, Siτ = {ai ∈ Aσi |∃ (a′τ1 , a−τ1 ,i ) ∈ Sτ × (×j∈I σ \{τ1 ,i} Aσj ),∀ aτ1 ∈ Aστ1 , uστ1 (a′τ1 , ai , a−τ1 ,i ) ≥ uστ1 (aτ1 , ai , a−τ1 ,i )}. Suppose Siτ = φ for some τ ∈ Σσ and i ∈ I τ \ {τ1 }. This means that any actions of τ1 in Sτ cannot be a best response to any action profiles of the others in Gσ , that is, Hτ1 (1) ∩ Sτ = φ. If τ = σ, then Hσ1 (1) ∩ Sσ = φ ⇒ a∗σ ∈ / Hσ1 (1) ⇒ a∗σ ∈ / Rσ1 (Gσ ). If τ = i1 · · · in σ with n ≥ 1, then Hi1 (1) ∩ Si1 ···in σ = φ 30

⇒ Hi2 (2) ∩ Si2 ···in σ = φ ⇒ ··· ⇒ Hin (n) ∩ Sin σ = φ ⇒ Hσ1 (1) ∩ Sσ = φ ⇒ a∗σ ∈ / Rσ1 (Gσ ). Thus, in any case, a∗σ ∈ / Rσ1 (Gσ ), but this contradicts a∗σ ∈ Rσ1 (Gσ ). Hence we have Siτ ̸= φ for any τ ∈ Σσ and i ∈ I τ \ {τ1 }. Therefore, there exists (aτ )τ ∈Σσ such that aτ ∈ Sτ for all τ ∈ Σσ . Then, in (aτ )τ ∈Σσ , ∀

τ ∈ Σσ ,∀ a ¯τ ∈ Aττ1 , uττ1 (aτ , a−τ ) ≥ uττ1 (¯ aτ , a−τ ),

where a−τ = (aiτ )i∈I τ \{τ1 } . That is, this (aτ )τ ∈Σσ is a best response hierarchy of Σσ . Since aσ = a∗σ , a∗σ is subjectively rationalizable. (proof of only if part) Suppose a∗σ ∈ Aσσ1 is a subjectively rationalizable action of viewpoint σ, that is, by definition, there exists a best response hierarchy of σ. Due to σ’s ICK, this implies that there exists (a∗τ )τ ∈Σσ such that a∗τ ∈ Aστ1 for all τ ∈ Σσ which satisfies ∀

τ ∈ Σσ ,∀ aτ ∈ Aστ1 , uστ1 (a∗τ , a∗−τ ) ≥ uστ1 (aτ , a∗−τ ),

where a∗−τ = (a∗iτ )i∈I σ \{τ1 } . We define Hi (t) for any i ∈ I σ with t = 1, 2, . . . in the same way as shown in the proof of if part above. Let us prove a∗σ ∈ Hσ1 (n) with some integer n ≥ 1. For any i1 · · · in σ ∈ Σσ , a∗i1 ···in σ ∈ Hi1 (0) because Hi1 (0) = Aσi1 . Then, ∀

i2 · · · in σ ∈ Σσ , a∗i2 ···in σ ∈ Hi2 (1)

⇒∀ i3 · · · in σ ∈ Σσ , a∗i3 ···in σ ∈ Hi3 (2) ⇒ ··· ⇒∀ in σ ∈ Σσ , a∗in σ ∈ Hin (n − 1) ⇒ a∗σ ∈ Hσ1 (n). a∗σ

∗ Since ∩∞ this holds for any∗ integer nσ ≥ 1, aσ ∈ Hσ1 (t) for any integer t ≥ 1, thus we have ∈ t=1 Hσ1 (t). Hence aσ ∈ Rσ1 (G ). ¤

By using the lemma, the next proposition shows that, in a best response hierarchy of viewpoint σ, when some viewpoint τ which is lower than σ or σ itself has ICK, the actions for all the viewpoints lower than τ (including τ itself) can be replaced by a rationalizable action of the lowest agent of τ in τ ’s subjective game. In other words, due to the proposition, the definition of subjective rationalizability can be redescribed under the condition of ICK. Proposition 3.2: Let σ ∈ Σ in H h = (I, (Gσ )σ∈Σ ) and suppose υ ∈ Σσ has ICK. a∗σ ∈ Aσσ1 is σ’s subjectively rationalizable action if and only if there exists ((a∗τ )τ ∈Σσ \Συ , a∗υ ) such that a∗τ ∈ Aττ1 for all τ ∈ Σσ \ Συ and a∗υ ∈ Aυυ1 which satisfies: • ∀ τ ∈ Σσ \ Συ ,∀ aτ ∈ Aττ1 , uττ1 (a∗τ , a∗−τ ) ≥ uττ1 (aτ , a∗−τ ), where a∗−τ = (a∗iτ )i∈I τ \{τ1 } , and • a∗υ ∈ Rυ1 (Gυ ).

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Proof : Let H h = (I, (Gσ )σ∈Σ ) be a hierarchical hypergame and σ ∈ Σ, and suppose υ ∈ Σσ has ICK. (proof of if part) Suppose there exists ((a∗τ )τ ∈Σσ \Συ , a∗υ ) such that a∗τ ∈ Aττ1 for all τ ∈ Σσ \ Συ and a∗υ ∈ Aυυ1 which satisfies both of the two conditions described in Proposition 3.2. Then, due to Lemma 3.1, aυ is υ’s subjectively rationalizable action in Gυ , that is, there exists (a∗ω )ω∈Συ such that a∗ω ∈ Aωω1 for all ω ∈ Συ , and ∀

ω ∈ Συ ,∀ aω ∈ Aωω1 , uωω1 (a∗ω , a∗−ω ) ≥ uωω1 (aω , a∗−ω ),

where a∗−ω = (a∗iω )i∈I ω \{ω1 } . Therefore ((a∗τ )τ ∈Σσ \Συ , (a∗ω )ω∈Συ ) is a best response hierarchy of σ, and hence a∗σ in ((a∗τ )τ ∈Σσ \Συ , a∗υ ) is σ’s subjectively rationalizable action. (proof of only if part) Suppose a∗σ ∈ Aσσ1 is σ’s subjectively rationalizable action, and let ∗ (aτ )τ ∈Σσ be a best response hierarchy of σ. Then, (a∗ω )ω∈Συ in it is a best response profile of viewpoint υ, thus a∗υ is υ’s subjectively rationalizable action. Since υ has ICK, a∗υ ∈ Rυ1 (Gυ ) due to Lemma 3.1. Hence ((a∗τ )τ ∈Σσ \Συ , a∗υ ) satisfies both of the two conditions in Proposition 3.2. ¤ The proposition makes it easier to calculate subjectively rationalizable actions under ICK. When we want to know viewpoint σ’s subjectively rationalizable action, instead of identification of a best response hierarchy which consists of infinite elements as shown in Definition 3.1, we just need to have rationalizable actions in subjective games of the highest viewpoints (in Σσ ) having ICK, and then identify the “reduced” best response hierarchy by using them as described in Proposition 3.2. The proposition also implies that every subjectively rationalizable action can be derived in this way. To see an example, reconsider agent i’s subjectively rationalizable action in Figure 3.1. Suppose now all the viewpoints which are one step lower than viewpoint i, namely viewpoint 1i, ..., mi, have ICK, that is, agent i considers every other agent thinks every agent sees the same game, every agent thinks every agent sees the same game, and so on.

Figure 3.2: Subjective rationalizability under ICK Proposition 3.2 implies that if a∗1i is agent 1’s rationalizable action in viewpoint 1i’s subjective game, ..., and a∗mi is agent m’s rationalizable action in viewpoint mi’s subjective game, then agent i’s best response in viewpoint i’s subjective game to (a∗1i , · · · , a∗mi ), i.e. a∗i , becomes a subjectively rationalizable action for agent i. We do not need a best response hierarchy which contains infinite elements in .

3.4

Example

In this section, we present an example of the concepts and findings presented so far. In particular, our focus is to illustrate how subjective rationalizability can give us new insights compared to existing solution concepts, as well as how the results obtained in the previous section can be applied when analyzing a hypergame. Let us consider a two-agent hypergame with I = (i, j). i’s view is given as viewpoint i’s subjective game, Gi , as shown in Table 3.2. i’s decision would depend on i’s expectation 32

about j’s choice, which depends on i’s view about how j perceives the game. Suppose that i thinks j thinks the game they play is not Table 3.2 but Table 3.3, that is, it is viewpoint ji’s subjective game, Gji . Gji is slightly different with Gi in terms of i’s utility. This means that i thinks j partly misperceives i’s utility function. Furthermore, we assume that i thinks j thinks i’s view is same as Gji , i thinks j thinks i thinks j’s view is same as Gji , and so on, that is, viewpoint ji has ICK. i\j a b c

x 2, 2 3, 3 0, 0

y 2, 3 3, 2 0, 0

z 0, 0 0, 0 1, 1

Table 3.2 : Viewpoint i’s subjective game i\j a b c

x 3, 2 2, 3 0, 0

y 2, 3 3, 2 0, 0

z 0, 0 0, 0 1, 1

Table 3.3 : Viewpoint ji’s subjective game Then, which action would i choose? If by following the conventional way of analysis we regard the situation as a two-level hypergame and apply conventional Nash-type solution concepts, the analysis, for instance, would go as follows: Gji has the unique Nash equilibrium, (c, z). Then, expecting j is going to choose z, i takes the best response to it in Gi , namely, c. In Gji , however, j’s other actions, x and y, appear rather attractive for j. Indeed not only z but also these two actions are subjectively rationalizable for viewpoint ji. Due to Lemma 3.1, given that viewpoint ji has ICK, a rationalizable action of j in Gji is a subjectively rationalizable action of viewpoint ji. It is easy to see that the three actions of j are rationalizable in this game as all of them survive iterated elimination of actions that cannot be best a response to any action of the opponent. Then, Proposition 3.2 implies that i’s action that can be a best response to some of these three actions of j in Gi is subjectively rationalizable for viewpoint i. Therefore, b and c are subjectively rationalizable7 . This means that if i makes decision according to the idea of subjective rationalizability, i may choose not only c but also b. Note that the solution concept is called subjective rationalizability, so it is not always true that any subjectively rationalizable action of viewpoint i can be a best response to some of j’s possible choices (viewpoint j’s subjectively rationalizable actions). For example, suppose j actually perceives the game shown in Table 3.4, Gj : j is unaware of i’s action c, and i misunderstands j’s utility function in Gi . If viewpoint j has ICK, since y is the only rationalizable action in Gj , it is the only subjectively rationalizable action for viewpoint j. Among viewpoint i’s two subjectively rationalizable actions, b is a best response to y, but c is not. Thus i’s action, c, is subjectively rationalizable but never gives i the highest utility from an objective point of view. i\j a b

x 3, 2 2, 2

y 2, 3 3, 3

z 0, 0 0, 0

Table 3.4 : Viewpoint j’s subjective game 7

Hence, as a process, we obtain the same result if we regard the situation as a two-level hypergame and apply rationalizablity (in the standard game theory) in the lower level. We here point out that our findings assures the agent’s choice, b or c in this case, is theoretically consistent with the idea of subjective rationalizability.

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3.5

Conclusion

The contribution of the present study is mainly twofold. First, we have proposed a new solution concept in hierarchical hypergames called subjective rationalizability. The concept is based on an idea that every agent takes a best response to expected choices of the others, every agent thinks every agent takes a best response to expected choices of the others, and so on. When an agent makes decision in a hypergame according to the idea, the agent’s choice can be predicted as her subjectively rationalizable action. Second, we have proved equivalence of subjective rationalizability and rationalizability under a condition called inside common knowledge. The result is useful to avoid the problem that it is practically impossible to calculate subjectively rationalizable actions in the way which its original definition shows, and makes the concept applicable for hypergame analysis. Furthermore, in the next chapter, we shall show that the concept of subjective rationalizability is essential when we discuss the relationships between hierarchical hypergames and Bayesian games. It will be shown that it is highly related to Bayesian Nash equilibrium of Bayesian games.

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Chapter 4

Hypergames and Bayesian Games: A Theoretical Comparison of the Models of Games with Incomplete Information In this chapter, we study the relationships between two independently developed models of games with incomplete information, hypergames and Bayesian games1 . We first show that any hypergame can naturally be reformulated in terms of Bayesian games in an unified way, and then prove that some equilibrium concepts defined for hypergames are in a sense equivalent to those for Bayesian games. Based on the results, we discuss carefully how each model should be used according to the analyzer’s purpose. Keywords: Bayesian games, Bayesian representation of hypergames.

4.1

Introduction

We introduced two models of games with incomplete information in Chapter 2: hypergames and Bayesian games. Hypergame theory deals with agents who may misperceive some components of the game (Bennett (1977)). It is the basic idea of hypergames that each agent is assumed to have her own subjective view of it, which is formulated as a normal form game called her subjective game, and make decisions based on it. In this way it allows agents to hold different perceptions about the game. On the other hand, Bayesian games have been proposed by Harsanyi (1967), who argued that incompleteness of information about anything are captured without loss of generality by subjective probability distributions for each agent over the set of possible states. In his way of modeling, such possibilities are modeled as types of each agent, and a game with incomplete information is reformulated as a game of complete information called a Bayesian game by introducing a set of types as well as each agent’s belief about them (in a form of probability distribution on the others’ types). The aim of this chapter is to compare the two models. In particular, we examine how hypergames can be differentiated from Bayesian games. Since they have been established and developed independently, the relationship has not been investigated rigorously enough. Let us illustrate our motivation with an example. Consider a two-agent interactive decision of agents 1 and 2, where 1 believes the game they play is prisoners’ dilemma (Table 4.1) while 2 believes chicken game (Table 4.2). The situation can be captured as a (simple) hypergame by defining each agent’s subjective game as each table shows. Since they both 1

The present chapter is largely based on Sasaki and Kijima (2012).

35

see the same set of agents as well as actions but suppose different utility functions, we say that they perceive correctly agents and actions but misperceive utility functions. 1\2 t b

l 3, 3 4, 1

r 1, 4 2, 2

Table 4.1: prisoners’ dilemma 1\2 t b

l 3, 3 4, 2

r 2, 4 1, 1

Table 4.2: chicken game In terms of its theoretical relation with Bayesian games, we have two key questions that motivate our study. First, is it possible to formulate the situation as a Bayesian game as well? The answer is: the standard formulation of Bayesian game would allow us to do so. In the example, two types are introduced for each agent: one type associates the agent with prisoners’ dilemma, and the other with chicken game. In Bayesian games, a type is characterized by subjective prior and utility function. In this case, for example, agent 1’s type associated with prisoners’ dilemma has a subjective prior (probability distribution) that assigns probability 1 to agent 2’s type associated with prisoners’ dilemma while probability 0 to the other type. It has the same utility function as in prisoners’ dilemma. We can define the other types in a similar way and hence construct a Bayesian game (we shall define formally the transformed game later). In the current study, we shall show that, in a similar way, any hypergames can naturally and uniquely reformulated in terms of Bayesian games and propose the general procedure which we call Bayesian representation of hypergames. In the transformation process, first each subjective game in a hypergame is extended in an unified way, and then, based on the extended subjective games for all the agents, a Bayesian game is constructed. The second question is, then, would equilibrium concepts for each model lead us to any different implications? To examine the problem, we investigate relations of equilibrium concepts for hypergames and Bayesian games. As a result, we shall argue that some equilibrium concepts for hypergames lead us to the same implications as equilibria for Bayesian games. The same thing can be said about hierarchical hypergames. In the generalization of Bayesian games, Mertens and Zamir (1985) claim that belief about the other’s belief, belief about the other’s belief about the other’s belief, and so on, can be captured in an “universal type set.” We propose the procedure of Bayesian representation of hierarchical hypergames based on their claim2 . After all, it seems that Bayesian games are general enough in the sense that any hypergames can be captured in terms of Bayesian games. But we point out that some equilibrium concept cannot be translated into any existing concept for Bayesian games and discuss carefully based on our analysis how each model can be differentiated and should be used according to the analyzer’s purpose. Since we need different formulations for simple hypergames and hierarchical hypergames, we first study the former in 4.2, and then the latter in 4.3.

4.2

Simple Hypergames and Bayesian Games

We introduced the formal definition of simple hypergames in 2.5.1. We use the same concepts as well as notations here. 2

For a brief guide of the claim of Mertens and Zamir (1985), see Myerson (1991, Sec. 2.9).

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4.2.1

Equilibria in Simple Hypergames

We first define two equilibrium concepts for simple hypergames which we use in the subsequent analysis. The first one is called hyper Nash equilibrium (Kijima (1996)). Definition 4.1 (Hyper Nash equilibrium): Let H = (I, (Gi )i∈I ) be a simple hypergame. a∗ = (a∗i , a∗−i ) ∈ ×i∈I Aii is a hyper Nash equilibrium of H iff ∀

i ∈ I, a∗i ∈ Ni (Gi ).

Let us denote the set of hyper Nash equilibria of H by HN (H). In a hyper Nash equilibrium, every agent chooses a Nash action. It can be interpreted as follows: if we assume every agent adopts Nash action as the decision criteria3 , that is, always chooses some Nash action, an outcome that obtains is necessarily a hyper Nash equilibrium (as long as every agent has at least one Nash action). Therefore the set of hyper Nash equilibria provides us with all the candidates of outcomes likely to happen under the assumption. In fact, by definition, HN (H) = ×i∈I N (Gi ). Next, the second equilibrium concept we use is best response equilibrium. Definition 4.2 (Best response equilibrium): Let H = (I, (Gi )i∈I ) be a hypergame. a∗ = (a∗i , a∗−i ) ∈ ×i∈I Aii is a best response equilibrium of H iff ∀

i ∈ I,∀ ai ∈ Aii , uii (a∗ ) ≥ uii (ai , a∗−i ).

Let us denote the set of best response equilibria of H by BE(H). A best response equilibrium is such an outcome in which each agent chooses a best response to the choices of the others (in each subjective game). Although the definition apparently looks like Nash equilibrium for normal form games, the implication is largely different. Best response equilibrium does not assure that, from a particular agent’s point of view, the other’s choices are also their best responses. Therefore, even if an agent takes a best response, she might consider that she should change her choice as some other agent who she thinks does not take a best response might change the choice. The notion of best response equilibrium refers to nothing more than the fact that every agent chooses a best response4 . Example: Consider a two-agent simple hypergame of 1 and 2, namely H = (I, Gi )i∈I with I = {1, 2}. Table 4.3 illustrates 1’s subjective game, G1 , where A11 = {a, b, c, d}, A12 = {p, q, r} and each entry expresses utilities for the both which reflect the utility functions perceived by 1, i.e., u11 and u12 . Similarly, Table 4.4 represents 2’s subjective game, G2 . In the hypergame, the both agents perceive the set of agents correctly but misperceive each other’s action set and utility function. Nash equilibria in each subjective game are given as N (G1 ) = {(a, p), (b, q)} while N (G2 ) = {(b, q)}. Hence HN (H) = {(a, q), (b, q)}. On the other hand, BE(H) = {(b, q), (c, r)}. 1\2 a b c d

p 3, 3 0, 0 0, 0 1, 1

q 0, 0 2, 2 0, 0 0, 0

r 0, 0 0, 0 1, −1 0, 0

Table 4.3: 1’s subjective game, G1 3

As we have seen in 2.2, from a game theoretical point of view, this assumption is not perfectly convincing. Best response equilibrium in simple hypergames is mathematically equivalent to the concept of group stability based on rationality in first-level hypergames provided in Wang et al. (1989). 4

37

1\2 a b c

p 3, 3 0, 0 5, 0

q 0, 0 2, 2 0, 0

r −5, 0 0, 0 −1, 1

s 0, 5 1, 0 0, 0

Table 4.4: 2’s subjective game, G2

4.2.2

Bayesian Representation of Simple Hypergames

We propose a general way to transform simple hypergames into Bayesian games that we call Bayesian representation of simple hypergames. We also give an example in the end. Harsanyi (1967) claims that any kinds of uncertainties about a game as well as perceptual differences among agents can be modeled in an unified way, which goes on as follows5 : • (Agents) Whether an agent is participating in the game can be converted into what the agent’s action set is, by allowing her only one action, “non-participation” (NP), when she is supposed to be out of the game. • (Actions) Whether a particular action is feasible for an agent can in turn be converted into what the agent’s utility function is, by saying that she will get some very low utility whenever she takes the action that is supposed to be infeasible. • (Utility functions) This way, any uncertainty or perceptual differences about agents as well as actions can be reduced to those about utility functions, if any. Then by regarding each possible utility function of each agent as a type of the agent, the game can be modeled as a Bayesian game. Let us call the above argument Harsanyi’s claim. When we apply it to situations represented as simple hypergames, it is interpreted as follows. For example, suppose first that agent i thinks that another agent j does not participate in the game, though j actually is in the game. Then the claim argues that i’s exclusion of j is game-theoretically equivalent to saying that i includes j in the set of the agents and allow j to use only one action, “nonparticipation.” This way allows every agent to see the common set of agents, which coincides with the set of all the agents actually involved in the hypergame6 . Next, suppose agent i thinks that another agent j’s particular action, aj , is not feasible for j, while it is actually included in j’s action set. Then Harsanyi’s claim argues that this is equivalent to saying that i considers that aj is surely in j’s action but gives j very low utility whenever j uses it. Consequently every agent sees in turn the same action set of a particular agent, which is the union of the agent’s action set originally conceived by each agent. As a result, perceptual differences in agents as well as actions are resolved, and those only in utility functions remain in our hand. Generally, according to Harsanyi’s claim, in any simple hypergames, each subjective game can be “extended” as the following. Definition 4.3 (Extended subjective games): Let H = (I, (Gi )i∈I ) be a simple hy¯ i = (I¯i , A¯i , u pergame. For any i ∈ I, a normal form game G ¯i ) is called agent i’s extended subjective game (induced from H) iff it satisfies all of the following conditions: • I¯i = I. ∪ ∪ • A¯i = ×j∈I A¯ij , where ∀ j ∈ I, A¯ij = k∈I Akj if j ∈ I k for any k ∈ I, A¯ij = k∈I Akj ∪{N P } otherwise. 5

For a brief guide of Harsanyi (1967), see e.g. Myerson (1991). One might think this way results in the set of agents becoming tremendously enormous because it requires one to include anyone. As is typical in the literature of Bayesian games, we ignore any agent who is regarded as a participant of the game by nobody and actually not in the game. Note also that we have assumed that the agent set in an agent’s subjective game never includes anybody who is actually not involved in the game. 6

38

¯ij : A¯i → ℜ. For any j ∈ I and a = (aj , a−j ) ∈ A¯i , u ¯ij (a) is defined • u ¯i = (¯ uij )j∈I , where u as follows, where c is a real constant bigger than −∞: (i) if I i = I,  i i  uj (a) if a ∈ A u ¯ij (a) = −∞ if aj ∈ / Aij   c otherwise (ii) if I i ̸= I,   uij ((al )l∈I i ) if j ∈ I i ∧ ak = N P for any k ∈ I \ I i ∧ (al )l∈I i ∈ Ai    −∞ if (j ∈ I i ∧ ak = N P for any k ∈ I \ I i ∧ aj ∈ / Aij ) u ¯ij (a) =  ∨(j ∈ / I i ∧ aj ̸= N P )    c otherwise ¯ = (I, (G ¯ i )i∈I ) is called the extended simple hypergame (induced Then the hypergame H i ¯ is agent i’s extended subjective game induced from H. from H) when for any i ∈ I, G ¯ and Gi is the original Conversely, we may say that H is the original simple hypergame of H i ¯ subjective game of G . Although the definition may look complicated, the underlying idea is simple: we just follow Harsanyi’s claim. In an agent’s extended subjective game, the agent set includes all the agents actually involved in the situation. Then the action set for a particular agent is given as the union of the agent’s action set in each agent’s original subjective game (and NP (non-participation) if at least one agent thinks the agent is out of the game). Utility functions are determined based on the next three principles. First, any outcomes modeled in the original subjective game assign the same utilities to each agent in its extension as well. Second, when someone takes an action that is not modeled in the original subjective game, the agent always gets extremely low utility, −∞. Third, in such cases, the other agents are supposed to get some utility c. ¯ i is the extended subjective game of Since the extension is unique up to c ∈ ℜ, we say G i i ¯ ¯ ¯ ¯ agent i. Let us denote I and A by I and A, respectively, because by definition they are identical for all i ∈ I. (Recall that Harsanyi’s claim argues any perceptual differences about agents and actions can be resolved. ) Henceforth we assume that, for any i, j ∈ I and a ∈ Ai , uij (a) > −∞, that is, in an original subjective game, utility one can obtain is always bigger than −∞. Then the next lemma assures that Nash equilibria in each (original) subjective game are “preserved” in its extension, and vice versa. Lemma 4.1 (Nash equilibria of extended subjective games): Let H = (I, (Gi )i∈I ) be ¯ i = (I, ¯ A, ¯ u a simple hypergame and G ¯i ) be the extended subjective game of i ∈ I. Then we have, for any i ∈ I, { N (Gi ) if I i = I i ¯ N (G ) = ¯ j )j∈I i ∈ N (Gi ) ∧∀ k ∈ I \ I i , ak = N P } otherwise. {a ∈ A|(a Proof : (i : case of I i = I) ¯ i ) ⊇ N (Gi )) Suppose a∗ = (a∗ )j∈I i ∈ N (Gi ), which means, (proof of N (G j ∀

j ∈ I i ,∀ aj ∈ Aij , uij (a∗ ) ≥ uij (aj , a∗−j ).

¯ i , I¯ = I i and A¯j ⊇ Ai for any j ∈ I. ¯ Then, for any j ∈ I, ¯ In G j ∀

aj ∈ A¯j ∩ Aij , u ¯ij (a∗ )(= uij (a∗ )) ≥ u ¯ij (aj , a∗−j )(= uij (aj , a∗−j )) 39

and ∀

aj ∈ A¯j \ Aij , u ¯ij (a∗ )(= uij (a∗ )) ≥ u ¯ij (aj , a∗−j )(= −∞)

hold. Hence ∀

¯ ∀ aj ∈ A¯j , u ¯ij (aj , a∗−j ). j ∈ I, ¯ij (a∗ ) ≥ u

¯ i ). This is equivalent to a∗ ∈ N (G i i ¯ ¯ i ), which means, (proof of N (G ) ⊆ N (G )) Next, suppose a∗ = (a∗j )j∈I¯ ∈ N (G ∀

¯ ∀ aj ∈ A¯j , u j ∈ I, ¯ij (a∗ ) ≥ u ¯ij (aj , a∗−j ).

i ¯ a∗ ∈ ¯ij (a∗ ) = −∞, therefore Now suppose ∃ j ∈ I, j / Aj . Then u ∃

aj ∈ A¯j , u ¯ij (aj , a∗−j )(≥ c) > u ¯ij (a∗ ).

¯ a∗ ∈ Ai . Then Thus ∀ j ∈ I, j j ∀

j ∈ I i ,∀ aj ∈ Aij , uij (a∗ )(= u ¯ij (a∗ )) ≥ uij (aj , a∗−j )(= u ¯ij (aj , a∗−j )).

¯ i ) = N (Gi ). This is equivalent to a∗ ∈ N (Gi ). Hence we have N (G i (ii : case of I ̸= I) ¯ i ) ⊇ {a ∈ A|(a ¯ j )j∈I i ∈ N (Gi ) ∧∀ k ∈ I \ I i , ak = N P }) Suppose a∗ = (proof of N (G ∗ ¯ l )l∈I i ∈ N (Gi ) ∧∀ k ∈ I \ I i , ak = N P }. In G ¯ i , I¯ ⊃ I i and A¯j ⊇ Ai for (aj )j∈I¯ ∈ {a ∈ A|(a j any j ∈ I¯ ∩ I i . Then ∀

j ∈ I¯ ∩ I i ,∀ aj ∈ A¯j ∩ Aij , u ¯ij (a∗ )(= uij ((a∗l )l∈I i )) ≥ u ¯ij (aj , a∗−j )(= uij (aj , (a∗l )l∈I i \{j} ))

because (a∗l )l∈I i ∈ N (Gi ), and ∀

j ∈ I¯ ∩ I i ,∀ aj ∈ A¯j \ Aij , u ¯ij (a∗ )(= uij ((a∗l )l∈I i )) ≥ u ¯ij (aj , a∗−j )(= −∞),

and moreover, ∀

j ∈ I¯ \ I i ,∀ aj ∈ A¯j , u ¯ij (a∗ )(= c) > u ¯ij (aj , a∗−j )(= −∞),

¯ ∀ aj ∈ A¯j , u ¯ i ). where c ∈ ℜ. Hence ∀ j ∈ I, ¯ij (a∗ ) ≥ u ¯ij (aj , a∗−j ), and this means a∗ ∈ N (G ¯ i ) ⊆ {a ∈ A|(a ¯ j )j∈I i ∈ N (Gi ) ∧∀ k ∈ I \ I i , ak = N P }) Suppose a∗ = (proof of N (G ∗ i ¯ ), which means, (a ) ¯ ∈ N (G j j∈I ∀

¯ ∀ aj ∈ A¯j , u j ∈ I, ¯ij (a∗ ) ≥ u ¯ij (aj , a∗−j ).

¯ik (N P, a∗−k )(= ¯ik (a∗ ) = −∞, which is smaller than u Now suppose ∃ k ∈ I \I i , a∗k ̸= N P . Then u / Aij . Then u ¯ij (a∗ ) = −∞, c). Thus ∀ k ∈ I \ I i , a∗k = N P . Next suppose ∃ j ∈ I¯ ∩ I i , a∗j ∈ therefore ∃

aj ∈ A¯j , u ¯ij (aj , a∗−j )(≥ c) ≥ u ¯ij (a∗ ).

Thus ∀ j ∈ I¯ ∩ I i , a∗j ∈ Aij . Then ∀

j ∈ I i ,∀ aj ∈ Aij , uij ((a∗k )k∈I i )(= u ¯ij (a∗ )) ≥ uij (aj , (a∗l )l∈I i \{j} )(= u ¯ij (aj , a∗−j )).

¯ l )l∈I i ∈ N (Gi ) ∧∀ k ∈ This is equivalent to (a∗k )k∈I i ∈ N (Gi ). Consequently, a∗ ∈ {a ∈ A|(a i i i ¯ ¯ I \ I , ak = N P }. Hence we have N (G ) = {a ∈ A|(al )l∈I i ∈ N (G ) ∧∀ k ∈ I \ I i , ak = N P }. By (i) and (ii), the lemma holds. ¤ 40

The lemma says, when the agent set in an agent’s original subjective game includes all the agents involved, the set of Nash equilibria in it coincides with that in its extension. Otherwise an outcome is a Nash equilibrium in an agent’s extended subjective game if and only if the choices of those agents included in the agent set in her original subjective game constitute a Nash equilibrium of it and the others all choose NP. Then an extended simple hypergame can be transformed into a Bayesian game by regarding each possible view as a type. We call the transformed game, or the reformulation process itself, the Bayesian representation of the hypergame. Definition 4.4 (Bayesian representation of simple hypergames): Let H = (I, (Gi )i∈I ) ¯ i = (I, ¯ A, ¯ u be a simple hypergame and G ¯i ) be the extended subjective game of i ∈ I. A b Bayesian game G (H) = (I, A, T, p, u) is called the Bayesian representation of H iff each elements of Gb (H) satisfies the following conditions: ¯ • I in H and I in Gb (H) are identical (and equal to I). ¯ • A = A. • T = ×i∈I b Ti . For all i, j ∈ I b , Ti = {tji |j ∈ I b }. tji ∈ Ti is a type of agent i whose view ¯j . is associated with G • p = (pi )i∈I b , where pi (·|ti ) is agent i’s subjective prior, which is a joint probability distribution on T−i for each ti ∈ Ti such that for any j ∈ I b , pi (t−i |tji ) = 1 if t−i = (tjk )k∈I b \{i} , pi (t−i |tji ) = 0 otherwise. • u = (ui )i∈I b , where ui : A × T → ℜ such that for any a ∈ A, tji ∈ Ti and t−i ∈ T−i , ui (a, (tji , t−i )) = u ¯ji (a). Recall that, in Bayesian games, a type is characterized by both subjective prior and utility function. Intuitively, we define tji ∈ Ti as agent i’s type who believes it is common ¯ j . Hence tj knowledge that the game they play is agent j’s extended subjective game, G i ¯j assigns probability 1 to a combination of types of the others each of which also perceives G while assigning probability 0 to any other combinations, and has the same utility function as ¯ j , i.e. u in G ¯ji . The subjective priors reflect the basic assumption of simple hypergames that every agent believes each own subjective game is common knowledge. Bayesian games can be formulated with objective priors instead of subjective priors. Since, ¯ i , we particularly say ti is in the original simple hypergame, agent i considers the game is G i agent i’s actual type for any i ∈ I. Based on the idea, we define objective priors of Bayesian representations of simple hypergames as follows: Definition 4.5 (Objective priors): Let Gb (H) = (I, A, T, p, u) be the Bayesian representation of a simple hypergame H. Then po is called the objective prior of Gb (H) iff for any t = (ti , t−i ) ∈ T , { 1 if ∀ i ∈ I, ti = tii o p (t) = 0 otherwise An objective prior reflects each agent’s actual view in the original hypergame. It assigns probability 1 to a combination of types each of which is the actual type of each agent, while probability 0 to any other combinations. We may also write the Bayesian representation of a hypergame H as Gb (H) = (I, A, T, po , u). It is easy to see that Bayesian representations of any simple hypergames do not allow the agents to have consistent subjective priors. Example: Reconsider the two-agent simple hypergame of the example in 4.2.2. According to Definition 4.3, each agent’s subjective game in the hypergame is extended as shown in ¯ 1 and G ¯ 2 denote the extended subjective games for Table 4.5 and 4.6, respectively. Let G 41

¯ be the extended simple hypergame. They now have the common action sets. each and H ¯ 1 , since agent 2’s action s is not modeled in With respect to utility values, for example, in G 1 its original, G , whenever 2 takes it, 2 gets utility −∞ while 1 gets utility c. It is easy to see that Lemma 4.1 holds, that is, each extended subjective game has the same set of Nash equilibria as its original. 1\2 a b c d

p 3, 3 0, 0 0, 0 1, 1

q 0, 0 2, 2 0, 0 0, 0

r 0, 0 0, 0 1, −1 0, 0

c, c, c, c,

s −∞ −∞ −∞ −∞

¯1 Table 4.5: 1’s extended subjective game, G 1\2 a b c d

p 3, 3 0, 0 5, 0 −∞, c

q 0, 0 2, 2 0, 0 −∞, c

r −5, 0 0, 0 −1, 1 −∞, c

s 0, 5 1, 0 0, 0 −∞, c

¯2 Table 4.6: 2’s extended subjective game, G Then its Bayesian representation Gb (H) = (I, A, T, p, u) is formulated as follows: • I = {1, 2}. • A = A1 × A2 , where A1 = {a, b, c, d} and A2 = {p, q, r, s}. • T = T1 × T2 , where T1 = {t11 , t21 } and T2 = {t12 , t22 }. • p = (p1 , p2 ), where for any i, j ∈ I, pi (t−i |tji ) = 1 if t−i = (tj−i ), pi (t−i |tji ) = 0 otherwise. • u = (u1 , u2 ), where for each i ∈ I, ui : A × T → ℜ such that for any a ∈ A, tji ∈ Ti and t−i ∈ T−i , ui (a, (tji , t−i )) = u ¯ji (a). ¯ j is We interpret each type tji (with i, j ∈ {1, 2}) as a type of agent i who believes G 1 1 common knowledge. Therefore, for example, t1 assigns probability 1 to t2 while probability 0 to t22 , and has the same utility function as u ¯11 . In this case, the actual types of each agent 1 2 o are t1 and t2 . Thus the objective prior p is defined as, for any t = (t1 , t2 ) ∈ T , { 1 if t1 = t11 and t2 = t22 o p (t) = 0 otherwise. Next, we show that, as with typical Bayesian games, a Bayesian representation can be illustrated as a game tree. Reconsider the example used in 4.1, namely, a two-agent simple hypergame of agents 1 and 2 in which 1’s subjective game is prisoners’ dilemma (Table 4.1) while that of 2 is chicken game (Table 4.2). Since they both perceive correctly the set of agents as well as actions for each agent, the extended hypergame induced it is same as the original.

42

Figure 4.1: The Bayesian representation The Bayesian representation of the hypergame can be described as Figure 4.1 shows. Nature moves first and determines types of the agents. For example, 21 means that agent 1’s type is t21 , that is, agent 1 whose view is associated with chicken game, while that of agent 2 is t12 , that is, agent 2 whose view is associated with prisoners’ dilemma. The objective prior describes probabilities of nature’s move. In this case, it assigns probability 1 to 12 while probability 0 to any other combinations of types. Following the nature’s move, each agent knows her own type but does not know the opponent’s type. The fact is expressed with the information sets described in the game tree: ω11 and ω12 are agent 1’s information sets while ω21 and ω22 are for agent 2. For example, if agent 1’s type is t11 , 1 faces the information set ω11 which contains two decision nodes. This means, as usual, that there agent 1 cannot tell at which node out of the two 1 actually is. But 1 has a belief about it which is represented by the subjective prior. In this case, t11 assigns probability 1 to agent 2’s type being t12 , while probability 0 to t22 . Likewise, in any other information set, the agent who faces it has a belief about the opponent’s type. Finally, utilities for each agent are determined by actions and types of the agents: the upper value is for agent 1 and the lower is for agent 2.

4.2.3

Relationships of Equilibrium Concepts

Next we investigate relationship between simple hypergames and those Bayesian representations, particularly by examining relations of their equilibria concepts. Our first result describes a deep relation between hyper Nash equilibrium and Bayesian Nash equilibrium7 . Recall that by introducing strategies that are functions which associate types of each agent with her actions, i.e. si : Ti → Ai for each i ∈ I, we can conduct equilibrium analysis of Bayesian games. Proposition 4.2 (Equilibria of simple hypergames and their Bayesian representations (with subjective priors)): Let H = (I, (Gi )i∈I ) be a simple hypergame and Gb (H) = (I, A, T, p, u) be its Bayesian representation. Then a∗ = (a∗i , a∗−i ) ∈ HN (H) iff there exists s∗ = (s∗i , s∗−i ) ∈ BN (Gb (H)) such that ∀ i ∈ I, s∗i (tii ) = a∗i . 7 Recall that in order to calculate Bayesian Nash equilibrium, utilities need to be cardinal. But we note that, in a Bayesian representation of a hyeprgame, we only use probability 0 or 1, therefore it would be no problem even if we regard utilities of agents as ordinal in the subsequent discussion including proofs in the propositions.

43

Proof : Suppose (s∗i , s∗−i ) ∈ BN (Gb (H)). This means, ∀ i ∈ I,∀ ti ∈ Ti ,∀ si ∈ Si , ∑ ∑ ui ((s∗i (ti ), s∗−i (t−i )), (ti , t−i ))pi (t−i |ti ) ≥ ui ((si (ti ), s∗−i (t−i )), (ti , t−i ))pi (t−i |ti ). t−i ∈T−i

t−i ∈T−i

In Gb (H), this is equivalent to: ∀ i, j ∈ I,∀ si ∈ Si , ui ((s∗i (tji ), (s∗k (tjk ))k∈I\{i} ), (tji , (tjk )k∈I\{i} )) ≥ ui ((si (tji ), (s∗k (tjk ))k∈I\{i} ), (tji , (tjk )k∈I\{i} )). ¯ i = (I, ¯ A, ¯ u Let G ¯i ) be the extended subjective game for each i ∈ I. Then the above statement is equivalent to: ∀

i, j ∈ I,∀ si ∈ Si , u ¯ji (s∗i (tji ), (s∗k (tjk ))k∈I\{i} ) ≥ u ¯ji ((si (tji ), (s∗k (tjk ))k∈I\{i} )).

¯ j ). This equivalent to, ∀ j ∈ I, (s∗i (tji ))i∈I ∈ N (G ∀ ∗ i ¯ Hence, i ∈ I, ai ∈ Ni (G ) iff there exists s∗ = (s∗i , s∗−i ) ∈ BN (Gb (H)) such that ∀ i ∈ ¯ i ) = Ni (Gi ) for any i ∈ I (due to Lemma 4.1), we have I, s∗i (tii ) = a∗i . Since Ni (G ∀

¯ i) i ∈ I, a∗i ∈ Ni (G

⇔∀ i ∈ I, a∗i = Ni (Gi ) ⇔ (a∗i )i∈I ∈ HN (H). Hence we have the proposition. ¤ The proposition refers to the relation between hyper Nash equilibrium and Bayesian Nash equilibrium. Precisely, it claims that if an outcome is a hyper Nash equilibrium in a simple hypergame, then its Bayesian representation has some Bayesian Nash equilibrium in which the actual type of each agent chooses the same action as in the hyper Nash equilibrium, and conversely, if a Bayesian representation of a hypergame has a Bayesian Nash equilibrium, then the actions chosen by the actual type of each agent in it must be a hyper Nash equilibrium in the original hypergame. Next let us consider Bayesian representations with objective priors (instead of subjective priors). Then we have the next proposition. Proposition 4.3 (Equilibria of extended simple hypergames and their Bayesian representations (with objective priors)): Let H = (I, (Gi )i∈I ) be a simple hypergame, ¯ be the extended simple hypergame induced from it, and Gb (H) = (I, A, T, po , u) be its H ¯ iff there exists s∗ = (s∗ , s∗ ) ∈ Bayesian representation. Then a∗ = (a∗i , a∗−i ) ∈ BE(H) i −i b ∀ ∗ i ∗ N (G (H)) such that i ∈ I, si (ti ) = ai . Proof : Suppose (s∗i , s∗−i ) ∈ N (Gb (H)). This means, ∀ i ∈ I,∀ si ∈ Si , ∑ ∑ ui ((s∗i (ti ), s∗−i (t−i )), t)po (t) ≥ ui ((si (ti ), s∗−i (t−i )), t)po (t). t∈T

t∈T

In Gb (H), this is equivalent to the fact that ∀ i ∈ I,∀ si ∈ Si , ui ((s∗i (tii ), (s∗j (tjj ))j∈I\{i} ), (tii , (tjj )j∈I\{i} )) ≥ ui ((si (tii ), (s∗j (tjj ))j∈I\{i} ), (tii , (tjj )j∈I\{i} )) ¯ i = (I, ¯ A, ¯ u Let G ¯i ) be the extended subjective game of each i ∈ I induced from H. Then the statement above is equivalent to: ∀

i ∈ I,∀ si ∈ Si , u ¯ii (s∗i (tii ), (s∗j (tjj ))j∈I\{i} ) ≥ u ¯ii (si (tii ), (s∗j (tjj ))j∈I\{i} ). 44

Thus, ∀ i ∈ I, ∀ ai ∈ A¯i , u ¯ii (a∗i , a∗−i ) ≥ u ¯ii (ai , a∗−i ) iff there exists s∗ = (s∗i , s∗−i ) ∈ N (Gb (H)) ∀ ∗ i ∗ such that i ∈ I, si (ti ) = ai . The former of the statement is by definition equivalent to, ¯ Hence we have the proposition. ¤ (a∗i , a∗−i ) ∈ BE(H). The proposition in turn refers to the relation between best response equilibrium of extended simple hypergames and Nash equilibrium of Bayesian games. Precisely, it says that an outcome is a best response equilibrium in an extended simple hypergame if and only if the Bayesian representation (with objective prior) has some Nash equilibrium in which the actual type of each agent chooses the same action as in the outcome. Since Proposition 4.3 refers to extended simple hypergames, we then try to specify how original hypergames and those Bayesian representations relate to each other. The following lemma would be useful for that purpose. Lemma 4.4 (Best response equilibria of simple hypergames): Let H = (I, (Gi )i∈I ) ¯ be the extended simple with Gi = (I i , Ai , ui ) for each i ∈ I be a simple hypergame and H ¯ Particularly the equality holds hypergame induced from it. Then we have BE(H) ⊆ BE(H). j ∀ i ∀ i if (i) i ∈ I, I = I, and (ii) i, j ∈ I, Aj ⊆ Aj . ¯ i = (I, ¯ A, ¯ u Proof : Let G ¯i ) be the extended subjective game of i ∈ I induced from H. Suppose ∗ ∗ (ai , a−i ) ∈ BE(H), which means, ∀

i ∈ I,∀ ai ∈ Aii , uii (a∗ ) ≥ uii (ai , a∗−i ).



i ∈ I,∀ ai ∈ A¯i , u ¯ii (a∗ ) ≥ u ¯ii (ai , a∗−i )

Then

because u ¯ii (a∗ ) = uii (a∗ ), and u ¯ii (ai , a∗−i ) = uii (ai , a∗−i ) if ai ∈ Aii , u ¯ii (ai , a∗−i ) = −∞ otherwise. ¯ Hence we have BE(H) ⊆ BE(H). ¯ This is equivalent to a∗ ∈ BE(H). ∀ i ∀ Next, let us assume H satisfies (i) i ∈ I, I = I, and (ii) i, j ∈ I, Ajj ⊆ Aij . Suppose ¯ which means, a∗ = (a∗i , a∗−i ) ∈ BE(H), ∀

i ∈ I,∀ ai ∈ A¯i , u ¯ii (a∗ ) ≥ u ¯ii (ai , a∗−i ).

Since ∀ i ∈ I,∀ ai ∈ A¯ii \ Aii ,∀ a−i ∈ A−i , u ¯ii (ai , a−i ) = −∞, we have ∀ i ∈ I, a∗i ∈ Aii . By (ii), ∀ ∗ i then i, j ∈ I, aj ∈ Aj . Thus ∀

i ∈ I,∀ ai ∈ Aii , uii (a∗ )(= u ¯ii (a∗ )) ≥ uii (ai , a∗−i )(= u ¯ii (ai , a∗−i )).

¯ This means a∗ ∈ BE(H). Under the conditions (i) and (ii), we have both BE(H) ⊆ BE(H) ¯ ¯ and BE(H) ⊆ BE(H), hence BE(H) = BE(H). ¤ The lemma claims that if an outcome is a best response equilibrium in a simple hypergame, then it is so in its extension as well. Moreover it refers to a sufficient condition under which the converse is also true: when, in the hypergame, every agent perceives the agent set correctly and at least all the actions available to the others, the converse is true. Consequently, we have the next proposition. Proposition 4.5 (Equilibria of simple hypergames and Bayesian representations (with objective priors)): Let H be a simple hypergame and Gb (H) = (I, A, T, po , u) be its Bayesian representation. If a∗ = (a∗i , a∗−i ) ∈ BE(H), then there exists s∗ = (s∗i , s∗−i ) ∈ N (Gb (H)) such that ∀ i ∈ I, s∗i (tii ) = a∗i . Particularly, if H satisfies the sufficient condition of Lemma 4.4, the converse also holds. Proof : The proposition is straightforward from Proposition 4.3 and Lemma 4.4. ¤ 45

Proposition 4.5 says that if an outcome is a best response equilibrium in a hypergame, then the Bayesian representation (with objective prior) has some Nash equilibrium in which the actual type of each agent chooses the same action as in the best response equilibrium. Particularly if the hypergame satisfies the sufficient condition of Lemma 4.4, the converse is also true. Example: By using the example in 4.2.2, let us illustrate the theoretical results presented above. First, consider Proposition 4.2 that refers to the relation between hyper Nash equilibrium and Bayesian Nash equilibrium. Recall, in our example, HN (H) = {(a, q), (b, q)}. On the other hand, Gb (H) has two Bayesian Nash equilibria, i.e., ((s1 (t11 ), s1 (t21 )), (s2 (t12 ), s2 (t22 ))) = ((a, b), (p, q)) and ((b, b), (q, q)). Since the actual types are t11 and t22 , we can see the proposition certainly holds. Next, Proposition 4.3 describes the relation between best response equilibrium and Nash equilibrium of Bayesian games. With the objective prior above, Gb (H) has 48 Nash equilibria, i.e., ((s1 (t11 ), s1 (t21 )), (s2 (t12 ), s2 (t22 ))) = ((b, x1 ), (x2 , q)), ((c, x1 ), (x2 , r)) and ((d, x1 ), (x2 , s)), ¯ = {(b, q), (c, r), (d, s)}, the proposition where xi can be any of agent i’s action. Since BE(H) holds here. Furthermore, given that BE(H) = {(b, q), (c, r)}, Proposition 4.5 also holds. In this case, H does not satisfy the sufficient condition of Lemma 4.4, and BE(H) is a proper ¯ subset of BE(H).

4.2.4

Discussion

Our interpretations of the results presented in 4.2.3 go as follows. If we are interested in hyper Nash equilibrium of hypergames, we would say that investigating Bayesian Nash equilibrium of those Bayesian representations leads to same implications. It is because, as Proposition 4.2 claims, an action profile that constitutes a hyper Nash equilibrium is always that taken by the actual types of each agent in some Bayesian Nash equilibrium, and vice versa. Likewise, if one wants to know best response equilibrium in a hypergame, Nash equilibrium of its Bayesian representation leads to the same implication as suggested by Proposition 4.5. Eventually, we conclude that any hypergames can be analyzed in terms of Bayesian games with those existing equilibria as long as our interest is in the two equilibrium concepts for hypergames8 . On the other hand, when we consider real interactive situations, Harsanyi’s claim often seems to be hard to accept, and in fact it still remains such controversial issues epistemic game theory focuses on9 . And more importantly to us, it appears incompatible with the idea of hypergames. But our result shows that we can ignore the problem as long as we analyze the two equilibrium concepts for hypergames. That is, even when it is not natural to accept Harsanyi’s claim, once we conduct the procedure of Bayesian representation as if we accept it, the analysis of the transformed game leads us to the same insight. After all, one might think Bayesian games are general enough, and thus we do not need any other models like hypergames for studying games with incomplete information. But we here would like to emphasize that there are at least two points about uniqueness of hypergame analyses. First, since hypergames are much simpler than Bayesian games, it would be a good choice to use the former in order to analyze such a situation that can be captured in terms of hypergames. In such cases, the results shown above assure that we can use the equilibrium concepts for simple hypergames when our interest is in equilibrium analysis of Bayesian 8

The converse obviously does not hold true: there are Bayesian games that cannot be transformed into and analyzed in terms of hypergames. Furthermore, even for a Bayesian game that is transformed from a hypergame, there might exist other hypergames that lead to the same Bayesian game as a result of those Bayesian representations. That is, although the transformation from a hypergame to a Bayesian game is unique, the converse may not. 9 See e.g. Binmore (2009) and Gilboa (2010).

46

games. Furthermore, recall that when we calculate equilibria for Bayesian games, we need to specify actions for all the types of each agent. On the other hand, equilibria analysis in hypergames requires us much less tasks. Second, as discussed above, we can use Bayesian games and the existing equilibrium concepts as long as we focus on hyper Nash equilibrium or best response equilibrium in simple hypergames, but this may not be the case when we want to analyze some other equilibrium concept. For example, stable hyper Nash equilibrium (Sasaki and Kijima, 2008), defined as an outcome that is a Nash equilibrium in every agent’s subjective game, cannot be captured by existing concepts of Bayesian games. In a simple hypergame, the set of stable hyper Nash equilibria is by definition always subset of hyper Nash equilibria, and similarly that of best response equilibria. Hence, if we are interested in its implications, hypergames can provide us with unique insights. For example, in the simple hypergame we used as an example in 4.2.1, Table 4.3 and 4.4, there are two hyper Nash equilibria, (a, q) and (b, q). But out of the two, only (b, q) is the stable hyper Nash equilibrium. It is called a stable hyper Nash equilibrium because the both agents see the outcome as a Nash equilibrium, thus they do not have incentives for deviation from it. On the other hand, with respect to another hyper Nash equilibrium, (a, q), it is not a Nash equilibrium in agent 2’s subjective game because a is not a best response against q for agent 1 from 2’s point of view. Then agent 2 may consider that agent 1 may deviate from the outcome, and if so, 2 also may change the choice: it is not stable in this sense. Our result shows that (a, q) and (b, q) are indifferent when we analyze Bayesian Nash equilibrium in the Bayesian representation, while hypergame analysis with the concept of stable hyper Nash equilibrium provides us with different implications about the two outcomes.

4.3

Hierarchical Hypergames and Bayesian Games

We introduced the formal definition of hierarchical hypergames in 2.5.2. We use the same concepts as well as notations here.

4.3.1

Equilibria in Hierarchical Hypergames

As with simple hypergames, we discuss two equilibrium concepts of hierarchical hypergames and study those relations to equilibria in Bayesian games. Among them one is subjective rationalizability we studied in Chapter 3. Let us denote the set of subjectively rationalizable actions of agent i in a hierarchical hypergame H h by Ri (H h ). The other one is best response equilibrium defined in the similar way of that for simple hypergames. Definition 4.6 (Best response equilibrium (for hierarchical hypergames)): Let H h = (I, (Gσ )σ∈Σ ) be a hierarchical hypergame. a∗ = (a∗i , a∗−i ) ∈ Ao is a best response equilibrium of H h iff ∀

i ∈ I,∀ ai ∈ Aii , uii (a∗ ) ≥ uii (ai , a∗−i ).

Let us denote the set of best response equilibria of H h by BE(H h ). Best response equilibrium for hierarchical hypergames has the same implications as that for simple hypergames: each agent chooses a best response to the choices of the others.

4.3.2

Bayesian Representation of Hierarchical Hypergames

We provide the procedure of transformations of hierarchical hypergames into Bayesian games in the similar manner as that of simple hypergames. First we define extension of subjective games in hierarchical hypergames. 47

Definition 4.7 (Extended subjective games): Let H h = (I, (Gσ )σ∈Σ ) be a hierarchical ¯ σ = (I¯σ , A¯σ , u hypergame. For any σ ∈ Σ, a normal form game G ¯σ ) is called viewpoint σ’s extended subjective game iff it satisfies all of the following conditions: • I¯σ = I. ∪ ∪ ′ ′ • A¯σ = ×i∈I A¯σi , where ∀ i ∈ I, A¯σi = σ′ ∈Σ Aσi if i ∈ I σ for any σ ∈ Σ, A¯σi = σ′ ∈Σ Aσi ∪ {N P } otherwise. • u ¯σ = (¯ uσi )i∈I , where u ¯σi : A¯σ → ℜ. For any i ∈ I and a = (ai , a−i ) ∈ A¯σ , u ¯σi (a) is defined as follows, where c is a real constant bigger than −∞: (i) if I σ = I,  σ σ  ui (a) if a ∈ A u ¯σi (a) = −∞ if ai ∈ / Aσi   c otherwise (ii) if I σ ̸= I,   uσi ((aj )j∈I σ ) if i ∈ I σ ∧ ak = N P for any k ∈ I \ I σ ∧ (aj )j∈I σ ∈ Aσ    −∞ if (i ∈ I σ ∧ ak = N P for any k ∈ I \ I σ ∧ ai ∈ / Aσi ) u ¯σi (a) =  ∨(i ∈ / I σ ∧ ai ̸= N P )    c otherwise ¯ h = (I, (G ¯ σ )σ∈Σ ) is called the extended hierarchical hypergame (induced from H h ). Then H The basic ideas of the extension is same as that defined for simple hypergames. Let us ¯ respectively. denote I¯σ and A¯σ by I¯ and A, We assume that, for any σ ∈ Σ, i ∈ I and a ∈ Aσ , uσi (a) > −∞. Then the next lemma assures that any subjectively rationalizable actions of any agents in a hierarchical hypergame are “preserved” in its extension, and vice versa. Lemma 4.6 (Subjective rationalizability in extended hierarchical hypergames): ¯ h be the extended hierarchical hypergame of a hierarchical hypergame H h = (I, (Gσ )σ∈Σ ). Let H ¯ h ) = Ri (H h ). Then we have, for any i ∈ I, Ri (H ¯ h ) ⊇ Ri (H h )) Suppose a∗ ∈ Ri (H h ), which means, there exists Proof : (i) (proof of Ri (H i ∗ ∗ (aσ )σ∈Σi such that aσ ∈ Aσσ1 for all σ ∈ Σi which satisfies ∀

σ ∈ Σi ,∀ aσ ∈ Aσσ1 , uσσ1 (a∗σ , a∗−σ ) ≥ uσσ1 (aσ , a∗−σ ),

¯ h , ∀ σ ∈ Σi , A¯σ ⊇ Aσσ . For any σ ∈ Σi , if aσ ∈ A¯σ \ Aσσ , where a∗−σ = (a∗jσ )j∈I σ \{σ1 } . In H 1 1 1 1 1 u ¯σσ1 (a∗σ , a∗−σ ) > u ¯σσ1 (aσ1 , a∗−σ ) = −∞, otherwise, i.e., if aσ1 ∈ Aσσ1 , u ¯σσ1 (a∗σ , a∗−σ ) ≥ u ¯σσ1 (aσ1 , a∗−σ ). Therefore, we have ∀

σ ∈ Σi ,∀ aσ ∈ A¯σ1 , u ¯σσ1 (a∗σ , a∗−σ ) ≥ u ¯σσ1 (aσ , a∗−σ ).

¯ h ). Hence Ri (H ¯ h ) ⊇ Ri (H h ). This means a∗i ∈ Ri (H h h ¯ ¯ h ), which means, there exists (ii) (proof of Ri (H ) ⊆ Ri (H )) Next, suppose a∗i ∈ Ri (H ∗ ∗ (aσ )σ∈Σi such that aσ ∈ A¯σ1 for all σ ∈ Σi which satisfies ∀

σ ∈ Σi ,∀ aσ ∈ A¯σ1 , u ¯σσ1 (a∗σ , a∗−σ ) ≥ u ¯σσ1 (aσ , a∗−σ ), 48

. Now suppose there exists σ ∈ Σi such that a∗σ ∈ / Aσσ1 . But if so, where a∗−σ = (a∗jσ )j∈I\{σ ¯ 1} ∀ σ ∗ σ for such σ, a−σ1 ∈ A¯−σ1 , u ¯σ1 (aσ , a−σ1 ) = −∞, which is smaller than u ¯σ1 (aσ1 , a−σ1 ) for any σ σ ¯ aσ1 ∈ Aσ1 (⊆ Aσ1 ), and this contradicts the requirement of subjective rationalizability, i.e., ∗ ∀a ∈ A σ ∗ ¯σ , u ¯σσ1 (aσ , a∗−σ ). Thus we have ∀ σ ∈ Σi , a∗σ ∈ Aσσ1 . Since we have σ 1 ¯σ1 (aσ , a−σ ) ≥ u σ ∀ σ ∗ σ assumed that for any σ ∈ Σ, ∀ j ∈ I σ , Ajσ j ⊆ Aj , for any σ ∈ Σ, k ∈ I \ {i}, akσ ∈ Ak . Therefore, for any σ ∈ Σi , ∀

aσ ∈ Aσσ1 , uσσ1 (a∗σ , a∗−σ )(= u ¯σσ1 (a∗σ , a∗−σ )) ≥ uσσ1 (aσ , a∗−σ )(= u ¯σσ1 (aσ , a∗−σ )).

¯ h ) ⊆ Ri (H h ). This means a∗i ∈ Ri (H h ). Hence Ri (H By (i) and (ii), we have the lemma. ¤ The lemma says that an action of an agent is subjectively rationalizable in a hierarchical hypergame if and only if it is subjectively rationalizable in its extension as well. Then we define Bayesian representations of hierarchical hypergames as follows10 : Definition 4.7 (Bayesian representation of hierarchical hypergames): Let H h = ¯ σ = (I, ¯ A, ¯ u (I, (Gσ )σ∈Σ ) be a hierarchical hypergame and G ¯σ ) be the extended subjective b h game of σ ∈ Σ. G (H ) = (I, A, T, p, u) is called the Bayesian representation of H h iff it satisfies all of the following conditions: ¯ • I in H h and I in Gb (H h ) are identical (and equal to I). ¯ • A = A. • T = ×i∈I Ti . For all i ∈ I and σ ∈ Σ, Ti = {tσi |σ ∈ Σ ∧ σ1 = i}, where tσi ∈ Ti is a type ¯σ. of agent i to whose view is associated with G • p = (pi )i∈I , where pi (·|ti ) is agent i’s subjective prior, which is a joint probability distribution on T−i for each ti ∈ Ti such that for any tσi ∈ Ti , pi (t−i |tσi ) = 1 if t−i = σ (tjσ j )j∈I\{i} , while pi (t−i |ti ) = 0 otherwise. • u = (ui )i∈I , where ui : A × T → ℜ such that for any a ∈ A, tσi ∈ Ti and t−i ∈ T−i , ui (a, (tσi , t−i )) = u ¯σi (¯ a), where a = a ¯. ¯ σ and Intuitively, we define tσi ∈ Ti as a type of agent i who believes that the game is G jσ σ ¯ any agent j(̸= i) sees G . Hence ti assigns probability 1 to a combination of types each of which is tjσ j for any j ̸= i while assigning probability 0 to any other combinations, and has ¯ σ , i.e. u the same utility function as that in G ¯σi . Note that those subjective priors directly reflect the hierarchy of perceptions in the extended hierarchical hypergame. Types of an agent are defined for those viewpoints σ whose lowest agent, σ1 , is the agent. Thus, for instance, ji tij / Ti . This is because, in the hierarchical hypergame, nobody thinks that agent i ∈ Ti but ti ∈ ji ¯ . We do not include such types that every agent thinks “impossible” in the type i sees G set of the Bayesian representation. As with Bayesian representations of simple hypergames, we call, for each i ∈ I, tii is the actual type of agent i. Note that the transformation into a Bayesian game is unique given a hierarchical hypergame. Objective priors of Bayesian representations of hierarchical hypergames are defined as follows: Definition 4.8 (Objective priors): Let Gb (H h ) = (I, A, T, p, u) be the Bayesian representation of a hierarchical hypergame H h . Then po is called the objective prior of Gb (H h ) iff for any t ∈ T , { 1 if ∀ i ∈ I, ti = tii po (t) = 0 otherwise 10

Our definition of type set in Bayesian representations of hierarchical hypergames is similar to the way of constructing universal belief space by Mertens and Zamir (1985). See also Myerson (1991, Sec. 2.9).

49

The objective prior assigns probability 1 to types each of which is the actual type of each agent, while assigns probability 0 to any other combinations of types. Again, the definition of objective priors tells us that Bayesian representations of any hierarchical hypergames do not allow the agents to have consistent subjective priors.

4.3.3

Relationships of Equilibrium Concepts

We define agent i’s strategy as si : Ti → Ai as usual, and let Si denote the set of agent i’s strategies. Then we shall show some propositions that describe the relations between hierarchical hypergames and those Bayesian representations. Our first result refers to the relation between equilibrium concepts of hierarchical hypergames and their Bayesian representations (with subjective priors). Proposition 4.7 (Equilibria of hierarchical hypergames and their Bayesian representations (with subjective priors)): Let H h = (I, (Gσ )σ∈Σ ) be a hierarchical hypergame and Gb (H h ) = (I, A, T, p, u) be its Bayesian representation. Then ∀ i ∈ I, a∗i ∈ Ri (H h ) iff there exists s∗ = (s∗i , s∗−i ) ∈ BN (Gb (H h )) such that ∀ i ∈ I, s∗i (tii ) = a∗i . ¯ σ = (I, ¯ A, ¯ u Proof : Let G ¯σ ) be the extended subjective game of σ ∈ Σ. Suppose (s∗i , s∗−i ) ∈ b h BN (G (H )), which means ∀ i ∈ I,∀ ti ∈ Ti ,∀ si ∈ Si , ∑ ∑ ui ((s∗i (ti ), s∗−i (t−i )), (ti , t−i ))pi (t−i |ti )) ≥ ui ((si (ti ), s∗−i (t−i )), (ti , t−i ))pi (t−i |ti ). t−i ∈T−i

t−i ∈T−i

In Gb (H h ), this is equivalent to: ∀ i ∈ I,∀ tσ ∈ Ti ,∀ si ∈ Si , jσ jσ σ σ ∗ jσ σ ui ((s∗i (tσi ), (s∗j (tjσ j ))j∈I\{i} ), (ti , (tj )j∈I\{i} )) ≥ ui ((si (ti ), (sj (tj ))j∈I\{i} ), (ti , (tj )j∈I\{i} )),

which is equivalent to: ∀ i ∈ I,∀ tσ ∈ Ti ,∀ si ∈ Si , u ¯σi (s∗i (tσi ), (s∗j (tjσ ¯σi ((si (tσi ), (s∗j (tjσ j ))j∈I\{i} ) ≥ u j ))j∈I\{i} )), which is equivalent to: ∀ i ∈ I,∀ σ ∈ Σ, ∀sσ1 ∈ Sσ1 , u ¯σσ1 (s∗σ1 (tσσ1 ), (s∗j (tjσ ¯σσ1 ((sσ1 (tσσ1 ), (s∗j (tjσ j ))j∈I\{σ1 } ) ≥ u j ))j∈I\{σ1 } )), which is equivalent to the fact that, for any i ∈ I, s∗σ1 (tσσ1 ) for each σ ∈ Σi constitutes a best ¯ h , the extension of H h , and hence ∀ i ∈ I, s∗ (ti ) ∈ Ri (H ¯ h ). response hierarchy in H i i ¯ h ) iff there exists (s∗ , s∗ ) ∈ BN (Gb (H h )) such that ∀ i ∈ After all, ∀ i ∈ I, a∗i = Ri (H i −i ∗ i ∗ h ¯ ) = Ri (H h ) for any i ∈ I (due to Lemma 4.6), we have ∀ i ∈ I, si (ti ) = ai . Since Ri (H I, a∗i = Ri (H h ). Hence we have the proposition. ¤ Proposition 4.7 states that subjective rationalizability of hierarchical hypergames and Bayesian Nash equilibrium of Bayesian games are in a sense “equivalent” to each other. Precisely, it claims that a hierarchical hypergame has such an outcome in which every agent chooses a subjectively rationalizable action for each if and only if its Bayesian representation has some Bayesian Nash equilibrium in which the actual type of each agent chooses the same action as the subjectively rationalizable action. On the other hand, if we consider Bayesian representations with objective priors, we have the next proposition. Proposition 4.8 (Equilibria of extended hierarchical hypergames and Bayesian representations (with objective priors)): Let H h = (I, (Gσ )σ∈Σ ) be a hierarchical hy¯ be its extension, and Gb (H h ) = (I, A, T, po , u) be the Bayesian representation pergame, H h ¯ h ) iff there exists s∗ = (s∗ , s∗ ) ∈ N (Gb (H h )) such that of H . Then a∗ = (a∗ , a∗ ) ∈ BE(H

∀i

∈ I, s∗i (tii ) = a∗i .

i

−i

i

50

−i

¯ σ = (I, ¯ A, ¯ u Proof : Let G ¯σ ) be the extended subjective game of σ ∈ Σ induced from H h . ∗ b ∗ Suppose (si , s−i ) ∈ N (G (H h )), which means ∀ i ∈ I,∀ si ∈ Si , ∑ ∑ ui ((s∗i (ti ), s∗−i (t−i )), t)po (t) ≥ ui ((si (ti ), s∗−i (t−i )), t)po (t). t∈T

t∈T

In Gb (H h ), this is equivalent to: ∀ i ∈ I,∀ si ∈ Si , ui ((s∗i (tii ), (s∗j (tjj ))j∈I\{i} ), (tii , (tjj )j∈I\{i} )) ≥ ui ((si (tii ), (s∗j (tjj ))j∈I\{i} ), (tii , (tjj )j∈I\{i} )), which is equivalent to: ∀ i ∈ I, ∀ si ∈ Si , u ¯ii (s∗i (tii ), (s∗j (tjj ))j∈I\{i} ) ≥ u ¯ii (si (tii ), (s∗j (tjj ))j∈I\{i} ). Thus ∀ i ∈ I, ∀ ai ∈ A¯i , u ¯ii (a∗i , a∗−i ) ≥ u ¯ii (ai , a∗−i ) iff there exists s∗ = (s∗i , s∗−i ) ∈ N (Gb (H h )) ∗ i ∀ ∗ such that i ∈ I, si (ti ) = ai . The former of the statement is by definition equivalent to ¯ h ). Hence we have the proposition. ¤ (a∗i , a∗−i ) ∈ BE(H Proposition 4.8 refers to the “equivalence” between best response equilibrium of extended hierarchical hypergames and Nash equilibrium of Bayesian games. It says that an extended hierarchical hypergame has a best response equilibrium if and only if its Bayesian representation (with objective prior) has some Nash equilibrium in which the actual type of each agent chooses the same action as the one in the best response equilibrium. In order to extend the result to (original) hierarchical hypergames, the following lemma would be useful. Lemma 4.9 (Best response equilibria of extended hierarchical hypergames): Let ¯ h be its extension. Then we have H h = (I, (Gσ )σ∈Σ ) be a hierarchical hypergame and H h h ∀ ¯ ). Particularly equality holds if (i) i ∈ I, I i = I, and (ii)∀ i, j ∈ I, Aj ⊆ BE(H ) ⊆ BE(H j

Aij . ¯ σ = (I, ¯ A, ¯ u Proof : Let G ¯σ ) be the extended subjective game of σ ∈ Σ induced from H h . ∗ ∗ Suppose (ai , a−i ) ∈ BE(H h ), which means, ∀

i ∈ I,∀ ai ∈ Aii , uii (a∗ ) ≥ uii (ai , a∗−i ).

Then we have ∀

i ∈ I,∀ ai ∈ A¯i , u ¯ii (a∗ ) ≥ u ¯ii (ai , a∗−i )

because u ¯ii (a∗ ) = uii (a∗ ), and u ¯ii (ai , a∗−i ) = uii (ai , a∗−i ) if ai ∈ Aii , while u ¯ii (ai , a∗−i ) = −∞ ∗ h h ¯ ). Hence we have BE(H ) ⊆ BE(H ¯ h ). otherwise. This is equivalent to a ∈ BE(H Next, let us assume H h satisfies (i) ∀ i ∈ I, I i = I, and (ii)∀ i, j ∈ I, Ajj ⊆ Aij . Suppose ¯ h ), which means, a∗ = (a∗i , a∗−i ) ∈ BE(H ∀

i ∈ I,∀ ai ∈ A¯i , u ¯ii (a∗ ) ≥ u ¯ii (ai , a∗−i ).

Since ∀ i ∈ I,∀ ai ∈ A¯ii \ Aii ,∀ a−i ∈ A−i , u ¯ii (ai , a−i ) = −∞, we have ∀ i ∈ I, a∗i ∈ Aii . By (ii), ∀ ∗ i then i, j ∈ I, aj ∈ Aj . Thus we have ∀

i ∈ I,∀ ai ∈ Aii , uii (a∗ )(= u ¯ii (a∗ )) ≥ uii (ai , a∗−i )(= u ¯ii (ai , a∗−i )).

This means a∗ ∈ BE(H h ). Under the conditions (i) and (ii), we have both BE(H h ) ⊆ ¯ h ) and BE(H ¯ h ) ⊆ BE(H h ), hence BE(H h ) = BE(H ¯ h ). ¤ BE(H The lemma means that if an outcome is a best response equilibrium in a hierarchical hypergame, then it is so in its extension as well. As with Lemma 4.4, it also specifies sufficient conditions for the converse to be true. 51

Consequently, we have the next proposition. Proposition 4.10 (Equilibria of hierarchical hypergames and Bayesian representations (with objective priors)): Let H h be a hierarchical hypergame and Gb (H h ) = (I, A, T, po , u) be its Bayesian representation. If a∗ = (a∗i , a∗−i ) ∈ BE(H h ), then there exists s∗ = (s∗i , s∗−i ) ∈ N (Gb (H h )) such that ∀ i ∈ I, s∗i (tii ) = a∗i . Particularly, if H h satisfies the sufficient condition of Lemma 4.10, the two statements are equivalent. Proof : The proposition is straightforward from Proposition 4.8 and Lemma 4.9. ¤ Proposition 4.10 says that if an outcome is a best response equilibrium in a hierarchical hypergame, then its Bayesian representation (with objective prior) has some Nash equilibrium in which the actual type of each agent chooses the same action as the one in the best response equilibrium. Particularly, if the hierarchical hypergame satisfies the sufficient conditions of Lemma 4.9, the converse is also true.

4.3.4

Discussion

We interpret the results in a similar manner to that of simple hypergames. Any hierarchical hypergames can be analyzed in terms of Bayesian games as long as our interest is in the two equilibrium concepts, subjective rationalizability and best response equilibrium. Even when it is not natural to accept Harsanyi’s calim or the idea of universal belief space by Mertens and Zamir, it is harmless to analyze Bayesian representations of hirerarchical hypergames as if we accept them. As with simple hypergames, it would be an advantage of choosing hierarchical hypergames that it is simpler than Bayesian games with universal belief space and thus easier to formulate interactive situations in question. Furthermore, if we consider such an equilibrium concept for hierarchical hypergames that captures the type of “stability” with which stable hyper Nash equilibrium deals in simple hypergames, it might provide us unique insights that cannot be captured by analyses with Bayesian games, though such concepts have not yet been proposed. In addition, we refer to another topic here: what brings about the difference between Proposition 4.2 and Proposition 4.7? In simple hypergames, hyper Nash equilibrium has the same implication as Bayesian Nash equilibrium, while so is subjective rationalizability in hierarchical hypergame. But since simple hypergames can be regarded as a special case of hierarchical hypergames, hierarchical hypergames where every agent has inside common knowledge, it seems that, in simple hypergames, it should not be hyper Nash equilibrium but be rationalizable actions of each agent that is related to Bayesian Nash equilibrium. The two propositions implies that, for a situation where every agent has ICK, whether we formulate it simple hypergames or hierarchical hypergames affect the implications of equilibrium analyses. This is because, in Bayesian representations of hierarchical hypergames, each viewpoint is modeled as each different type whether or not it has ICK. Suppose an agent i has individual common knowledge in a hierarchical hypergame. Then, for j ̸= i, the agent’s two types, tii and tiji i , are allowed to take different actions in spite of their views’ being identical. The same goes for any tiσ i with σ ∈ Σi . But this is impossible in Bayesian representations of simple hypergames because they assume only one type is associated with each view. Hence the difference comes from different settings of Bayesian representation of each model. In general, a type of an agent in Bayesian games is characterized by its subjective prior and utility function. Therefore, if some two types have same components for both of them, we may regard the two as one unified type rather than different types. If interested, one may redefine Bayesian representation of hierarchical hypergames (or possibly simple hypergames) with taking into account these.

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Chapter 5

Decision Making and Perceptual Views in Repeated Hypergames: A Game Theoretical Characterization of Systems Intelligence The chapter provides a new model called repeated hypergames that explicitly deals with situations in which agents interact repeatedly with each subjective view about the game1 . Each period of such iterative interactions is assumed to be expressed as a simple hypergame. An agent’s behavior depends on her subjective game and, at the same time, she may revise it according to an outcome of each period. The notion of behavioral domain is adopted to describe both behaviors and revisions of subjective games. Particularly the framework will be applied to a certain class of hypergames called systems of holding back, where misperceptions prevent achievements of Pareto-optimal stationary state. We examine how differences of behavioral domains affect in such situations. The idea is inspired by a recently developed systemic concept called systems intelligence. Keywords: hypergames, repeated hypergames, behavioral domain, systems intelligence, systems of holding back.

5.1

Introduction

The repeated hypergame model we propose in this chapter deals with situations in which agents interact with each other repeatedly and each period of the interactions is expressed as a simple hypergame. At each period, an agent makes a decision according to her current subjective game, and depending on the outcome of the period, she might revise the subjective view before going to the next period2 . In this chapter, by “hypergames” we mean simple hypergames. The concept of behavioral domain is adopted to describe her possible choices. It is given as a non-empty subset of her action set depending on the game structure. Thus it may contain two or more actions, and it is assumed that she never chooses those actions excluded from it. For instance, one can define an agent’s behavioral domain as her Nash actions. Then, it tells us that the agent never chooses non-Nash actions. Behavioral domain is also used as the criterion for an agent to revise her subjective game. That is, she is supposed to 1

The present chapter is based on Sasaki and Kijima (2010). In this respect, our repeated hypergames are essentially different from repeated games in the standard game theory, in which agents always see the same game from the beginning to the end (e.g. repeated prisoner’s dilemma). 2

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have perception about the other’s behavioral domain, and if the other agent takes an action outside of it, then she is assumed to revise her subjective game so that her “new view” can explain the unexpected choice. In this way, a repeated hypergame is played. Then, our interest is in how behavioral domains can affect “stationary states” of repeated hypergames. By a stationary state we mean such a state in which every agent is willing to change neither her choice nor perception about the game. For the purpose, we define and compare two types of agents. One type called a normal agent takes some Nash actions every time as is typically assumed in the hypergame literature3 . They might be called rational in this sense. On the other hand, the second type which we call a proactive agent, in addition to Nash actions, may also take what we define as trial actions. A proactive agent may choose an action that is not apparently rational in her current subjective game, but take it hoping that it might improve the situation in the long run4 . Behaviors of proactive agents are inspired and characterized by a recently developed systemic concept, systems intelligence (H¨am¨al¨ainen and Saarinen (2004)), which is introduced in the next section. As an application of our framework, we define the main targets of SI called system of holding back (SHB), which are problematic and ubiquitous social phenomena caused primarily by discrepancies of perceptions among agents, in terms of hypergames. Then we shall show that agents can get over SHB only if at least one of them is proactive. Hence our study would contribute researches of SI itself, which has not provided any formal foundations. There are several notable related works. In game theory, updates of belief in repeated games have been studied in some learning models such as Kalai and Lehrer (1993) and Fudenberg and Levine (1993, 1998). Beliefs in those models, however, mean probability distributions on the other’s choices, while, in our model, agents may update formulations of games per se. Keneko’s (1987) conventionally stable sets discuss similar situations with ours: agents, starting with restricted views about the game, interact many times and adjust their behaviors according to past outcomes. Agents in his model lack clear views about the game while, in hypergames, agents are supposed to have tentative but clear views. Inductive game theory (Kaneko and Matsui (1999); Kaneko and Kline (2008)) also have a similar motivations to ours but its focus is on how one’s subjective view is created rather than how such a view can be revised. Although a single-agent model, Gilboa and Schemeidler (1995, 2001) have proposed case-based decision theory, in which an agent makes a decision depending on her previous decisions and their consequences. In hypergame theory, Kijima’s (1996) I-PALM studies dynamic changes of subjective views of agents in terms of hypergames but it describes explicitly neither how they may be revised nor how a certain situation can change into another. Inohara (2000) studies exchanges of information among agents in hypergames and those effects on agents’ subjective views, while we deal with view revisions based on outcomes of games. There are some learning models proposed (e.g. Takahashi et al. (1999); and Putro et al. (2000)) but they fall into adaptive learning or evolutionary approaches. Although not a hypergame model, Feinberg (2007) studies how an agent may update her view about another agent’s action set. Following the introduction, 5.2 reviews original concepts on systems intelligence and systems of holding back. 5.3 provides the general framework of repeated hypergames, which are then analyzed and applied to systems of holding back in 5.4. Finally 5.5 discusses on several topics regarding the model and analysis.

3

As discussed in Chapter 2 and 3, this is a somewhat strong assumption in the standard game theory. Although these kinds of off-the-equilibrium plays have studied in the standard game theory, they are usually interpreted as “errors” (thus probabilities of taking such actions are assumed to be negligible, e.g. trembling hand perfect equilibrium by Selten (1975)). In contrast, we introduce trial actions as deliberate choices of what we call proactive agents. In this sense, our idea is rather close to “experimentations” in a learning model provided by Fudenberg and Levine (1998, Ch. 7). 4

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5.2

Systems Intelligence and Systems of Holding Back

Systems intelligence (SI), a systemic concept coined by H¨am¨al¨ainen and Saarinen (2004), is originally referred to as “an intelligent behavior in the context of complex systems involving interaction and feedback” (H¨am¨al¨ainen and Saarinen (2006)). It regards the systems perspective as essential while focusing on some distinctive features which have been downplayed in the course of systems thinking5 . Here we summarize its claims. SI’s emphasis is on actions rather than thinking. “What can intelligent choice mean when you cannot step outside and sort out the options and their systemic impacts?” (ibid.) – this is its key question, whereas systems thinking, they say, requires one to step outside the system and identify it. What SI suggests is not to describe fully a system in question but just to seek its systemic leverage. Its fundamental assumption is that people adjust to what they believe is the system thus the “true” system might hide something that they are not recognizing. Based on the view, it stands on optimism for changes that even a micro-behavioral change can work as a systemic leverage – a trigger that can lead to a drastic change in the system. It encourages people to accept such a big picture and admit the possibility that what they see as the system might not reflect the truth, then argues such kind of “humbleness” would make the micro-behavioral change possible. Therefore SI is not expertise but something that human beings possess inherently. That is, what is needed to be intelligent in the sense of SI is not to learn some new knowledge or methodology but just an awareness. The main targets of SI are paradoxical and ubiquitous social phenomena called systems of holding back (SHB), which are such human interactive systems that have got into “mutually aggregating spirals which lead people to hold back contributions they could make because others hold back contributions they could make” (H¨am¨al¨ainen and Saarinen (2007)). In SHB, people involved actually picture a same desire internally yet nobody behaves to achieve it. Consequently the “common desire” does not result, and instead they obtain a somewhat less desirable outcome. Let us see three example cases of SHB6 . They will be formulated as a hypergame in the subsequent analysis. Case 1 (an arms race): Suppose two countries, A and B, are in a complicated bilateral relation. They both have two options for military operation, namely arms reduction and military expansion, and aspire to a peaceful relation in which they both reduce their arms. However, A does not trust the opponent and believes that B wants to gain the military superiority in any cases. As long as B implements military expansion, A reluctantly does so as well. But actually B also sees the relation in a completely symmetric way, and this is why B expands its military. As a result, they both want peace yet an endless arms race happens. Case 2 (value co-creation in services): Let us consider SHB of a service provider (e.g. a restaurant, a cafe, etc.) and its customers. The provider wants to offer the best service while the customers want to receive it as well. In order to achieve such a good relationship, value co-creation process plays an important role (Spohrer and Maglio (2009)). That is, not only the provider should listen to the customers’ voices proactively but also the customers should give suggestions for betterment. However, the provider believes the customers are totally satisfied with the current service and there is no need to listen to their voices, and becomes too lazy to do so as long as the customers do not give any ideas, suggestions or complaints. Actually the customers believe, seeing the provider being lazy, it is not willing to improve the service and it is no use giving any suggestions to it. Again, they both want the best service yet value co-creation does not work out. 5 See e.g. Klir (1991) for general systems theory. For a comprehensive course of systems thinking, see Jackson (2003) for example. In particular, SI’s basic idea is highly relevant to systems dynamics (Senge (1990)), though SI differs from it in that SI does not require a full description of a system in question. 6 H¨ am¨ al¨ ainen and Saarinen (2004, 2006) give some more examples.

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Case 3 (a story of a seasoned couple): SHB can be seen here and there in our daily life. Imagine a matured couple, and both of them want to spend a romantic life together. Suppose, however, the husband believes that the wife is now totally different from what she used to be when they were young, and is no longer romantic. He holds back his romantic attitude and behavior as long as she looks so, but the same is true for her. Then, again, they both want the romantic life yet it does not come true. Likewise, a myriad of SHB can be noted in our society. Obviously one plausible reason for generating SHB would be differences in people’s subjective views about the situation. In the third example, the husband may be adjusting to his wife in his mind who is not romantic while she may be adjusting to him in her mind who is not romantic, and, as a result, he holds back, and vice versa. Then they fall into SHB. In general, one of the most serious problems of SHB is that people inside are not aware of the fact that they are in such a system: the husband may even think he is doing the best he can do. It makes it difficult for them to brake away from SHB without any “conscious efforts.” As mentioned above, SI suggests people be “humble” enough. Then, for example, the husband may accept the view that even a micro-behavioral change may change something and try to be romantic, expecting it can work as a systemic leverage. The change, if successful, may result in a drastic and sustainable betterment of the system i.e., the couple. In our repeated hypergame framework, we define two types of agents: normal agents and proactive agents. We intend they represent non-SI agents and SI agents, respectively.

5.3

Repeated Hypergame Model

We formulate the framework of repeated hypergames. Consider situations where agents interact with each other repeatedly and assume that the interaction in each period is described as a hypergame. We call the whole process of such iterative interactions a repeated hypergame. In each period of it, an agent’s choice depends on her current subjective game, and, according to the outcome, she may revise the subjective game itself. Those concepts introduced in 5.3.1-5.3.4 will be summarized with an example in 5.3.5.

5.3.1

Preparations for Modeling

Prior to introducing the repeated hypergame model, we provide some settings. The study focuses on two-agent (simple) hypergames in which the agents may misperceive only the other’s utility function while perceive correctly the set of agents as well as their actions. Assumption 5.1 (Hypergames in focus): We henceforth consider only two-agent hypergames played by agents p (she) and q (he). We assume that, in any hypergame H = ((p, q), (Gp .Gq )) with Gp = (I p , Ap , up ) and Gq = (I q , Aq , uq ), I p = I q = {p, q}, Apq = Aqq and Aqp = App . Accordingly, we simplify some notations. We write Ap = App and Aq = Aqq , and A = Ap × Aq . Let us denote each agent s subjective game simply as Gp = (I, A, up ) and q q G = (I, A, u ) with I = {p, q} and a hypergame as H = (Gp , Gq ). The following concept of base games is convenient for discussing “objective” descriptions of hypergames. Definition 5.1 (Base game): Let H = (Gp , Gq ) be a hypergame with Gp = (I, A, up ) and Gq = (I, A, uq ). A normal form game G = (I, A, u) is called the base game of H iff up = upp and uq = uqq . Let us denote the base game of a hypergame H by BGH .

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The base game of a hypergame is interpreted as an “imaginary” game that would have been the game if no agent had had any misperceptions in the situation. It is always given uniquely under Assumption 5.1. By comparing a hypergame and its base game, we can examine how misperceptions of agents can affect the situation. Note that the agents themselves cannot induce the base game: it is just an analytical tool for us. We adopt stable hyper Nash equilibrium (Sasaki and Kijima, 2008) to describe a “stationary state,” where every agent is willing to change neither her choice nor perception about the game. It is defined as such an outcome of a hypergame that is perceived as a Nash equilibrium by every agent. Definition 5.2 (Stable hyper Nash equilibrium): Let H = (Gp , Gq ) be a hypergame with Gp = (I, A, up ) and Gq = (I, A, uq ). Then a∗ ∈ A is called a stable hyper Nash equilibrium iff a∗ ∈ N (Gp ) and a∗ ∈ N (Gq ). We have the following lemma with respect to equilibria of a hypergame and its base game7 . Lemma 5.1 (Stable hyper Nash equilibria and Nash equilibria of base games): In a hypergame H, SHN (H) ⊆ N (BGH ). Proof : Let H = (Gp , Gq ) be a hypergame with Gp = (I, A, up ) and Gq = (I, A, uq ). Suppose a∗ = (a∗p , a∗q ) ∈ SHN (H). It means a∗ ∈ N (Gp ) and a∗ ∈ N (Gq ). This implies ∀

ap ∈ Ap , upp (a∗ ) ≥ upp (ap , a∗q )



aq ∈ Aq , uqq (a∗ ) ≥ uqq (a∗p , aq ).

and Since BGH = (I, A, (up , uq )) with up = upp and uq = uqq , it is equivalent to: ∀

ap ∈ Ap , up (a∗ ) ≥ up (ap , a∗q )



aq ∈ Aq , uq (a∗ ) ≥ uq (a∗p , aq ).

and This means a∗ ∈ N (BGH ). Hence we have the lemma. ¤ The lemma means that if an outcome in a hypergame is a stable hyper Nash equilibrium, then it is a Nash equilibrium in its base game. The implication is that any outcome that is not a Nash equilibrium in the base game cannot become a stationary state in the long run.

5.3.2

Rules for Behaviors and View Revisions

We here set the process of repeated hypergames, that is, how to act and how to revise subjective views. The notion of behavioral domain is adopted to determine the rules. Definition 5.3 (Behavioral domain): Let H = (Gp , Gq ) be a hypergame. ∆p (Gp ) ∈ 2Ap \{φ} and ∆q (Gp ) ∈ 2Aq \{φ} are called the behavioral domains of agent p and q perceived by agent p (i.e. in Gp ). Symmetrically we define ∆p (Gq ) and ∆q (Gq ). Behavioral domain of an agent is a non-empty subset of her actions. We define how an agent chooses an action and revises her view by using it. The next assumption specifies them. Assumption 5.2 (Behaviors and view revisions in repeated hypergames): Let H = (Gp , Gq ) be a hypergame at one period with Gp = (I, A, up ) and Gq = (I, A, uq ). Suppose a = (ap , aq ) ∈ A is obtained at the period. Then we assume the following two rules: 7

The lemma holds generally in n-agent hypergames in which all the agents perceive correctly the set of agents as well as the set of actions of every agent. See Sasaki and Kijima (2008).

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R1 ap ∈ ∆p (Gp ) and aq ∈ ∆q (Gq ). ˆ p ) iff aq ∈ ˆ p = (I, A, (upp , u R2 Agent p updates upq to u ˆpq such that aq ∈ ∆q (G / ∆q (Gp ), where G ˆpq )). q Agent q updates up in a symmetric way. R1 refers to choices of agents. Instead of providing a particular decision rule which always suggests one action the agent should choose (e.g. expected utility maximization), we specify actions the agent never chooses. R1 says that an agent always chooses some action in her behavioral domain (in her own subjective game). For example, if it is given as her Nash actions, then she takes an action from the set of her Nash actions. Note that R1 tells nothing about which one out of those she chooses if there are two or more Nash actions. R2 provides the necessary and sufficient condition about when an agent revises her subjective game as well as the condition which the update must satisfy. We assume that agent p expects that the opponent always chooses some action within his behavioral domain (perceived by herself). Therefore if she observes that his choice is outside of it, then she would feel a sort of “cognitive dissonance.” Then we assume, if and only if so, she revises his utility function in her subjective game so that the update can resolve the dissonance and the new view can explain the unexpected choice. Note that the way of such an update that satisfies R2 may not be unique.

5.3.3

Reachability of hypergames

Based on the assumption, we define reachability of hypergames. If a hypergame can change into another hypergame at one time, the latter is said to be directly reachable from the former. Definition 5.4 (Direct reachability of hypergames): Let H = (Gp , Gq ) be a hypergame ˆ = (G ˆ p, G ˆ q ) is directly reachable with Gp = (I, A, up ) and Gq = (I, A, uq ). A hypergame H ˆ iff there exists (ap , aq ) ∈ ∆p (Gp ) × ∆q (Gq ) such that aq ∈ from H (with H ̸= H) / ∆q (Gp ) or q ap ∈ / ∆p (G ), and both of the following two conditions hold: ˆ p ), where G ˆ p = (I, A, (upp , u 1. If aq ∈ / ∆q (Gp ), then aq ∈ ∆q (G ˆpq )). Likewise, if ap ∈ / q q q q q ˆ ˆ ∆p (G ), then ap ∈ ∆p (G ), where G = (I, A, (ˆ up , uq )). p p p ˆ = G . Likewise, if ap ∈ ∆p (Gq ), then G ˆ q = Gq . 2. If aq ∈ ∆q (G ), then G Let us denote the set of directly reachable hypergames from H by Φ1 (H). We interpret the set of directly reachable hypergames as the whole candidates of results ˆ ∈ Φ1 (H) means that a of one-time view revisions by agents from a hypergame. Thus, H ˆ hypergame H may change into another hypergame H at one time. Note that H itself is not included in Φ1 (H) By connecting directly reachable hypergames, reachability of hypergames can be defined. ˆ is reachable from a Definition 5.5 (Reachability of hypergames): A hypergame H ˆ iff we have a sequence (H1 , ..., Ht )(t ≥ 2) such that H1 = H, Ht = H ˆ hypergame H (̸= H) and for any integer k (1 ≤ k ≤ t − 1), Hk+1 ∈ Φ1 (Hk ). Let us denote the set of reachable hypergames from H by Φ(H). The set of reachable hypergames gives the whole candidates of future hypergames for a ˆ ∈ Φ(H) means that a hypergame H may become another hypergame specific hypergame. H ˆ H at some period in future.

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5.3.4

Two Types of Agents

We introduce two types of agents: normal agents and proactive agents. Intuitively, the former is like an agent typically assumed in the hypergame literature, and always takes some Nash action. On the other hand, by the latter type we represent the significance of systems intelligence. This type may choose non-Nash action. We first define normal agents. Definition 5.6 (Normal agents): Let Gp = (I, A, up ) be agent p’s subjective game in a hypergame. She is called a normal agent iff, { Np (Gp ) if N (Gp ) ̸= φ ∆p (Gp ) = Ap otherwise, and

{ Nq (Gp ) if N (Gp ) ̸= φ ∆q (Gp ) = Aq otherwise. The symmetric definition goes for agent q. A normal agent is such an agent whose perception of behavioral domains for both are given as Nash actions if any, otherwise the action set itself8 . That is, this type of agent always takes some Nash action, if any, and believes that the opponent also always chooses some Nash action. If the subjective game has no Nash equilibrium, the behavioral domain is instead given as the whole action set for each agent. Next, we define the other type, proactive agents. We characterize them by the following notion of trial actions. That is, we suppose a proactive agent takes Nash actions like normal agents but also may choose trial actions. Definition 5.7 (Trial actions): Let Gp = (I, A, up ) be agent p’s subjective game in a hypergame. Then a ¯p ∈ Ap is called her trial action iff N (Gp ) ̸= φ and a ¯p satisfies the following two conditions: 1. a ¯p ∈ / Np (Gp ), 2. There exists aq ∈ Aq such that upp (¯ ap , aq ) > upp (a∗ ) for any a∗ ∈ N (Gp ), and upp (¯ ap , aq ) ≥ p up (ap , aq ) for any ap ∈ Ap . The symmetric definition goes for trial actions of agent q. Let us denote the set of trial actions of agent p and q with each subjective game being Gp and Gq by Tp (Gp ) and Tq (Gq ), respectively. Trial actions are defined for the agent who perceives the subjective game only when it has at least one Nash equilibrium. It is such an action that is not the agent’s Nash action (1), and, together with a certain choice of the opponent, gives the agent a higher utility than any Nash equilibria, and is the best response to it (2). The definition appears complicated but the idea is rather intuitive, which is based on the philosophy of SI. Let us explain it with an example. Suppose agent p has the subjective game of Table 5.1. She recognizes one Nash equilibrium, (c, z). This means that the outcome is the only candidates of a stationary state in the repeated interactions from her current viewpoint. But once she accepts the possibility of misperceptions as SI suggests, she might come up with a different idea. She now may consider that there might exist some other candidates, some of 8 Hence, if the both subjective games have at least one Nash equilibrium and the both agents are normal, then an outcome that obtains is necessarily a hyper Nash equilibrium. Again, the setting is not perfectly reasonable because one may argue that we should adopt not Nash action but rationalizable action for defining a behavioral domain. If one is interested in doing so, then one can do it.

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which are more preferable for her than any of the Nash equilibria. If so, it would be natural to consider she might make efforts to achieve them. p\q a b c d

x 3, 3 0, 0 5, 0 0, 0

y 0, 0 2, 2 0, 0 0, 0

z 0, 0 0, 5 1, 1 0, 0

Table 5.1: trial action ((c, z) is the Nash equilibrium while b is agent p’s trial action.) We assume she may try such an action that constitutes such outcomes. It is reasonable to assume that they should satisfy at least two conditions. First, it should be preferable for her to the Nash equilibrium because if not, she, in the first place, would not have any incentive to achieve the outcome. Thus, d cannot be her trial action because it leads to her a lower utility than (c, z) whatever agent q chooses. Second, in those outcomes, her choice needs to be the best response to the opponent’s choice because if not, it cannot be a stationary state9 . Thus a cannot be her trial action, either, because, although (a, x) gives her a higher utility than (c, z), a is not a best response to x, thus she herself has an incentive to deviate from the outcome. Consequently, only b can be her trial action. The two requirements are represented in the second condition of Definition 5.7. Then we define proactive agents as follows. Definition 5.8 (Proactive agents): Let Gp = (I, A, up ) be agent p’s subjective game in a hypergame. She is called a proactive agent iff, { Np (Gp ) ∪ Tp (Gp ) if N (Gp ) ̸= φ p ∆p (G ) = Ap otherwise, and

{ Nq (Gp ) if N (Gp ) ̸= φ ∆q (Gp ) = Aq otherwise. The symmetric definition goes for agent q. Proactive agents differ from normal agents only in that the former includes, in addition to Nash actions, trial actions in their own behavioral domains. That is, they may choose not only Nash actions but also trial actions. We assume that the type of an agent remains the same throughout the process of a repeated hypergame and is either normal or proactive. We show some general properties of the repeated hypergame framework. The next statement assures that when an agent, whether normal or proactive, needs to revise her subjective game due to R2, she can always create a new subjective game that satisfies the requirement. Lemma 5.2 (Existence of revised views): Let Gp = (I, A, up ) be agent p’s subjective game in a hypergame. Then, for any aq ∈ Aq , there exists such an utility function of agent ˆ p ), where G ˆ p = (I, A, (upp , u q, u ˆpq , that aq ∈ Nq (G ˆpq )). Proof : Suppose a′q ∈ Aq . Let a′p ∈ arg maxap ∈Ap upp (ap , a′q ). Consider agent q’s utility ˆ p ), where function in Gp , u ˆpq , such that a′q ∈ arg maxaq ∈Aq u ˆpq (a′p , aq ). Then (a′p , a′q ) ∈ N (G ˆ p = (I, A, (upp , u G ˆpq )). Since such u ˆpq always exists, we have the lemma. ¤ The lemma means that, in a given subjective game of agent p, any action of agent q can be a Nash action by allowing her to revise properly his utility function. It holds symmetrically 9

Hence, in our definition, “cooperation” in prisoner’s dilemma is not a trial action.

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for Gq as well. Now with the lemma, we can say that the process of a repeated hypergame is “well-defined” in the sense that it never stops. With respect to base games in repeated hypergames, the next lemma is obvious in our settings. Lamma 5.3 (Base game in reachable hypergames): Let H be a hypergame. Then we ˆ ∈ Φ(H), BG ˆ = BGH . have ∀ H H ¯ = (G ¯ p, G ¯ q ) ∈ Φ1 (H). By Definition Proof : Let H = (Gp , Gq ) be a hypergame and H p q p q p ¯ and G ¯ q also include upp and uqq , 5.4, if G and G include up and uq , respectively, then G ˆ ∈ Φ1 (H). Then by respectively, hence BGH¯ = BGH . Therefore BGHˆ = BGH for any H Definition 5.5 and mathematical induction, we have the lemma. ¤ The lemma says that any reachable hypergames from a hypergame have the common base game. Thus, if we start with a particular hypergame, the base game never changes. It is because we assume that each agent’s utility function (in the agent’s own subjective game) does not change throughout a repeated hypergame.

5.3.5

Example

Let us summarize the concepts we introduced in the section with an example. Consider a two-agents hypergame of agents p and q, namely H = (Gp , Gq ). Table 5.2a and 5.2b represent their subjective games. We have N (G1 ) = N (G2 ) = {(b, r)}, hence SHN (H) = {(b, r)}. Note that they both perceive correctly the agents and actions for each but misperceive each opponent’s utility function. p\q t b

l 4, 2 3, 1

r 1, 4 2, 3

Table 5.2a: p’s subjective game, Gp p\q t b

l 2, 4 4, 1

r 1, 3 3, 2

Table 5.2b: q’s subjective game, Gq The base game, BGH , is given as the game of Table 5.3. Then we have N (BGH ) = {(t, l), (b, r)} and P N (BGH ) = {(t, l)}. Lemma 5.1 holds here. p\q t b

l 4, 4 3, 1

r 1, 3 2, 2

Table 5.3: the base game By comparing the base game with the hypergame, we can see that, if the both agents are normal, then it is the misperceptions that prevent them from achieving the Pareto-optimal Nash equilibrium. To understand it, suppose both agent p and q are normal. That is, ∆p (Gp ) = {b} and ∆q (Gq ) = {r}, thus, due to R1, the only outcome that can obtain is (b, r). Furthermore, since ∆q (Gp ) = {r} and ∆p (Gq ) = {b}, according to R2, neither of them revise the subjective game. Necessarily, the hypergame does not have any reachable hypergame, i.e., Φ1 (H) = φ (and hence Φ(H) = φ). As a result, (b, r) persists as a stationary state. 61

Then let us consider the possibility of the agents’ being proactive. Notice that they both have a trial action, that is, Tp (Gp ) = {t} and Tq (Gq ) = {l}. Now suppose agent p is proactive while agent q remains normal. Then since ∆p (Gp ) = {t, b}, she now may choose t, but once she does so, he is forced to revise his view because t ∈ / ∆p (Gq ). Lemma 5.2 implies that he can always create a new view that satisfies the requirement of R2. In fact, the game of Table 5.3 is one of the candidates of it because there t is in agent p’s behavioral domain perceived by him. Therefore, at least a hypergame in which agent p’s subjective game remains Gp while that of agent q is the game of Table 5.3 is directly reachable from H, that is, in Φ(H). In fact, the hypergame H falls into a class that we define as systems of holding back in the next section. We shall come back to the example again and analyze it more later.

5.4

Systems of Holding Back as Hypergames

Next we define systems of holding back (SHB) in terms of hypergames and analyze it with the framework of repeated hypergames. We shall present some general results with an example.

5.4.1

Definition and Results

We define SHB as follows: Definition 5.9 (Systems of holding back): A hypergame H = (Gp , Gq ) is called a system of holding back (SHB) iff H satisfies the following three conditions: 1. SHN (H) ̸= φ. 2. SHN (H) ∩ P N (BGH ) = φ. 3. Np (Gp ) ⊆ Np (Gq ) and Nq (Gq ) ⊆ Nq (Gp ). The definition represent the essential properties of SHB described in 5.2 and should be interpreted as follows. Intuitively, SHB is a hypergame in which the agents can get caught in an undesirable (in the sense of Pareto efficiency) stationary state from which they cannot get away if they are all normal. The first and second conditions above say that SHB has at least one stable hyper Nash equilibrium and hence a stationary state but any of them are not Pareto Nash equilibrium of its base game. The third condition means that each agent’s Nash action in each own subjective game is always the agent’s Nash action in the opponent’s subjective game, too. This implies that if the agents are both normal, then any possible choice of each opponent are perceived as expected by each agent. Note that these description are all from an analyzer’s point of view. The agents themselves cannot know whether or not the situation they face is SHB. Next, we present some general properties of SHB in terms of our repeated hypergame framework. First, suppose every agent is normal. Then the next proposition is trivial due to the definition of SHB. It claims that SHB does not have any reachable hypergames when the both agents are normal. Proposition 5.4 (Stability of SHB involving normal agents): Let H = (Gp , Gq ) be a hypergame that is SHB. If both agent p and q are normal, then Φ1 (H) = φ. Proof : Suppose the both agents are normal agents in a hypergame H = (Gp , Gq ) that is SHB. The first condition in Definition 5.9 implies that N (Gp ) ̸= φ and N (Gq ) ̸= φ and thus the third condition in Definition 5.9 implies that ∆p (Gp ) ⊆ ∆p (Gq ) and ∆q (Gq ) ⊆ ∆q (Gp ). Hence, Φ1 (H) = φ. ¤ The proposition implies that if the agents who are both normal start with SHB, or once the situation they play becomes SHB at one period, then the hypergame never changes. Since 62

SHB can find a stationary state which is not Pareto-optimal from an objective viewpoint, they may get caught in it forever. Then we analyze our main concern, that is, how becoming an proactive agent works in SHB. In the rest of the present subsection, we assume that agents cannot be indifferent between any two different outcomes. Formally, in any hypergame H = (Gp , Gq ), for any a, a′ ∈ A, a ̸= a′ implies upp (a) ̸= upp (a′ ), upq (a) ̸= upq (a′ ), uqp (a) ̸= uqp (a′ ) and uqq (a) ̸= uqq (a′ ). Then the next lemma refers to existence of trial strategy for both agent in SHB. Lemma 5.5 (Existence of trial actions): Let H = (Gp , Gq ) be a hypergame. If H is SHB, then Tp (Gp ) ̸= φ and Tq (Gq ) ̸= φ, and in particular, P Np (BGH ) ⊆ Tp (Gp ) and P Nq (BGH ) ⊆ Tq (Gq ). Proof : Let H = (Gp , Gq ) be a hypergame that is SHB. Since SHN (H) ̸= φ (by the first condition in Definition 5.9) and SHN (H) ⊆ N (BGH ) (by Lemma 5.1), N (BGH ) ̸= φ and hence P N (BGH ) ̸= φ. Let a∗ = (a∗p , a∗q ) ∈ P N (BGH ). Suppose that a∗p ∈ Np (Gp ). Then, by the third condition in Definition 5.9, a∗p ∈ Np (Gq ). Now, since (a∗p , a∗q ) ∈ N (BGH ), which means ∀ aq ∈ Aq , uqq (a∗ ) ≥ uqq (a∗p , aq ), there is only a∗ that can be a Nash equilibrium including a∗p in Gq , therefore, a∗ ∈ N (Gq ). Then, since a∗q ∈ Nq (Gq ), a∗q ∈ Nq (Gp ). Based on the same discussion above, a∗ ∈ N (Gp ). Then, a∗ ∈ SHN (H). However, since a∗ ∈ P N (BGH ), this contradicts the second condition of Definition 5.9. Hence we have a∗p ∈ / Np (Gp ). SHN (H) ̸= φ implies N (Gp ) ̸= φ. By the definition of Pareto Nash equilibrium, upp (a∗ ) > upp (a∗∗ ) for any a∗∗ ∈ N (Gp ), and upp (a∗ ) ≥ upp (a∗p , aq ) for any aq ∈ Aq . Consequently, a∗p ∈ Tp (Gp ). Since the symmetric discussion goes for a∗q , we have the lemma. ¤ The lemma claims that if a hypergame is SHB, then each agent has at least one trial action. Furthermore, each agent’s Pareto Nash action in its base game is always the agent’s trial action. Then, the next proposition make clear the roles of types of agents. Compare it to Proposition 5.4. Proposition 5.6 (Instability of SHB involving proactive agents): Let H = (Gp , Gq ) be a hypergame that is SHB. If at least one agent is proactive, then Φ1 (H) ̸= φ. Proof : Let H = (Gp , Gq ) be a hypergame that is SHB and a∗ = (a∗p , a∗q ) ∈ P N (BGH ) (which is not empty as shown in the proof of Lemma 5.5). Suppose suppose agent p is proactive. Then a∗p ∈ Tp (Gp ) by Lemma 5.5, thus a∗p ∈ ∆p (Gp ). Suppose a∗p ∈ Np (Gq ). As shown in the proof of Lemma 5.5, if this is the case, then a∗ ∈ N (Gq ). Thus a∗q ∈ Nq (Gq ), which implies a∗q ∈ Nq (Gp ) by the third condition of Definition 5.9. Then a∗ must be a Nash equilibrium in Gp as well. But these imply a∗ ∈ SHN (H) / Np (Gq ) and hence contradict the second condition of Definition 5.9. Therefore, we have a∗p ∈ q ∗ / ∆p (G ) whether agent q is normal or proactive. Then, by Definition 5.4 and and hence ap ∈ Lemma 5.2, Φ1 (H) ̸= φ. Since the discussion holds for the case of agent q being proactive, we have the proposition. ¤ According to the proposition, any SHB has at least one directly reachable hypergame if at least one of the agents is proactive. It implies that even when the hypergame they play at one period is SHB, if at least one agent is proactive, they the situation can change. Finally we present our main proposition. Proposition 5.7 (Pareto-optimization by being proactive): Let H be a hypergame ˆ ∈ Φ(H) such that a∗ ∈ that is SHB. If at least one agent is proactive, then there exists H ˆ for some a∗ ∈ P N (BGH ). SHN (H) 63

Proof : Let H = (Gp , Gq ) be a hypergame that is SHB with Gp = (I, A, up ) and Gq = (I, A, uq ), a∗ = (a∗p , a∗q ) ∈ P N (BGH ) and a′ = (a′p , a′q ) ∈ SHN (H). Note that such a∗ exists (by the first condition of Definition 5.9), a′ exists (as mentioned in the proof of Lemma 5.5), and a∗ ̸= a′ (by the second condition of Definition 5.9). Now suppose agent p is a proactive agent. Then, as shown in the proof of Proposition 5.6, a∗p ∈ ∆p (Gp ), and a∗p ∈ / ∆p (Gq ) regardless of agent q’s type. On the other hand, since ′ p ′ aq ∈ Nq (G ) and aq ∈ Nq (Gq ), a′q ∈ ∆q (Gp ) and a′q ∈ ∆q (Gq ). Hence there exists a ˆ q ) ∈ Φ1 (H), where G ˆ q = (I, A, (ˆ hypergame (Gp , G uqp , uqq )) in which u ˆqp (a∗ ) ≥ u ˆqp (ap , a∗q ) for q ∗ q ∗ ˆ q ). any ap ∈ Ap . Note that since uq (a ) ≥ uq (ap , aq ) for any aq ∈ Aq , a∗ ∈ N (G ˆ q ). On the other hand, as shown in the proof of Lemma 5.5, Therefore, now a∗q ∈ ∆q (G ˆ q ) are same as Lemma 5.3 describes.), a∗q ∈ / Nq (Gp ) (Note that the base game of H and (Gp , G ∗ p ˆ q ) (e.g. a∗ and hence aq ∈ / ∆q (G ). Since there exists ap ∈ ∆p (Gp ) such that ap ∈ ∆p (G p ˆ p, G ˆ q ) ∈ Φ1 ((Gp , G ˆ q )), where G ˆ p = (I, A, (upp , u or a′p ), there exists a hypergame (G ˆpq )) and ˆ p ). Since upp (a∗ ) ≥ upp (ap , a∗q ) for any ap ∈ Ap , a∗ ∈ N (G ˆ p ). a∗q ∈ Nq (G ˆ = (G ˆ p, G ˆ q ). Then a∗ ∈ SHN (H) ˆ (because a∗ ∈ N (G ˆ p ), a∗ ∈ N (G ˆ q )) and H ˆ ∈ Let H Φ(H) (by Definition 5.5). Since the symmetric discussion also goes for the case of agent q being proactive, we have the proposition. ¤ The proposition claims that, if at least one of the agents is proactive, then any SHB has a reachable hypergame that has a stable hyper Nash equilibrium which is a Pareto Nash equilibrium in the base game. This means that even if a hypergame at one period is SHB, if at least one agent is proactive, then, in future, the hypergame can change into another hypergame that can find a stationary state that is Pareto-optimal among those candidates of stationary states from an objective viewpoint.

5.4.2

Example

Reconsider the hypergame H presented in 5.3.5. By definition, it is SHB. We have already seen that Lemma 5.5 and Proposition 5.6 hold in the example: t for agent p and l for agent q are both trial actions as well as Pareto Nash actions in the base game, and the hypergame has a directly reachable hypergame if at least one of them is a proactive agent. Suppose that agent p is proactive, and now the hypergame they play is one in which her subjective game remains Gp (Table 5.2a) while agent q sees the game of Table 5.3 as his subjective game. Let us denote the current subjective game of agent q by Gˆq . Recall that ˆ q ), is directly reachable from H. the hypergame, (Gp , G ˆ q ) = {t, b}, any possible choice In this hypergame, since now ∆p (Gp ) = {t, b} and ∆p (G of agent p does not leads to agent q’s revise of his subjective game. On the other hand, ˆ q ) because l ∈ Nq (G ˆ q ), while l ∈ l ∈ ∆q (G / ∆q (Gp ) because l ∈ Nq (Gp ). Therefore, agent q’s possible choice, l, in turn, has agent p revise her subjective game. Again, the base game is ˆp, G ˆ p ) ∈ Φ1 ((Gp , G ˆ q )), where G ˆ p = BGH . one of the candidates of her new view. Then (G p q ˆ ,G ˆ ) by H. ˆ Then we have H ˆ ∈ Φ(H) and (t, l) ∈ SHN (H) ˆ as Proposition Let us denote (G 5.7 has suggested. The implication of the analysis goes as follows. Suppose the agents start with H, or has reached it at one period. If the agents are both normal, they play {b, r} forever, which is not desirable from an objective point of view. Otherwise, if at least one of them is proactive, the situation can change and they may achieve another stationary state, {t, l}, which is Pareto-optimal.

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5.4.3

Applications

By “re-labeling” of agents and their actions of the example hypergame above, we can formulate the three examples of SHB presented in 5.2 as hypergames (Table 5.4).

agents

actions

1 2 t b l r

Case 1 an arms race country A country B arms reduction military expansion arms reduction military expansion

Case 2 value co-creation a service provider its customer listening not listening suggestion no suggestion

Case 3 a seasoned couple a husband a wife romantic not romantic romantic not romantic

Table 5.4: Three examples of SHB By using Case 1, let us see how our model and findings can be applied to a realistic situation. Table 5.5 expresses the game of the two countries, A and B. We name each outcome as shown in the matrix. A\B arms reduction military expansion

arms reduction peace A’s superiority

military expansion B’s superiority arm race

Table 5.5: A game of two countries, A and B First, look at the base game, i.e., Table 5.3. It can be seen as an imaginary game that they would play if they have no misperceptions. A’s preference is, in order of preferable outcomes, peace, A’s superiority, arms race and B’s superiority. B’s preference is given symmetrically. Then, peace and arms race are the two Nash equilibria of the game, while the former is the only Pareto Nash equilibrium. Next, suppose they both misperceive the opponent’s preference like the example story. Then Table 5.2a can represent A’s subjective game, in which A believes B’s preference is, in order, B’s superiority, arms race, peace and A’s superiority. Likewise, let Table 5.2b be B’s subjective view. The hypergame consists of the two subjective games. Since they both see the arms race as the only Nash equilibrium in each subjective game, it becomes the unique stable hyper Nash equilibrium of the hypergame. According to Definition 5.9, the hypergame is SHB, therefore, as Proposition 5.4 argues, if they both are what we call normal agents, they cannot find a way out of the SHB and would fall into an endless arms race as a stationary state. Lemma 5.5 implies that arms reduction is a trial action for both A and B. Then, as Proposition 5.7 suggests, if at least one of them is proactive, the situation can change and peace, the Pareto Nash equilibrium of the base game, may be achieved as a stationary state. For example, if A is proactive, then it may try arms reduction. It is not a rational choice in A’s current view but A hopes that the behavioral change could somehow change the stalemated relation. Then we know it leads to B’s update of the subjective game so that A’s arms reduction becomes its rational choice (in B’s view). Based on the updated view, B may also implement arms reduction, which is now in its behavioral domain regardless of B’s type. It, in turn, can lead to A’s perceptional change. Finally, peace may become a stable hyper Nash equilibrium in the new hypergame, that is, they may successfully establish a peaceful relation. The same story can be applied to Cases 2 and 3. 65

5.5

Discussion

Finally we discuss several issues on our framework. Connections to Systems Intelligence Let us illustrate how our model reflects SI’s original philosophies and ideas. In particular, we emphasize three points in line with the distinctive features of SI outlined in 5.2. First, in our framework, a proactive agent need not describe fully the whole system, that is, the hypergame in question. She just sees her own subjective game and acts based on it. Although she admits possibility of misperceptions, she knows nothing more. We have showed even without knowing the whole structure, micro-behavioral change of proactive agents by choosing trial actions can result in betterment of the system. This is exactly the core of SI, which differentiates it from systems thinking. In fact, this is critically different from conventional hypergame studies per se. In the hypergame literature, analyses have been conducted in the form of, say, ex-post accounts, what-if analyses and interventions from an outsider’s viewpoint, all of which require efforts for full descriptions. On the other hand, our approach suggests the agents to act better but does not require such descriptive efforts. Second, our model takes optimism for changes. A proactive agent doubts that her view is absolutely correct. It is this kind of doubt that makes it possible for her to choose some sort of off-the-equilibrium plays i.e., her trial actions. Indeed, in hypergames, agents may misperceive the situation, and the fact assures the possibility that the trial actions work as systemic leverages. In contrast, such a doubt cannot arise in games with common knowledge in standard game theory, in which agents know (not believe) the others see the same game. Even in the hypergame literature, explicitly or implicitly, agents are supposed to believe each subjective game is common knowledge among all of them. Hence if they are rational in light of their own perceptions, they would behave like what we call normal agents. Third, what is needed for one to be SI is just an awareness. Since we assume that the agent’s action set is given a priori, what does matter is her behavioral domain. An awareness of a proactive agent that systemic leverage might exist in the complement of the narrow domain that a normal agent would have is crucial. Indeed, the only difference between our normal agents and proactive agents is whether or not the agent includes trial actions in her behavioral domain. SI argues human beings innately possess intelligence for this kind of awareness. SI may not work in some cases Our framework clarifies when SI (i.e., being proactive) works and when it does not. Accordingly, we are not too optimistic to claim that proactive agents within our model always can improve situations they face. We emphasize that our findings such as Proposition 5.7 hold if the hypergame is an SHB. Hence, if it is not, SI may not cope with it. Recall that it is the essential property of SHB that the agents cannot know from inside whether or not the situation is an SHB. For example, let us consider a two-agent hypergame where the both agents see Table 5.2b as each subjective game. Then (b, r) is not only the unique stable hyper Nash equilibrium of the hypergame but also the only Nash equilibrium in the base game. This implies that even if either of them is proactive, in any reachable hypergame from it, any other outcome cannot be a stable hyper Nash equilibrium (due to Lemma 5.1). However, the hypergame is not an SHB in the first place. This example may be compatible with our intuition that even SI cannot always change the situation to whatever she wants. Further research We note several topics for further research on repeated hypergames. First, although we assume a game at each period is expressed as a simple hypergame, it would be possible to extend it to some other model such as hierarchical hypergames, or 66

extensive hypergames. Second, it would be needed to characterize behavior of what we call proactive agents in a more rigorous way. That is, we need a sound reason in the decision theoretic context that an agent may choose a trial action, or some apparently irrational action, hoping that it can improve the situation in the long run. This challenge would be highly relevant to modeling self-awareness of unawareness10 . We, in real life, can be aware of the possibility that we might be unaware of something (but do not know what it is). As SI has suggested, this king of an awareness would let the agent take such an action. Finally, whatever model we apply to express an agent’s subjective view, how the agent may revise it is crucial in analysis of repetitive interactive situations. Some game theorists have recently tried to apply the theory of belief revision (Alchourron, G¨ardenfors and Makinson (1985); and G¨ardenfors (1992)) originally developed in the field of artificial intelligence to modeling epistemology of agents in games11 . The simple revision rule we impose in our model is along this line, but further rigorous discussions are needed in order to deal with view revisions in a more appropriate way.

10 The significance of modeling self-awareness of unawareness has been pointed out by several researchers (e.g. Dekel, Lipman and Rustichini (1998); and Li (2009)). 11 There are also several psychological studies showing that how an infant learns about the world is close to belief revision rather than Bayesian updates. See e.g. Gintis (2010).

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Chapter 6

Modeling Interpretations of Actions in Extensive Hypergames The chapter extends hypergame model to extensive forms and discusses “interpretations of actions” in terms of it1 . By interpretation of an action we mean how an agent, observing the other’s choice, constructs her decision problem, hence we need the model to be extended into extensive forms. We shall show that a stationary state of an interactive decision situation can depend largely on how an agent interprets the leader’s action. Keywords: extensive hypergames, Stackelberg hypergames, (stable) hyper Stackelberg equilibrium, interpretations of actions, intentions.

6.1

Introduction

Hypergame theory has not provided any models of extensive form games2 . In the present chapter, we suggest a hypergame model of extensive forms, which we call extensive hypergames, and illustrate that it enables us to analyze “interpretations of actions”, that is, how an agent formulates her decision problem according to another agent’s choice, as well as how such interpretations can affect the outcome of the game. The epistemological assumption is similar to that of simple hypergames in the sense that we do not consider “hierarchy of perceptions” with which hierarchical hypergames deal, that is, in an extensive hypergame, each agent is assumed to believe that her own subjective view is common knowledge. Extensive hypergame is a natural extension of simple hypergames. Each agent’s subjective view, i.e. subjective game, is given as an extensive form game. As stated in Chapter 2, we only study extensive form games with perfect information with no randomizing devices. This task, however, brings about a tricky problem. Unlike hypergames based on normal form games, agents may not be able to “play” the game in some extensive hypergame. This is because perceptions about the structure of the game tree might be different among the agents. To avoid the problem, we here restrict our attention to a simple class of extensive hypergames called Stackelberg hypergames. A Stackelberg hypergame is such an interactive decision situation where agents may misperceive the opponent’s utility function in a Stackelberg game. We note that it is sufficient to discuss interpretation of actions. In the first half of the chapter (6.3), we introduce Stackelberg hypergames and examine some properties of them. As natural extensions of hyper Nash equilibrium and stable hyper Nash equilibrium (Sasaki and Kijima (2008)), which are considered as a prediction of one-shot decision making and a stationary state in a simple hypergame, respectively, we define hyper Stackelberg equilibrium and stable hyper Stackelberg equilibrium. In a hyper Stackelberg 1

The idea of the present chapter has been seen in Sasaki (2011). The only exception we know is Wang’s (1996) model which studies a sort of signaling games. It deals with the receiver’s interpretation of the sender’s signal as a strategy of the agent. 2

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equilibrium, the leader first chooses her backward-induction solution in her own subjective game, and then the follower chooses his best response to it in his own subjective game. Since their subjective views might be different, from the leader’s point of view, the follower’s choice may not be the expected one. If so, the leader would have “cognitive dissonance” (Festinger (1957)) and may be urged to change the perception about the game. On the other hand, a stable hyper Stackelberg equilibrium is defined as such an outcome perceived as a Stackelberg equilibrium by the both agents. There the agents do not have any reasons to change their choice as well as perceptions about the game: hence we call such a state as “stationary”. We show that a stable hyper Stackelberg equilibrium might include an agent’s misperception and be different with a state that would have been stationary if the agents had had no misperceptions. Next, in the second half (6.4), we explicitly deal with interpretations of actions in Stackelberg hypergames. We care only about interpretations of the leader’s actions because our interest is in how an agent’s action can affect the other’s way of formulating the decision problem. If some two different actions of the leader lead to same subgames, an agent, the leader or the follower, might think these actions are “indifferent” in the sense that which one to be chosen by the leader makes no difference in the follower’s reasoning. If the situation is a Stackelberg hypergame, the two agents may have different opinions about whether or not some two actions are indifferent, that is, they might interpret the leader’s actions in different ways. We study how such a difference of interpretation about the leader’s actions can affect the situation. In order to formulate such interpretations, we introduce the concept of interpretation functions and reduction of subjective games by them. An interpretation function of an agent is a mapping from the leader’s action set to another set, which should be considered as “intentions” of the actions for the agent3 . We require that if intentions of some two actions are identical, then they must lead to same subgames. For example, when one would like to greet someone, there would be several ways to achieve the purpose. If she considers to raise a hand, to bow one’s head, and to say “Hello.” are all indifferent to achieve the intention, those actions are all mapped into one intention, “to greet someone.” Then, by using interpretation functions, we reduce a Stackelberg hypergame to an intention-based hypergame in which an agent’s action is described in terms of intentions. We assume that if the situation is played repetitively, the leader may take any actions to achieve a certain intention when she considers she should choose it because they are indifferent from her point of view: to greet someone, she may raise a hand at one period, but say “Hello.” at the next period. Hence this modeling provides us with unique insights particularly when our interest is in stationary states. We shall show that a stationary state can depend largely on how an agent interprets the leader’s action. Several related works are noted. Although no hypergame literatures have dealt with extensive form games, there are some literatures that provide similar game theoretical frameworks. For example, games with unawareness have provided models in which agents may perceive a decision problem as different extensive form games (e.g. Halpern and Rego (2006); and Feinberg (2010)). In particular, Feinberg’s model of dynamic games with unawareness is similar to ours but how to define an agent’s subjective view is different: it defines a subjective view as each decision node, thus each view cannot evaluate rationality of choices taken previously. In another topic, inductive game theory has dealt with a similar problem, and especially information protocols proposed by Kaneko and Kline (2008) are somewhat similar to our extensive hypergames, while their focus is on how agents construct views about the 3

The distinction between actions and intentions and its role in our society have been studied in sociology such as Max Weber’s interpretative sociology, and also applied to linguistics to study relations between actual speeches and those intentions (e.g. Grosz and Sidner (1986); and Dipert (1993)). If we take it seriously, the distinction would be not fixed but relative. For example, to greet someone can be an action to achieve some higher intention. But we do not go into the problem in our study.

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game rather than how they act based on the subjective views and may modify them. The concept of interpretation function can be seen in early hypergame studies (e.g. Bennett (1980); and Kijima (1996)), though they deal only with normal form hypergames. Moreover an interpretation function in those studies is defined as a mapping from an agent’s action in the agent’s own subjective game to the agent’s action in another’s subjective game. Therefore it cannot deal with such a situation in which an agent thinks some two actions of herself are indifferent while another agent interprets them in different ways. Finally, with respect to interpretation of actions in our use, Kobayashi (2012) provides the most similar framework, which is called interactive decision making with subjective frames. Subjective frame is a very close concept to our interpretation function. His model focuses on what we call stationary states, and unlike our model, does not take into account an agent’s perception about the other’s utility. Our study can be seen as providing an additional requirement to stability in Kobayashi’s model that an agent should consider the stationary state is consistent with rationality of the other agents.

6.2

Extensive Hypergames

We first show the general model of extensive hypergames. We use the same notations as those introduced in 2.2.3. Definition 6.1 (Extensive hypergames): H e = (I, (Gei )i∈I ) is called an extensive hypergame, where I is the finite set of agents and Gei = (I i , W i , P i , ui ) is an extensive form game (with perfect information with no nature’s move) called agent i’s subjective game , where: • I i is the finite set of agents perceived by agent i. • W i is the set of finite sequences each of which is called a history perceived by agent i, which satisfies the following properties: – Each component of a history perceived by agent i, h = (a1 , · · · , an ) ∈ W i , is an action taken by an agent. – If (a1 , · · · , an ) ∈ W i and m < n, then (a1 , · · · , am ) ∈ W i . – h0 ∈ W i , where h0 is the empty sequence, which is called the initial history. – A history (a1 , · · · , an ) ∈ W i is called a terminal history perceived by agent i iff there is no an+1 such that (a1 , · · · , an+1 ) ∈ W i . Let us denote the set of terminal histories perceived by agent i by Z i . • P i : W i \ Z i → I i is the agent assignment function perceived by agent i. • ui = (uij )j∈I , where uij : Z i → ℜ is agent j’s utility function perceived by agent i. Extensive hypergame is a natural extension of simple hypergames to extensive form games. In an extensive hypergame, each agent is assumed to perceive the situation as an extensive form game and consider the game structure common knowledge among all the agents (whom the agent perceives). In agent i’s subjective game, after each nonterminal history h ∈ W i , agent P i (h) chooses an action. Let us denote the set of actions available to the agent at the point (in agent i’s view) by Ai (h). The “general” definition brings about a complicate problem. In simple hypergames in which subjective games are given as normal form games, the game can always be “played” even though someone’s utility may not be defined on the outcome. On the other hand, it is not assured that every extensive hypergame can be played, that is, the agents may not reach any terminal histories in extensive hypergames. For example, consider a two-agent extensive hypergame shown in Figure 6.1 (utilities are dropped in the figure). The left-hand side is agent i’s subjective game while the right-hand side is that of agent j. In their views, agent i thinks agent j moves first, while agent j thinks it the other way around. The situation is quite puzzling because how such a game is played is unclear: rather, the game may never start. 70

Figure 6.1: An extensive hypergame that cannot be “played”

6.3

Stackelberg Hypergames

Although it would be an interesting research topic to investigate further properties of the general framework of extensive hypergame and develop it, we here do not treat such problems. Instead, in the subsequent analysis, we restrict our attention to a simple class of extensive hypergames, called Stackelberg hypergames, as it is sufficient to discuss “interpretation of actions,” which motivates our study.

6.3.1

The Framework

A Stackelberg hypergame is a two-agent extensive hypergame in which the both agents see the situation as a Stackelberg game. Furthermore, it is assumed that their perceptions about the distinction between the leader (she) and the follower (he) is same, and they both perceive the opponent’s action set correctly. Definition 6.2 (Stackelberg hypergames): Let H e = (I, (Gei )i∈I ) be an extensive hypergame and Gei = (I i , W i , P i , ui ) be agent i’s subjective game. H e is called a Stackelberg hypergame when it satisfies all of the following conditions: • I = {1, 2}, and I 1 = I 2 = I, where 1 and 2 are perceived by the both agents as the leader and the follower, respectively, i.e., for any i ∈ I, P i (h0 ) = 1, and P i (a1 ) = 2 for any a1 ∈ Ai (h0 ). • W 1 = W 2. • For any i ∈ I, for any a1 , a′1 ∈ Ai (h0 ), Ai (a1 ) = Ai (a′1 ). • For any i ∈ I, for any (a1 , a2 ) ∈ W i , there is no a3 such that (a1 , a2 , a3 ) ∈ W i . In other words, a Stackelberg hypergame is a Stackelberg game in which the agents may misperceive the opponent’s utility function. We denote, in each agent i’s (= 1,2) view, the action set of the leader by A1 and that of the follower lead by the leader’s choice by A2 . In a Stackelberg hypergame, the leader first chooses an action from A1 , say a1 , and then, informed of the choice of a1 , the follower chooses an action from A2 . Hence the set of “realizable” terminal histories corresponds to A1 × A2 . Let us denote it by A, and we refer to a ∈ A as an outcome. Therefore, henceforth we simply describe a Stackelberg hypergame as H e = (Ge1 , Ge2 ) with Ge1 = (I, A, u1 ) and Ge1 = (I, A, u2 ). In a Stackelberg hypergame, unlike standard Stackelberg games, each agent makes decision according to each subjective game, i.e., utility functions perceived subjectively in this case. As will be discussed below, this makes it unique to study Stackelberg hypergames.

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6.3.2

Equilibria

We discuss two types of equilibrium concepts for Stackelberg hypergame. First let us consider what would happen in an one-shot game. In a Stackelberg hypergame, the leader takes such an action that is a backward-induction solution in her own subjective game. That is, she believes that her choice maximizes her utility function given that the follower always chooses a best response to the leader’s choice. Then, on the other hand, the follower, observing the leader’s choice, chooses such an action that maximizes his utility function in his subjective game. Our first equilibrium concept called hyper Stackelberg equilibrium provides us with such a prediction of one-shot Stackelberg hypergames based on the idea. Definition 6.3 (Hyper Stackelberg equilibrium): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame and Gei = (I, A, ui ) be agent i’s (=1, 2) subjective game. a∗ = (a∗1 , a∗2 ) ∈ A is called a hyper Stackelberg equilibrium iff ∃

a2 ∈ A2 , (a∗1 , a2 ) ∈ S(Ge1 )



a2 ∈ A2 , u22 (a∗ ) ≥ u22 (a∗1 , a2 ).

and

Let us denote the set of hyper Stackelberg equilibria in H e be HS(H e ). In a hyper Stackelberg equilibrium, the leader chooses such an action that can constitute a Stackelberg equilibrium in her subjective game, and the follower chooses a best response to it. Figure 6.2 illustrates an example, where I = {1, 2} and A1 = A2 = {u, d}. Since, in the leader’s point of view, (u, u) is the Stackelberg equilibrium, she chooses u. Then, the follower, informed of 1’s choice of u, chooses a best response to it in his own subjective game, that is, d. After all, an outcome (u, d) is realized in this case, and it is the only hyper Stackelberg equilibrium. It is easily shown that, in a Stackelberg hypergame, there always exists hyper Stackelberg equilibrium, and in particular, it is unique when the agents are not indifferent between any two outcomes.

Figure 6.2: Hyper Stackelberg equilibrium If our interest is only in analyses of one-shot games, hyper Stackelberg equilibrium is sufficient. But it is not the case if we want to know equilibrium in the sense of “stable” states when the game is not one-shot and the agents interact multiple times. In the example above, following the realization of (u, d), agent 1 must consider something is wrong, for she had expected that agent 2 would choose not d but u. While setting aside whether she actually changes her perception, this “cognitive dissonance” would give her a motivation to reconsider about her perception about the game. For example, she might revise her view about agent 2’s utility function and create a new subjective game shown in Figure 6.3. She has partly changed the opponent’s utility function. If she acts according to the new 72

view, she would now choose d as its her backward-induction solution. Thus the first outcome (u, d) can no longer be called an “equilibrium.”

Figure 6.3: Revised subjective game of agent 1 On the other hand, agent 2 does not feel this kind of cognitive dissonance since he considers that agent 1 acts as a result of backward-induction inference and he himself takes a best response to it: (u, d) is the Stackelberg equilibrium in agent 2’s subjective game. Let us consider what if his subjective game is not the game above but that shown in Figure 6.4. In this case, after agent 1’s choice of u, agent 2 would choose d which maximizes his utility given it, so it is same that the predicted outcome is (u, d). Now agent 2 also faces cognitive dissonance since he had expected agent 1 would choose d. We do not specify what he would do then, but emphasize that he, facing the problem, might change his perception about the game. If one’s subjective game is revised, the original hypergame “collapses” by definition. Again, (u, d) cannot be an “equilibrium” of the hypergame.

Figure 6.4: Another subjective game of agent 2 After all, in order for an outcome to be an “equilibrium,” it is required that the both agents conclude at the state that it is consistent with each agent’s acting according to backwardinduction solution4 . Based on such observations, we propose our second equilibrium concept, stable hyper Stackelberg equilibrium. Definition 6.4 (Stable hyper Stackelberg equilibrium): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame and Gei = (I, A, ui ) be agent i’s (=1, 2) subjective game. a∗ ∈ A is called a stable hyper Stackelberg equilibrium iff ∀

i ∈ I, a∗ ∈ S(Gei ).

Let us denote the set of hyper Stackelberg equilibria in H e be SHS(H e ). Stable hyper Stackelberg equilibrium is an outcome perceived by the both agents as a Stackelberg equilibrium in each subjective game. We interpret the concept not as a prediction 4

Recall that common knowledge of rationality implies the agents play according to backward induction (Aumann (1995)). Strictly speaking, the statement here assumes that we do not drop the assumption that the both agents believe it is common knowledge that they both are rational. Although some game theoretical literature deal with agents who strategically use irrational choices, that is, agents who may go against backward-induction solutions (Reny (1992)), we do not consider such possibilities.

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of an one-shot game but as such a stationary state where the both agent have no incentive to change not only their choices but also their perceptions about the game. For example, consider a Stackelberg hypergame shown in Figure 6.5. In this hypergame, (u, u) is the Stackelberg equilibrium in both subjective games, and hence the stable hyper Stackelberg equilibrium of the hypergame. Since the outcome is consistent with common belief of rationality, they both are unwilling to change their decision making as well as perceptions about the game. Although the utility functions of the agents are different in each subjective game, such a difference is “preserved” in the equilibrium. Note that we assume that, for an agent, utilities of the others are unobservable. Thus we do not consider cognitive dissonance about them. In the hypergame below, when an outcome (u, u) is played, agent 1 gets utility as much as 3, while agent 2 thinks agent 1 gets utility 4. We assume that agent 2 never notices such a difference and therefore it does not bring about cognitive dissonance to him.

Figure 6.5: Stable hyper Stackelberg equilibrium Unlike hyper Stackelberg equilibrium, not all Stackelberg hypergames have stable hyper Stackelberg equilibrium5 . For instance, the hypergame shown in Figure 6.2 has no stable hyper Stackelberg equilibrium because there does not exist such an outcome that is perceived as a Stackelberg equilibrium by the both agents. Furthermore, the following proposition that describes the relation between hyper Stackelberg equilibrium and stable hyper Stackelberg equilibrium is straightforward from the definitions. Proposition 6.1 (Hyper Stackelberg equilibrium and stable hyper Stackelberg equilibrium): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame. Then we have SHS(H e ) ⊆ HS(H e ). Proof : Let, in a Stackelberg hypergame H e , a∗ = (a∗1 , a∗2 ) ∈ SHS(H e ), that is, a∗ ∈ S(Ge1 ) and a∗ ∈ S(Ge2 ). a∗ ∈ S(Ge1 ) implies ∃

a2 ∈ A2 , (a∗1 , a2 ) ∈ S(Ge1 ),

and a∗ ∈ S(Ge2 ) implies ∀

a2 ∈ A2 , u22 (a∗ ) ≥ u22 (a∗1 , a2 ).

Hence we have the proposition. ¤

5

Recall that we do not consider mixed extensions.

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6.3.3

Effects of Misperceptions

From an analyzer’s (which may also be referred to as “objective”) point of view, it is possible to study how discrepancies in perceptions can affect decision making of agents as well as equilibria of hypergames. For this purpose, we need to consider what the game would have been like if there had not been such misperceptions at all. We define such an “imaginary” game as the base game. Definition 6.5 (Base game): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame and Gei = (I, A, ui ) be agent i’s (=1, 2) subjective game. A Stackelberg game Ge = (I, A, u) is called the base game of the hypergame if u1 = u11 and u2 = u22 . Let us denote the base game of Stackelberg hypergame H e by BG(H e ). The base game of a Stackelberg hypergame is defined as such a Stackelberg game in which agent 1’s utility function is same as that in her own subjective game in the hypergame and the same goes for agent 2. It is interpreted as the game the agents would have played if they had perceived the opponent’s utility function correctly6 . Therefore we can study effects of misperceptions by comparing equilibria in a hypergame and its base game. For example, the base game of the hypergame shown in Figure 6.2 is illustrated in Figure 6.6. Recall that the hyper Stackelberg equilibrium of the hypergame was (u, d). On the other hand, the Stackelberg equilibrium of the base game is (d, u). It is interpreted that if they had had no misperceptions, they would have played (d, u), but as they in fact misperceive partly the game structure and play the hypergame of Figure 6.2, the realized outcome is (u, d). In the Stackelberg equilibrium of the base game, agent 1 and 2 would get utilities 3 and 2, respectively, while, in the hyper Stackelberg equilibrium of the hypergame, they get 1 and 4, respectively. In this case, due to misperceptions, agent 1 gets lower utility while agent 2 gets higher utility. It can be easily shown that, in general, both the leader and the follower might lose or gain utility in a hyper Stackelberg equilibrium compared to that in a Stackelberg equilibrium of the base game.

Figure 6.6: The base game of the hypergame of Figure 6.2 Recall that the outcome (u, d) is not a stable hyper Stackelberg equilibrium in the hypergame above. Next let us consider relations between stable hyper Stackelberg equilibrium and Stackelberg equilibrium in the base game. Let us reconsider about the hypergame of Figure 6.5, which has the stable hyper Stackelberg equilibrium, (u, u). Its base game is depicted in Figure 6.7, and has the Stackelberg equilibrium, (d, d). It is interpreted that if they had had no misperceptions, they would have played (d, d), but due to the misperceptions, (u, u) becomes the stationary state in the sense that they both have no reasons to change their decision making as well as perceptions about the game7 . In the Stackelberg equilibrium of the base game, utilities for agent 1 and 2 are 4 and 2, respectively. On the other hand, in the stable hyper Stackelberg equilibrium, their 6 Note that it is assumed here that, when there is a discrepancy of perceptions about agent i’s utility function between agent i and another agent j, it is j rather than i who misperceives it. 7 Compare the result to Lemma 5.1. As the lemma implies, in simple hypergames, a stable hyper Nash equilibrium is always a nash equilibrium in the base game. On the other hand, the example here shows that, in

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utilities are 3 and 4, respectively. This means that agent 1 gets lower utility while agent 2 gets higher utility as a result of such misperceptions.

Figure 6.7: The base game of the hypergame of Figure 6.5 In general, in a Stackelberg hypergame, the follower’s utility might be lower or higher in a stable hyper Stackelberg equilibrium compared to that in a Stackelberg equilibrium in the base game. On the other hand, the leader’s utility becomes always lower or the same at most, as the next proposition shows. Proposition 6.2 (The leader’s utility in a stable hyper Stackelberg equilibrium): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame and BG(H e ) be its base game. If a∗ ∈ SHS(H e ) and a∗∗ ∈ S(BG(H e )), then u1 (a∗∗ ) ≥ u11 (a∗ ). Proof : Let, in a Stackelberg hypergame H e , a∗ = (a∗1 , a∗2 ) ∈ SHS(H e ), and, in the base game BG(H e ), a∗∗ ∈ S(BG(H e )). Since a∗ ∈ S(Ge2 ), ∀

a2 ∈ A2 , u22 (a∗ ) ≥ u22 (a∗1 , a2 ).

Therefore, in the base game, as u2 = u22 , ∀

a2 ∈ A2 , u2 (a∗ ) ≥ u2 (a∗1 , a2 ).

Let A′ = {(a1 , a2 )|∀ a′2 ∈ A2 , u2 (a1 , a2 ) ≥ u2 (a1 , a′2 )} in the base game. Then we have a∗ ∈ A′ . Now let us assume u1 (a∗∗ ) < u11 (a∗ ). Then, as u1 = u11 , u1 (a∗∗ ) < u1 (a∗ ) in the base game. This implies ∃

a ∈ A′ , u1 (a∗∗ ) < u1 (a),

but this contradicts a∗∗ ∈ S(BG(H e )). Hence we have u1 (a∗∗ ) ≥ u11 (a∗ ). ¤ The proposition says, in other words, if we allow agents to misperceive the opponent’s utility function in a Stackelberg game, the leader never benefits by the misperception (when our interest is not in one-shot games but in stationary states). In particular, the leader may make a loss only when she herself misperceives the follower’s utility function. For, when the leader perceives the follower’s utility function correctly, i.e., u12 = u22 , Ge1 corresponds to BG(H e ), and hence her utility in a stable hyper Stackelberg equilibrium in H e is equal to that in a Stackelberg equilibrium in BG(H e ). In a related matter, the next proposition says that if at least one agent perceives the opponent’s utility function correctly (and neither of the agents are indifferent between any two outcomes8 ), the stable hyper Stackelberg equilibrium in a hypergame corresponds to the Stackelberg equilibrium in the base game. Proposition 6.3 (Misperceptions and utilities in a stable hyper Stackelberg equilibrium): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame and BG(H e ) be its base game. an extensive hypergame, even such an outcome that is not an equilibrium in the base game can be a stationary state. 8 We say agent i is not indifferent between any two outcomes iff for any a, a′ ∈ A with a ̸= a′ , ui (a) ̸= ui (a′ ). If we drop the indifference assumption, the proposition may not hold, though it is a trivial matter.

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Suppose neither of the agents are indifferent between any two outcomes. If a∗ ∈ SHS(H e ), a∗∗ ∈ S(BG(H e )) and u12 = u22 or u21 = u11 , then a∗ = a∗∗ . Proof : Let, in a Stackelberg hypergame H e , a∗ ∈ SHS(H e ), and, in the base game BG(H e ), a∗∗ ∈ S(BG(H e )). Suppose neither of the agents are indifferent between any two outcomes. If u12 = u22 , then Ge1 = BG(H e ). Thus S(Ge1 ) = S(BG(H e )). Since the Stackelberg equilibrium is unique in BG(H e ), a∗∗ is the only Stackelberg equilibrium in Ge1 . If a∗ ̸= a∗∗ , then Ge1 has another Stackelberg equilibrium, but this contradicts the uniqueness of Stackelberg equilibrium in the game. Therefore we have a∗ = a∗∗ . Likewise, the symmetric inference goes for the case of u21 = u11 . Hence we have the proposition. ¤

6.3.4

Application

To see an application of our framework, let us consider a bilateral relationship between two countries, A and B (a variation of an application, Case 1 (arms race), in Chapter 5). The matter is their military operations, and suppose they both have two options, namely arms reduction (AR) and military expansion (ME). As for their preferences, suppose the following: they both aspire to a peaceful relation in which they both reduce their arms. On the other hand, the last thing they want to do is to concede military superiority to the opponent, that is, to let only the opponent to choose military expansion. The next thing they want to avoid is arms race in which they both implement military expansion. Furthermore we assume that their choices are not simultaneous but sequential: A first makes a decision, and then, informed of A’s choice, B chooses one of the two options. This assumption would be natural if, for example, A is a larger country and has a greater influence on the region than B. If the both countries perceive the game as described above, the situation can be formulated as a normal Stackelberg game as shown in Figure 6.8. The Stackelberg equilibrium of the game is (AR, AR), that is, the best situation for the both countries is achieved9 .

Figure 6.8: The Stackelberg game of the two countries Now let as consider what if we allow them to misperceive the other’s utility function. Suppose that A does not trust B and believes that B’s military superiority is the best outcome for B, while B perceives the game in the same way as above. Then the situation can be described as a Stackelberg hypergame in which A’s subjective game is Figure 6.9 while that of B is Figure 6.8. The game of Figure 6.8 can also be regarded as the base game of the hypergame. A’s misperception is seen in that uB (AR, AR)> uB (AR, ME) in the base game A while uA B (AR, AR)< uB (AR, ME) in A’s subjective game. 9 If their choices are supposed to be simultaneous, the corresponding game is known as a “stag hunt game,” in which not only (AR, AR) but also (ME, ME) is a Nash equilibrium. Thus coordination between the equilibria becomes a serious matter (Skyrms (2004)). On the other hand, such a problem does not arise in the Stackelberg game.

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Figure 6.9: A’s subjective game in the Stackelberg hypergame of the two countries The hypergame has the only hyper Stackelberg equilibrium, (ME, ME). This means that due to the misperception of A, a different outcome is achieved, and what is more, it is worse for the both countries. The outcome is as expected by A as it is the Stackelberg equilibrium in A’s subjective game. On the other hand, it is unexpected for B. B had expected that A would choose AR because it is the backward-induction solution, therefore B has a cognitive dissonance which might force B to change the perception about the game. Although our framework does not specify how it is revised, B may update the subjective view to the one shown in Figure 6.10 so as to A’s choice being consistent with the idea of backward induction. If so, (ME, ME) is the Stackelberg equilibrium in the revised B’s subjective game as well, and hence it is the stable hyper Stackelberg equilibrium in the hypergame. After all, (ME, ME) can last as the stationary state, though they could have built a peaceful relationship if they had had no misperceptions. As Proposition 6.2 suggests, A, the leader, obtains lower utility value there than that in the Stackelberg equilibrium of the base game.

Figure 6.10: Revised B’s subjective game

6.4 6.4.1

Interpretation of Actions Concepts

In a Stackelberg game, if some two different actions of the leader lead to same subgames, an agent, the leader or the follower, might think these actions are indifferent in the sense that which one to be chosen by the leader makes no difference in the follower’s reasoning. If the situation is a Stackelberg hypergame, the two agents may have different opinion about whether or not some two actions are indifferent, that is, they might interpret the leader’s actions in different ways. In this section, we study how such a difference of interpretation about the leader’s actions can affect the situation. First we introduce the concept of interpretation function to formulate an agent’s interpretation about the leader’s actions. Definition 6.6 (Interpretation functions): Let Gei = (I, A, ui ) be agent i’s (=1, 2) subjective game in a Stackelberg hypergame H e = (Ge1 , Ge2 ). A function f i : A1 → X i , 78

where X i is a finite set which is interpreted as the set of the leader’s intentions perceived by agent i, is called agent i’s interpretation function (on the leader’s action set). It satisfies the following condition: for any a1 , a′1 ∈ A1 , if f i (a1 ) = f i (a′1 ), then, for any a2 ∈ A2 , ui1 (a1 , a2 ) = ui1 (a′1 , a2 ) and ui2 (a1 , a2 ) = ui2 (a′1 , a2 ). An interpretation function of an agent is a mapping from the leader’s action set to another set, which we consider as “intentions” of the actions from the agent’s point of view. Intuitively, when some actions are indifferent for an agent, she gives one common intention to them, and the interpretation function describes her opinion in this respect. We naturally require that if intentions of some two actions are identical, then they must lead to same subgames. When we define an equivalence relation on the leader’s action set based on whether some two actions are mapped into one common element by the function, its images can also be regarded as the labels of the equivalence classes10 . By using interpretation functions, we consider reduction of subjective games. In the reduction process, if two or more actions are indifferent in the sense stated above in one’s subjective game, those actions are merged into one action. Definition 6.7 (Reduction of subjective games by interpretation functions): Let Gei = (I, A, ui ) be agent i’s (=1, 2) subjective game in a Stackelberg hypergame H e = ¯ ei = (Ge1 , Ge2 ) and f i be agent i’s interpretation function. Then a Stackelberg game G i i ei i ¯ (I, A , u ¯ ) is called agent i’s reduced subjective game of G by f iff it satisfies both of the following conditions: ∪ • A¯i = A¯i1 × A¯i2 , where A¯i1 = a1 ∈A1 f i (a1 ) and A¯i2 = A2 . • u ¯i = (¯ uij )j∈I , where, for any j ∈ I and a ¯ = (f i (a1 ), a2 ) ∈ A¯i , u ¯ij (¯ a) = uij (a1 , a2 ). The game which is generated as a result of such a reduction is seen as the agent’s subjective view modeled based not on original actions but on her interpretation of the actions. Therefore we refer to such a reduced view as her intention-based subjective game. In particular, we may refer to an action of the leader in a intention-based subjective game as a intention: f i (a) is an intention (interpreted by agent i) of an action a. On the other hand, we refer to the original subjective game as the action-based subjective game. For example, suppose an agent i’s (action-based) subjective game is illustrated as the lefthand side one in Figure 6.11. In this game, the leader has three actions, and suppose that two out of them, m and d, are interpreted as indifferent. Then agent i’s interpretation function is: f i (m) = f i (d)(̸= f i (u)). As a result of the reduction, the two actions are integrated into one action as described in the right-hand side game, which is the agent’s intention-based subjective game.

Figure 6.11: Reduction of a subjective game An equivalence relation is a binary relation ∼ satisfying reflexibility, symmetry and transitivity. In a set X, an equivalence class of an element a is defined as the set, {x ∈ X|a ∼ x}. 10

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The next lemma assures that if an outcome is a Stackelberg equilibrium in an agent’s action-based subjective game, then the corresponding outcome in its intention-based subjective game is also a Stackelberg equilibrium, and vice versa. It is easily shown that the lemma holds in the example of Figure 6.11: (m, d) and (d, d) are the Stackelberg equilibria in the action-based subjective game, while (f (m)(= f (d), d)) is the Stackelberg equilibrium in the intention-based subjective game. Lemma 6.4 (Preservation of Stackelberg equilibria in reduction of subjective games): Let Gei = (I, A, ui ) be agent i’s (=1, 2) subjective game in a Stackelberg hyper¯ ei = (I, A¯i , u game H e and G ¯i ) be the reduced subjective game of agent i whose interpretation ¯ ei ). function is f i . Then, a∗ = (a∗1 , a∗2 ) ∈ S(Gei ) iff (f i (a∗1 ), a∗2 ) ∈ S(G ¯ ei ). This means, (a) (f i (a∗ ), a∗ ) ∈ A¯′i and (b) for any Proof : Suppose (f i (a∗1 ), a∗2 ) ∈ S(G 1 2 ′i i i ∗ ∗ i ¯ a ∈ A ,u ¯1 (f (a1 ), a2 ) ≥ u ¯1 (a), where A¯′i = {(f i (a1 ), a2 ) ∈ A¯i |∀ a′2 ∈ A¯i2 , u ¯i2 (f i (a1 ), a2 ) ≥ i ′ i u ¯2 (f (a1 ), a2 )}. (a) is equivalent to: ∀

a2 ∈ A¯i2 , u ¯i2 (f i (a∗1 ), a∗2 ) ≥ u ¯i2 (f i (a∗1 ), a2 )

⇔∀ a2 ∈ A2 , ui2 (a∗1 , a∗2 ) ≥ ui2 (a∗1 , a2 ) ⇔ a∗ ∈ A′i , where A′i = {(a1 , a2 ) ∈ A|∀ a′2 ∈ A2 , ui2 (a1 , a2 ) ≥ ui2 (a1 , a′2 )}. On the other hand, (b) is equivalent to: ∀

a ∈ A′i , ui1 (a∗1 , a∗2 ) ≥ ui1 (a).

Hence (a∗1 , a∗2 ) ∈ S(Gei ). Since the converse is also true, we have the lemma. ¤ Then reduction of hypergames is defined by reducing each agent’s subjective game in the manner stated above. Definition 6.8 (Reduction of Stackelberg hypergames): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame and f i be agent i’s (=1, 2) interpretation function. Then a Stack¯ e = (G ¯ e1 , G ¯ e2 ) is called the reduced Stackelberg hypergame of H e by f 1 elberg hypergame H 2 ei ¯ and f iff, for each agent i, G is the reduced subjective game of Gei by f i . The hypergame created by the reduction is a combination of each agent’s intention-based subjective game, and thus it is referred to as a intention-based hypergame. Note that a intention-based may not be a Stackelberg hypergame as perceptions about the leader’s action set might be different between the agents in it. In contrast, we may refer to the original hypergame as action-based.

6.4.2

Equilibria

Next let us study some properties of the framework with interpretation functions, introducing some equilibrium concepts. First we consider one-shot decision making. If an agent makes a decision based on an intention-based subjective game, we need a solution concept that enables us to predict an outcome which would be achieved in the (one-shot) hypergame. Hyper Stackelberg equilibrium, however, cannot be applied to it as it is because it may not be a Stackelberg hypergame (note that the leader’s action set might be different in each subjective game in an intention-based hypergame). So we propose an alternative analytical tool. If the leader makes a decision in an intention-based subjective game, she would calculate a Stackelberg equilibrium in it and choose some action whose intention constitutes the equilibrium. That is, if some a ¯∗1 ∈ A¯11 is a backward-induction solution in her intention-based subjective game, she would choose an action a∗1 ∈ A1 such that f 1 (a∗1 ) = a ¯∗1 . Then, on the 80

other hand, interpreting a∗1 as f 2 (a∗1 ), the follower would take a best response to it in his intention-based subjective game. Based on such an observation, we define intention-based hyper Stackelberg equilibrium as a solution concept of intention-based hypergames as follows: Definition 6.9 (Intention-based hyper Stackelberg equilibrium): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame with Gei = (I, A, ui ) being agent i’s (=1, 2) subjective game ¯ e = (G ¯ e1 , G ¯ e2 ) be the reduced Stackelberg hypergame of it by interpretation in it and H 1 2 ¯ ei = (I, A¯i , u functions f and f with G ¯i ) being agent i’s intention-based subjective game. ∗ ∗ ∗ 1 2 ¯ e iff a ¯ = (¯ a1 , a ¯2 ) ∈ A¯1 × A¯2 is called an intention-based hyper Stackelberg equilibrium of H ∃

¯ e1 ) a ¯2 ∈ A¯12 , (¯ a∗1 , a ¯2 ) ∈ S(G

and for some a∗1 ∈ A1 such that f 1 (a∗1 ) = a ¯∗1 , ∀

a ¯2 ∈ A¯22 , u ¯22 (f 2 (a∗1 ), a ¯∗2 ) ≥ u ¯22 (f 2 (a∗1 ), a ¯2 ).

¯ e be HS(H ¯ e ). Let us denote the set of intention-based hyper Stackelberg equilibria in H An intention-based hyper Stackelberg equilibrium is interpreted as a predicted outcome in an one-shot game. It is described in terms of an intention-based hypergame, thus defined on the outcome space of it, A¯11 × A¯22 . Note that in an intention-based hypergame, each agent may interpret an outcome in different ways. Therefore when the leader chooses an action a∗1 as intention f 1 (a∗1 ), the follower gives a intention to it as f 2 (a∗1 ), and takes a best response to ∗ 1 ∗ 1 ′∗ it. If the leader’s another action a′∗ 1 has the same intention as a1 for her, i.e., f (a1 ) = f (a1 ), 2 ′∗ then she may choose a′∗ 1 , and if so, the follower interprets it as f (a1 ) and responds to it. 2 ∗ 2 ′∗ The follower’s best responses to f (a1 ) and f (a1 ) might be different. Thus when the leader acts according to her intention-based subjective game, which action to choose to achieve a intention might become crucial. Then, however, we have the next proposition. It claims that if an outcome is a hyper Stackelberg equilibrium in a Stackelberg hypergame, the corresponding outcome is a intention-based hyper Stackelberg equilibrium in its reduced hypergame, and vice versa. Proposition 6.5 (Intention-based hyper Stackelberg equilibrium and hyper Stack¯ e = (G ¯ e1 , G ¯ e2 ) elberg equilibrium): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame and H be its intention-based hypergame. Then we have (a∗1 , a∗2 ) ∈ HS(H e ) iff (f 1 (a∗1 ), a∗2 ) ∈ ¯ e ). HS(H Proof : Suppose a∗ = (a∗1 , a∗2 ) ∈ HS(H e ). This means ∃

a2 ∈ A2 , (a∗1 , a2 ) ∈ S(Ge1 )



a2 ∈ A2 , u22 (a∗ ) ≥ u22 (a∗1 , a2 ).

and

These conditions are equivalent to: ∃

¯ e1 ) a2 ∈ A¯12 , (f 1 (a∗1 ), a2 ) ∈ S(G



a2 ∈ A¯22 , u ¯22 (f 2 (a∗1 ), a∗2 ) ≥ u ¯22 (f 2 (a∗1 ), a2 ),

and

respectively. In particular, the equivalence of the first condition is due to Lemma 6.4. ¯ e ), and we have the proposition. ¤ Hence a∗ ∈ HS(H The proposition implies that whether or not an agent captures the game as intentionbased does not matter in order to predict which outcome would be obtained in one-shot 81

interactive decision making. Therefore one may think to construct intention-based models might be redundant. But next we shall show that this is not the case for analysis of stationary states when the agents interact with one another multiple times according to each own intention-based subjective game. If the leader considers she should choose a ¯∗1 ∈ A¯11 in her intention-based subjective game, then she may choose any action a∗1 ∈ A1 such that f 1 (a∗1 ) = a ¯∗1 in the 1 1 repetitive process because all of such actions are indifferent for her. If f (x) = f (y) = a ¯∗1 for x, y ∈ A, she may choose x at one period but may choose y at the next period. Then, a stationary state requires, (i) from the leader’s point of view, for any choices of such actions that achieves the intention a ¯∗1 which is her backward-induction solution, she must get always same responses from the follower which is his best response, and (ii) from the follower’s point of view, the leader must always take some action that has a same intention which is her backward-induction solution (in the follower’s intention-based subjective game), and he himself takes a best response to it. Based on such an observation, we define intention-based stable hyper Stackelberg equilibrium which we consider as a stationary state of a repetitive interaction. Definition 6.10 (Intention-based stable hyper Stackelberg equilibrium): Let H e = ¯ e = (G ¯ e1 , G ¯ e2 ) be the reduced Stackelberg (Ge1 , Ge2 ) be a Stackelberg hypergame and H ¯ ei = (I, A¯i , u hypergame of it by interpretation functions f 1 and f 2 with G ¯i ) being agent i’s ∗ ∗ ∗ 1 2 ¯ ¯ (=1, 2) intention-based subjective game. a ¯ = (¯ a1 , a ¯2 ) ∈ A1 × A2 is called an intention-based stable hyper Stackelberg equilibrium iff for any a1 , a′1 ∈ A1 such that f 1 (a1 ) = f 1 (a′1 ) = a ¯∗1 , ∀

¯ ei ) i ∈ I, (f i (a1 ), a ¯∗2 ) ∈ S(G

and f 2 (a1 ) = f 2 (a′1 ). ¯ e be SHS(H ¯ e ). Let us denote the set of intention-based hyper Stackelberg equilibria in H An intention-based stable hyper Stackelberg equilibrium is interpreted as a stationary state in the sense described above. It is such an outcome that is regarded as a Stackelberg equilibrium in intention-based subjective games of the both agents, and this condition must hold whatever actions the leader chooses to achieve the intention which is the backwardinduction solution in her own intention-based subjective game. Furthermore, it requires that the follower interpret all such actions of the leader in the same way, for otherwise from the follower’s viewpoint, the leader may be seen as changing her choice from time to time. Then we have the next proposition. Proposition 6.6 (Intention-based stable hyper Stackelberg equilibrium and stable hyper Stackelberg equilibrium): Let H e = (Ge1 , Ge2 ) be a Stackelberg hypergame and ¯ e = (G ¯ e1 , G ¯ e2 ) be the reduced Stackelberg hypergame of it by interpretation functions f 1 H 2 ¯ e ), then (a∗ , a∗ ) ∈ SHS(H e ). and f . If (f 1 (a∗1 ), a∗2 ) ∈ SHS(H 1 2 ¯ e ). Thus (f 1 (a∗ ), a∗ ) ∈ S(G ¯ e1 ) and (f 2 (a∗ ), a∗ ) ∈ Proof : Suppose (f 1 (a∗1 ), a∗2 ) ∈ SHS(H 1 2 1 2 e2 ∗ ∗ e1 ∗ ∗ ¯ S(G ). Then, by Lemma 6.4, we have (a1 , a2 ) ∈ S(G ) and (a1 , a2 ) ∈ S(Ge2 ), which means (a∗1 , a∗2 ) ∈ SHS(H e ), and thus we have the proposition. ¤ The proposition says that if an outcome is an intention-based stable hyper Stackelberg equilibrium, the corresponding outcome in the action-based hypergame is a stable hyper Stackelberg equilibrium in it. Although the proposition itself is straightforward from the definitions, we rather emphasize that the converse may not hold. That is, if an outcome is a stable hyper Stackelberg equilibrium in a Stackelberg hypergame, the corresponding outcome in the intention-based hypergame may not be a stationary state, i.e., an intentionbased stable hyper Stackelberg equilibrium. This implies that at which level, actions or intentions, an agent makes decisions may become crucial to discuss stationary states. 82

6.4.3

Application

As an application of the framework, let us reconsider the game between the two countries we studied in the previous section. We modify the game setting and suppose now that A, the leader, has three actions, arms reduction with no use of military technology (ARN), arms reduction with peaceful use of military technology (ARU), and military expansion (ME). Furthermore, we suppose the both countries perceive the opponent’s utility function correctly in the intention-based hypergame, but how to interpret A’s action is different between them. To be more precise, the action-based hypergame is depicted in Figure 6.12. Then, by reduction of each agent’s subjective game, the intention-based hypergame is obtained as shown in Figure 6.13. Based on each subjective game in the action-based hypergame, we set each interpretation function as: f A (ARN ) = f A (ARU ) = AR, f A (M E) = M E for A, while f B (ARN ) = AR, f B (ARU ) = f B (M E) = M E for B, where AR means arms reduction. That is, from A’s point of view, ARN and ARU are indifferent and the both options mean arms reduction. On the other hand, the two options of A are not indifferent for B. B interprets ARN as A’s arms reduction but ARU as A’s military expansion.

Figure 6.12: The action-based hypergame

Figure 6.13: The intention-based hypergame As depicted in Figure 6.13, the subjective games of the two countries are identical in the intention-based hypergame. This implies, if they had always interpreted A’s action in the same way, the situation would have been quite simple and ended up a peaceful relation where they both choose arms reduction: in fact this is a normal extensive form game in the standard game theory. But the situation we now tackle with is not that simple. The intention-based hypergame has two intention-based hyper Stackelberg equilibria, (AR, AR) and (AR, ME). To see it, 83

first A chooses AR as it is the backward-induction solution in A’s intention-based subjective game. Then, which action, ARN or ARU, A chooses to achieve the intention is not clear because the two actions are indifferent for it. But since B’s best response to each of these two actions are different, that is, B considers it should choose AR if A chooses ARN while ME if A chooses ARU, the both outcomes are equally likely to happen. As Proposition 6.5 suggests, (ARN, AR) and (ARU, ME) are the hyper Stackelberg equilibria in the action-based hypergame. As of stationary states of the situation, the action-based hypergame has one stable hyper Stackelberg equilibrium, (ARN, AR), but the intention-based hypergame has no intentionbased stable hyper Stackelberg equilibrium (see Proposition 6.6). This implies that if the agents perceive the situation as the action-based hypergame shows, (ARN, AR) can be a stationary state, but if they act according to each subjective game in the intention-based hypergame, the situation cannot be stable as it is: they need to change their choices or perceptions about the game. This is because, as mentioned above, in A’s choice of AR, it may equally choose ARN and ARU, which are followed by different responses of B. For example, if (ARU, ME) is played, then A would feel cognitive dissonance and may change the perception about the game. However, how A update its subjective view is not clear. For example, A may revise B’s utility function in A’s intention-based subjective game as shown in Figure 6.14. Then, in the revised game, A would choose ME. A’s choice of ME, in turn, may urge B to revise the subjective view. If B then revise the view as shown in Figure 6.15, (ME, ME) becomes the Stackelberg equilibrium there, and hence (ME, ME) becomes the intention-based stable hyper Stackelberg equilibrium.

Figure 6.14: Revised A’s intention-based subjective game 1 (A has revised B’s utility function)

Figure 6.15: Revised B’s intention-based subjective game Let us consider another example of A’s revision of the subjective view. A may conclude that ARN and ARU are not indifferent and the two options would lead different subgames. 84

Then A may revise the subjective view as shown in Figure 6.16. Now since f A (ARN ) ̸= f A (ARU ) ̸= f A (M E), the action-based subjective game and the intention-based one become identical for A. Then, if B’s intention-based subjective game is same as the previous one, (ARN, AR) becomes the intention-based stable hyper Stackelberg equilibrium. The outcome gives the both countries higher utility values than in (ME, ME).

Figure 6.16: Revised A’s intention-based subjective game 2 (A has revised how to interpret its own actions)

6.5

Discussion

We have proposed extensive hypergame model and, in particular, studied a simple class of it, Stackelberg hypergames. We finally note several topics on further research of the framework. First, we have assumed that an agent believes each own subjective view is common knowledge among all the agents, just as is assumed in simple hypergames. One natural extension of the model is to take into account hierarchy of perceptions. Likewise, we may consider perception about interpretation functions of the others. In reality, we often make decisions expecting other people may have different interpretations of a particular action. Second, research on the general framework of extensive hypergames would be challenging. As we have discussed, extensive hypergames have a tricky problem that the game may not be able to play until the end when the agents have different perceptions about the game tree. It would be useful to develop the model with getting over such problems in order for it to analyze various decision problems. In terms of interpretation of actions in our sense, it would be interesting to study such situations in which according to the leader’s action, the follower may solve various subgames where not only utility functions but also agents or action set may be different.

85

Chapter 7

Concluding Remarks 7.1

Conclusion

In 1.2, we have presented three main aims of our study. We here conclude how they have been accomplished so far. The first aim is “modeling subjective formulations of games.” Although we have adopted hypergame models for the purpose throughout the thesis, we first argued that several topics on theoretical foundations of hypergames remained a matter of further study. In particular, we have discussed two issues in the thesis. One is on rational behaviors of agents in (one-shot) hierarchical hypergames. In Chapter 3, we proposed a new solution concept called subjective rationalizability and examined its properties. An action of an agent is called subjectively rationalizable when the agent thinks it can be a best response to the other’s choices, each of which the agent thinks each agent thinks is a best response to the other’s choices, and so on. The concept, however, is apparently impractical because we need to know each agent’s choice in infinite hierarchy of a perception for calculating it. In order to make the concept applicable for hypergame analysis, we then proved that it is equivalent to rationalizability in the standard game theory under a condition called inside common knowledge (Lemma 3.1 and Proposition 3.2). The other issue is on a relationship between hypergames and Bayesian games. In Chapter 4, we compared the two independently developed models of games with incomplete information. We first showed that, according to Harsanyi’s claim, any hypergame can naturally be reformulated in terms of Bayesian games in an unified way called Bayesian representation of hypergames. Note that the converse obviously does not hold: there are Bayesian games that cannot be expressed as hypergames. Then we proved that some equilibrium concepts defined for hypergames can provide us with the same implications as equilibria for Bayesian games. For instance, a hyper Nash equilibrium in a simple hypergame always consists of such actions that the actual type of each agent, types of the agents to which the objective probability 1 is assigned, take in a Bayesian Nash equilibrium in the corresponding Bayesian game, and vice versa (Proposition 4.2). Based on the results, we discussed how each model should be used according to the analyzer’s purpose. We concluded that since stable hyper Nash equilibrium cannot be transformed into any existing equilibrium concepts in Bayesian games, so if our interest is in its implication, we need hypergame modeling rather than Bayesian games. Our second aim is “modeling revisions of subjective views.” In Chapter 5, we proposed a new framework called repeated hypergames in order to deal with view revisions of agents explicitly. In a repeated hypergame, agents interact repetitively, and may update each subjective view according to an outcome of a game at each period, which is supposed to be given as a simple hypergame. We introduced the concept of behavioral domain, which is given as a subset of an agent’s action set, to specify how the agent would act and may update the subjective game. That is, it is supposed that an agent may choose any action in the domain and never chooses any actions not included in it, and if and only if an agent observes another 86

agent’s choice is one outside the agent’s behavioral domain, she updates her view so that the new subjective game can explain the “unexpected choice.” Based on such assumptions, we defined reachable hypergames as the whole candidates of hypergames the agents may play in future. In these settings, a stationary state is referred to as a stable hyper Nash equilibrium in such a hypergame that has no reachable hypergames, that is a hypergame that would never change. The framework has been applied to a certain class of hypergames called systems of holding back, where agents believe they themselves do the best from each agent’s point of view but misperceptions prevent achievements of Pareto-optimal stationary state (from an objective view). We examined how differences of behavioral domains of agents may work in such situations, and showed that apparently irrational behavior can lead to the betterment of such situations (Proposition 5.7), while if the agents always choose rational actions, i.e., Nash actions, the situation never be improved. The idea has been inspired by a recently developed systemic concept called systems intelligence, and our study can be considered as a game theoretic characterization of it. Finally, our third aim is “modeling interpretations of actions.” For the purpose, we, in Chapter 6, first proposed a new framework called extensive hypergames, which is an extension of hypergame models to extensive forms. Having restricted our attention to a simple class of it called Stackelberg hypergames, that is, such situations in which an agent may misperceive utility functions of the other agents in Stackelberg games, we examined their properties. We defined two kinds of equilibrium concepts for Stackelberg hypergames: hyper Stackelberg equilibrium and stable hyper Stackelberg equilibrium. The former is considered as a prediction of an one-shot hypergame while the latter is as a stationary state of the situation when the game is played repeatedly. Furthermore we introduced the notion of base game as the game that would have been played if the agents had had no misperceptions. By comparing equilibria in the hypergame and its base game, we analyzed how misperceptions affect in Stackelberg hypergames. For example, it has been shown that, in a Stackelberg hypergame, the leader never be better off in the stationary state than in the base game (Proposition 6.2). Then we explicitly formulated interpretations of actions in the sense of an agent’s belief in what kind of subgame follows an action of the leader in a Stackelberg hypergame. By introducing interpretation functions, we examined how interpretations of the leader’s action can influence an outcome. An interpretation function is a mapping from actions to what we call intentions. Then it has been shown that whether agents formulate the situation at the level of actions or intentions can be critical to determine stationary states. That is, when an outcome is an intention-based hyper Stackelberg equilibrium, the corresponding actions constitute action-based hyper Stackelberg equilibrium, while the converse may not be true (Proposition 6.6). Therefore, when a game is considered as a mental model of an agent, how she identifies the set of alternatives is very crucial to determine equilibria.

7.2

Further Research

We have already mentioned technical topics on further research in each chapter, such as extension or generalization of each framework. In addition, we consider our study on modeling subjectivity and interpretations of human beings can potentially be applied to several areas. Here we give some examples. Since our framework studies how an order, i.e. a stationary state, emerges in our society depending on its history, comparative institutional analysis initiated by Aoki (2001) obviously shares a quite close spirit to ours. Linguistics and communication study would be also candidates of applications. In particular, a certain class of utterance called speech acts (Austin (1962); and Searle (1975)) can be characterized formally by our framework of extensive hypergames. Finally, it would also contribute to a kind of decision theory that particularly 87

encourages people to care about formulation of decision problems rather than merely solving them. As discussed in Chapter 5, systems intelligence is in this category. Another notable example in this line would be Keeney’s (1996) value-focused thinking. Although the frameworks we proposed in the thesis such as repeated hypergames and extensive hypergames are ambitious endeavors and there are still many problems remaining, hopefully they play a role as cornerstones to be got over to enrich studies on decision theoretical modeling of human subjectivity and interpretations.

88

Appendix A. List of mathematical symbols 1. Logic p∧q p∨q ¬p iff p⇒q p⇔q ∀ ∃

p and q p or q not p if and only if p implies q p iff q, i.e. (p ⇒ q) ∧ (q ⇒ p) the universal quantifier, i.e. “for all” the existential quantifier, i.e. “for some”

2. Sets {a, b, c} {x|p(x)} a∈A a∈ /A A⊆B A⊂B A*B A∪B A∩B A\B ¬A ∪ ∩λ∈Λ Aλ λ∈Λ Aλ A×B ×λ∈Λ Aλ A−λ 2A f :A→B |A| φ ℜ

a set with elements a, b and c the set of x for which p(x) is true a is an element of a set A a is not an element of A A is a subset of B, i.e. x ∈ A ⇒ x ∈ B A is a proper subset of B, i.e. A ̸= B ∧ (x ∈ A ⇒ x ∈ B) A is not a subset of B the union of A and B, i.e. {x|x ∈ A ∨ x ∈ B} the intersection of A and B, i.e. {x|x ∈ A ∧ x ∈ B} the difference set of A and B, i.e. {x|x ∈ A ∧ x ∈ / B} the complement of A {x|∃ λ ∈ Λ, x ∈ Aλ } {x|∀ λ ∈ Λ, x ∈ Aλ } the product set of A and B, i.e. {(x, y)|x ∈ A ∧ y ∈ B} the product set of Aλ for all λ ∈ Λ ×λ′ ∈Λ\{λ} Aλ′ the power set of A, i.e. {X|X ⊆ A} a function, or a mapping, from A to B the cardinality (number of the elements) of A the empty set the set of real numbers

89

B. List of symbols used in the dissertation G Ge Gb H BGH Hh He N (G) Ni (G) P N (G) P Ni (G) Ri (G) S(Ge ) BN (Gb ) N (Gb ) HN (H) BE(H) Ri (H h ) BE(H h ) HS(H e ) SHS(H e )

a normal form game an extensive form game a Bayesian game a simple hypergame the base game of a simple hypergame a hierarchical hypergame an extensive hypergame the set of Nash equilibria of a normal form game G the set of agent i’s Nash actions in a normal form game G the set of Pareto Nash equilibria of a normal form game G the set of agent i’s Pareto Nash actions in a normal form game G the set of agent i’s rationalizable actions in a normal form game G the set of Stackelberg equilibria of an extensive form game Ge the set of Bayesian Nash equilibria of a Bayesian game Gb the set of Nash equilibria of a Bayesian game Gb the set of hyper Nash equilibria of a simple hypergame H the set of best response equilibria of a simple hypergame H the set of agent i’s subjectively rationalizable actions in a hierarchical hypergame H h the set of best response equilibria of a hierarchical hypergame H h the set of hyper Stackelberg equilibria of an extensive hypergame H e the set of stable hyper Stackelberg equilibria of an extensive hypergame H e

90

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