Bargaining with incomplete information: Evolutionary stability in …nite populations Kai A. Konrady

Florian Morathz

January 2, 2016

Abstract This paper considers evolutionarily stable strategies (ESS) in a take-it-orleave-it o¤er bargaining game with incomplete information. We …nd responders reject o¤ers which yield a higher positive material payo¤ than their outside option. Proposers, in turn, may make more attractive o¤ers than in the perfect Bayesian equilibrium. E¢ ciency-enhancing trade may break down even when the responder has no private information. Overall, the probability of trade and ex post e¢ ciency is lower in the ESS than in the corresponding perfect Bayesian equilibrium. The results are observationally equivalent to behavioral explanations such as in-group favoritism and a preference for punishing sel…sh proposers but are driven by concerns about relative material payo¤ in …nite populations. Keywords: Evolutionary stability; Finite population; Take-it-or-leave-it offer bargaining; Asymmetric information JEL Codes: C73; C78; D82

Financial support from the German Research Foundation (DFG, grant no. SFB-TR-15) is gratefully acknowledged. y Corresponding author. Max Planck Institute for Tax Law and Public Finance, Marstallplatz 1, 80539 Munich, Germany; e-mail: [email protected]. z Max Planck Institute for Tax Law and Public Finance, Marstallplatz 1, 80539 Munich, Germany; e-mail: ‡[email protected].

1

1

Introduction

We study evolutionary stability of ultimatum bargaining behavior in a context with incomplete information and within a …nite population. The core analysis considers an evolutionary setup with a sequence of generations of individuals. In each period a given generation constitutes a population of time-invariant and …nite size. Members of a given generation interact in a state game. The outcomes of these interactions determine their evolutionary …tness, which determines the population dynamics. In each of these state games the agents are grouped in pairs of an (informed) seller and an (uninformed) buyer and interact in an ultimatum bargaining game with incomplete information. We derive the evolutionarily stable o¤er-making and o¤er-responding behavior in this game. Also, we ask how this behavior relates to the outcome of a perfect Bayesian equilibrium (PBE). We …nd that the evolutionarily stable price o¤ers are higher than the o¤ers in the PBE, but the responder is more reluctant to accept than in the PBE. Overall, trade becomes less likely for evolutionarily stable strategies (ESS) than for PBE behavior. We also consider several extensions. One of these is bargaining if the informed player - the seller in our context - makes an o¤er and the uninformed buyer accepts or rejects. In the context of evolutionarily stable strategies this case has some surprising features, and mutually bene…cial trades in terms of material payo¤s may not be realized. This contrasts with a corresponding PBE in this case which implements full e¢ ciency. Another extension considers the importance of trading within, rather than across populations. The research question and our analysis relate to two main areas of research: the study of bargaining, particularly under incomplete information; and the study of evolutionary stability. The ultimatum bargaining game has been studied theoretically and empirically (see Güth et al. 1982 and Güth and Kocher 2014 for a recent survey) and has been used as one of the major tools to describe the settlement of a distributional con‡ict. Incomplete information has been identi…ed as a main source of ine¢ ciency in such con‡icts (see, e.g., Chatterjee and Samuelson 1983 and Myerson and Satterthwaite 1983). Bargaining under incomplete information may systematically fail to reach an e¢ cient outcome. We consider how the outcome of bargaining games with incomplete information is modi…ed if players pursue evolutionarily stable strategies, based on the characterization of evolutionary stability in …nite populations

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developed by Scha¤er (1988) and by Nowak et al. (2004). In the PBE of ultimatum bargaining with incomplete information, the uninformed buyer as proposer maximizes expected material payo¤. When making a price o¤er he faces the following trade-o¤. If he o¤ers a high price, this may increase the likelihood of acceptance, but it also increases what the buyer has to pay in case trade takes place. If the price o¤er is below the seller’s reservation price, this leads to failure of trade, even though mutually bene…cial trade would be possible. Evolutionary …tness as the objective that drives players’behavior brings in a di¤erent logic of optimization. We …nd that trade becomes even less likely in …nite populations when players apply evolutionarily stable strategies. Our main analysis demonstrates this result under the assumption of uniformly distributed reservation prices of the sellers; Section 4.2 discusses the logic of evolutionarily stable behavior for the case of a more general probability distribution. Previous research in this area has described a large set of bargaining games that di¤er from ultimatum bargaining (see, e.g., Osborne and Rubinstein 1990 for an overview). Some of these develop a very di¤erent kind of ‘population dynamics’ than in our framework. An important example is Wolinsky (2000), who studies the dynamics and steady-state properties of a population of workers in a multi-period framework with individual wage negotiations, assuming Nash bargaining. We focus on ultimatum bargaining for several reasons. First, repeated interaction between players in a setup with incomplete information increases complexity along dimensions which are orthogonal to our research question. Second, ultimatum bargaining is the underlying game in some of the seminal contributions on the role of incomplete information for the breakdown of trade in PBE. We analyze whether this breakdown becomes more or less likely if players follow evolutionarily stable strategies. Evolutionary stability is a concept that has been developed to describe and predict population dynamics in biology contexts. Seminal contributions by Maynard Smith and Price (1973) and Maynard Smith (1974) consider such dynamics in the context of very large populations. This concept assumes that individual players follow some behavioral pattern, which is called their evolutionary strategy. Individuals who follow the same pattern are of the same ‘type.’ Individuals who follow di¤erent patterns are of di¤erent types. Assuming that types who have a higher material payo¤ than other types reproduce faster, thereby succeeding to spread their own type in the next generation, this concept studies the evolutionary success of types. In populations with very many individuals, a single individual’s behavior has an impact that is too 3

insigni…cant to have an in‡uence on the average performance of the aggregate population. Hence, in such contexts behavioral patterns that are evolutionarily stable and maximization of absolute material payo¤ typically coincide. Work of Scha¤er (1988) highlighted that this perspective changes in small populations (see also Alós-Ferrer and Ania 2005, Nowak et al. 2004 and Tarnita et al. 2009). In …nite populations, single individuals may increase their own reproductive success not only by advancing their own material resources, but also by reducing those of their competitors. In …nite populations, the latter reduces the average material resources of other members of the population with whom the individual competes for reproductive success. As Scha¤er (1988) and a number of applications demonstrate, players who apply evolutionarily stable strategies are more “aggressive”than if they simply maximized their own material payo¤ as in a standard PBE. The evolutionarily stable strategy may yield a lower own material payo¤ if it yields an even lower average material payo¤ of the remainder of the population.1 However, as has been shown by Eaton, Eswaran and Oxoby (2011) and Konrad and Morath (2012), evolutionarily stable strategies may also be described by in-group favoritism: behavior that could mistakenly be interpreted as altruism. The bargaining context studied here has a structure that may provide opportunity for the emergence of such in-group favoritism. The population is grouped into small subgroups that consist of pairs of players, the respective players in each pair interact as an in-group when bargaining with each other, and there are gains from trade. Our results show, however, that considerations of evolutionary stability lead to less trade and lower total material payo¤s within a subgroup. In the bargaining context evolutionarily stable strategies (ESS) make agreement less likely, compared to the PBE with players who maximize their absolute material payo¤. The responder behaves “tougher” than in the PBE. As a seller he rejects some o¤ers for which the o¤ered price exceeds his own material valuation of keeping the good. In this case he sacri…ces his own material reward. The proposer behaves more generously and o¤ers a higher price in the ESS than in the PBE. This may look like in-group favoritism, but from the perspective of the proposer it may be seen as an accommodating reaction to the tough behavior of the responder. And the accommodating behavior is not su¢ cient to overcome the primary, e¢ ciency1

Related to this, a literature based on a seminal paper by Mui (1995) highlights the role of envy. Evolutionarily stable strategies in small populations can actually provide an economic underpinning for envy, and the strength of such emotions may relate to group size and the type of interaction.

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reducing e¤ect of tough responder behavior. Thus, the probability of trade and the expected material payo¤s of players are lower for evolutionarily stable strategies than in the PBE. This e¤ect is strongest in small populations and becomes less pronounced if the population size is larger.2 We also show that a similar logic applies for evolutionarily stable strategies if the informed seller makes an ultimatum price o¤er to the uninformed buyer. Much related research has been pursued on evolutionarily stable strategies in the context of bargaining. Seminal work by Gale, Binmore and Samuelson (1995) and Nowak, Page and Sigmund (2000) considers evolutionary analyses of the ultimatum game. Binmore, Piccione and Samuelson (1998) analyze alternating-o¤er bargaining and Ellingsen (1997) considers the evolutionary stability of “obstinate” types in a Nash demand game. Abreu and Sethi (2003) study a bargaining model with behavioral types that never concede to the opponent’s demand, assuming type unobservability. For work on the evolutionary stability of trading mechanisms (instead of strategies and preferences, respectively) see, for instance, Lu and McAfee (1996). Learning and stochastic convergence has also been studied in other contexts. Seminal contributions include Young (1993) and Kandori et al. (1993). There are also recent studies that are more closely related to our analysis. Heifetz and Segev (2004) consider the strategic role of “toughness” in a bargaining context. This approach exploits the strategic commitment value of a toughness preference of the responder if the preference type is observed by the proposer.3 Their bargaining framework is related to ours, but the strategic bene…t of being tough relies on type observability in their framework. A proposer who observes that his responding coplayer is “tough”accommodates for the toughness and makes a more generous o¤er. Hence, observed toughness has a strategic e¤ect which bene…ts the tough player. In our framework, types are unobserved and therefore have no strategic commitment 2

We show that evolutionarily stable behavior converges toward Nash equilibrium behavior if the population size goes to in…nity, a result which is also obtained in various contexts for the case of complete information. See Ania (2008) and Hehenkamp, Possajennikov and Guse (2010) for discussions on the equivalence of Nash equilibria and evolutionary stable equilibria in …nite populations. 3 For an evolutionary underpinning of di¤erent types of (other-regarding) preferences based on their strategic e¤ects see also Sethi (1996), Bester and Güth (1998), Koçkesen, Ok and Sethi (2000), Dufwenberg and Güth (2000), Possajennikov (2000), Ok and Vega-Redondo (2001), Sethi and Somanathan (2001) and Dekel, Ely and Yilankaya (2007). For a recent contribution that addresses the evolutionary stability of preferences under incomplete information in an assortative matching game see Alger and Weibull (2013).

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value. Huck and Oechssler (1999) consider a reduced form of the standard ultimatum o¤er game with perfect and complete information. They allow for a minimal set of possible proposer strategies: either o¤er a fair split or o¤er zero. Their evolutionary stability criterion is not identical but is closely related to the criterion developed by Scha¤er (1988). Huck and Oechssler …nd that behavior by which players reject zero o¤ers is evolutionarily stable and can be generated as the subgame-perfect equilibrium behavior of players with behavioral preferences. We widen the scope to the full continuous action space for take-it-or-leave-it bargaining; also, we depart from Huck and Oechssler (1999) by considering incomplete information.4 Our focus is on the well-known breakdown of trade in the PBE due to incomplete information and we ask whether evolutionarily stable behavior may widen or narrow the range in which trade takes place successfully. The paper is also related to the literature that considers comparison of Nash equilibrium and evolutionarily stable strategies in …nite populations more generally and in di¤erent contexts. Examples are problems of tax competition (Wagener 2013), contests (Leininger 2003; Hehenkamp, Leininger and Possajennikov 2004; Leininger 2009; Boudreau and Shunda 2012; Wärneryd 2012), common pool problems (Atzenho¤er 2010) and oligopoly theory (Scha¤er 1989). The study of evolutionarily stability in the ultimatum game stands in line with this literature and also reveals a tendency for players to be spiteful or to exhibit more aggressive behavior. The consideration of asymmetric roles of players of proposer and responder in the ultimatum game highlights an important e¤ect: Evolutionarily stable responder behavior may become so much more aggressive that the o¤er maker accommodates and concedes, even though this player also maximizes relative material payo¤. We proceed as follows. The next section describes the framework and a criterion for evolutionary stability. Section 3 considers one-sided incomplete information and o¤ers made by the uninformed player. Section 4 discusses the inverse case with o¤ers made by the informed player as well as the robustness of our results. Section 5 concludes. 4

The indirect approach (in which evolution operates on objective functions) evokes di¢ cult questions when it addresses incomplete information (see Konrad and Morath 2012 for a discussion and a …rst approach). Also, there is a well-known multiplicity of behavioral preferences that typically support the same strategy choice in the direct approach. Huck and Oechssler (1999) consider fairness or a preference for punishment. Bar-Gill (2006) considers optimism in bargaining problems. Other important candidates are in-group favoritism and out-group spite (Eaton, Eswaran and Oxoby 2011).

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2

The framework

The setup. We study an evolutionary framework with a …nite population of n = 2m players in each generation. The state game consists of bilateral bargaining interactions within randomly formed subgroups of two players each. Within each two-player subgroup, nature randomly assigns the roles of seller and buyer. Each player has the same probability of being assigned the role of seller or buyer.5 The seller owns a resource which, if kept, gives him material value vS . This material value vS is an independent draw from a uniform distribution on the unit interval [0; 1] and the value of vS is the seller’s private information. The buyer attributes a material value vB to obtaining possession of the resource, which is commonly known. We study a benchmark case with vB = 1 in which trade is always e¢ cient in terms of aggregate material payo¤. The two players interact in take-it-or-leave-it o¤er bargaining: One player makes a price o¤er and the other player accepts or rejects this price o¤er. In the main analysis we focus on the case in which the uninformed player (the buyer in our case) has the role of proposer and the informed player (the seller in our case) is the responder, as this scenario is known to cause ine¢ ciency in a standard perfect Bayesian game.6 In each matched pair of players the buyer proposes buying the resource for a payment b 2 [0; 1). The seller as responder accepts or rejects this o¤er. If the o¤er is accepted, the buyer pays b and receives the resource. This ends the game, with material payo¤s equal to vB b = 1 b for the buyer and equal to b for the seller. If the o¤er is rejected, the seller keeps the resource and this ends the game. The material payo¤s are zero for the buyer and vS for the seller in this case. Evolutionary strategies consist of a vector (b; (vS )) with two components. Here, the function (vS ) : [0; 1] ! [0; 1) is a function of own valuation of possession of the resource and describes a threshold strategy: A seller/responder with a valuation vS of keeping the good accepts an o¤er 5

To avoid an ex post asymmetry stemming from this assignment, we could allow each individual to be teamed up twice in each generation, once as a proposer and once as a responder. This would add complexity to the analysis. However, it is not essential for our results, as the equilibrium concept focuses on expected relative payo¤. 6 In Section 4.1 we discuss a reversal of the roles of proposer and responder and assume that the informed seller makes a price o¤er and the uninformed buyer is the responder.

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b if and only if b (vS ). Note that, if (vS ) > vS , the seller chooses a threshold for acceptance that is higher than his material valuation and, hence, sacri…ces own material payo¤ by rejecting o¤ers b 2 [vS ; (vS )]. The opposite holds for (vS ) < vS . We restrict consideration to pure strategies and strictly increasing threshold functions .7 Let us also de…ne the function VS ( ) as the inverse function of (vS ) where it exists. The function VS (b) determines the upper end of the range vS 2 [0; VS (b)] for which the seller/responder is willing to accept a given o¤er b of the buyer. De…ning evolutionary stability. For a de…nition of evolutionarily stable strategies (ESS) in …nite populations and one-step mutations, suppose that 2m 1 players follow a given strategy = (b; ) that determines a player’s actions in the di¤erent roles as functions of the valuations. We distinguish between one-step proposer mutations, i.e., a buyer/proposer who makes an o¤er that deviates from b, and one-step seller/responder mutations, i.e., deviations from (vS ). We denote such deviations from the candidate evolutionarily stable strategy by a superscript “M .” Hence, a mutant strategy along the buyer dimension would be described by an o¤er bM , and M (vS ) would denote a possible mutant strategy in the seller dimension.8 Denote by ( M ; M ) the strategy pro…le if one individual chooses M but all remaining individuals choose . Focusing on evolutionary …tness in …nite populations within the concept of Scha¤er (1988), we are interested in the players’ex ante expected material payo¤s. To derive these, we …rst consider the timing of the resolution of uncertainty and of the players’ moves in a given state game. Each of the n = 2m players is programmed to follow a strategy (b; (vS )), which de…nes the player’s ‘type.’ We allow (vS ) to be a function of the player’s own valuation of keeping the good, but we assume that co-players’speci…c types cannot be observed. This rules out that buyers/proposers or sellers/responders condition their programmed action on their respective co-player’s type.9 It also makes sure that our …ndings are not driven by 7

Note that the assumption of well-behaved equilibrium strategies is important for the set of equilibria, as there may be further equilibria involving discontinuous acceptance functions; see also the related discussion in Chatterjee and Samuelson (1983) on the corresponding case in which players maximize material payo¤. 8 Since players are either assigned the role of a buyer/proposer or the role of a seller/responder, we could equivalently allow for simultaneous deviations of a mutant player in both dimensions, which would require minor adaptations in the proofs below but yield the same conclusions. 9 Evolutionary game theory has considered the role of “greenbeards,”the phenomenon that individuals condition their behavior on the (observable) type of their co-player. This type of conditional play is the basis for much of the results that allow for other-regarding behavior in the equilibrium.

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Figure 1: Time structure of the state game the ‘strategic’value of players’types. The evolutionary stability of a strategy is assessed on the basis of expected payo¤s conditional on what is known in stage 0: The distribution of strategies in the population is known, but matching has not taken place and the roles of proposer or responder have not yet been assigned. Also, only the distribution of the idiosyncratic valuation vS of the resource is known at stage 0. For vB , the valuation vB = 1 is common knowledge. Note that players are symmetric at this stage. We de…ne by E0 i ( i ; i ) the expected material payo¤ of player i given the state of information in stage 0. Similarly, E0 i ( i ; i ) denotes the average expected material payo¤ of the other players in the population under strategy pro…le ( i ; i ). At stage 1, players learn the group and the role (seller/responder or buyer/proposer) they are assigned to. At stage 2, a player in the role of the seller learns his vS ; then, the players take actions according to their strategies (b; (vS )). Figure 1 summarizes the time structure. We can now use the equilibrium condition in Scha¤er (1988) to de…ne: De…nition 1 The strategy is an evolutionarily stable strategy (“ESS”) if there is no one-step mutation M from such that E0

M

(

M

;

M)

E0

M

(

M

;

M)

> 0:

(1)

As discussed brie‡y in the introduction, evolutionary stability is a concept that This type of phenomenon is not the basis for our results. Note further that the o¤er b chosen by a player in the proposer role is independent of the responder type of the player who responds. This, in turn, eliminates “strategic” incentives that have also been prominent in explaining other-regarding preferences and deviations from Nash behavior in the evolutionary equilibrium, e.g., in Robson (1990), Güth and Yaari (1992), Banerjee and Weibull (1993), and Bester and Güth (1998). See also Ely and Yilankaya (2001) and Dekel, Ely and Yilankaya (2007) on observable versus non-observable preference types and the relation to Nash behavior.

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has been developed to describe and predict population dynamics in biology contexts. Seminal contributions by Maynard Smith and Price (1973) and Maynard Smith (1974) consider such dynamics in the context of very large populations. In populations with very many individuals, a single individual typically has no signi…cant impact on the material payo¤ of the average player in the remaining population; for this reason, E0 M ( M ; M ) becomes a constant with respect to the possible mutations M . Maximization of material payo¤ and maximization of own …tness tend to coincide. This perspective changes in small populations where single individuals can in‡uence E0 M ( M ; M ). They may increase their own reproductive success not only by advancing their own material resources, but also by reducing the average material resources of other members of the population with whom the individual competes for reproductive success. A second di¤erence between the original concept of evolutionary stability in large populations and Scha¤er’s concept is about population dynamics. The laws of large numbers cannot be applied in small populations. The composition of …nite populations develops stochastically. The ‘…xation probability’for any type, i.e., the probability that a single mutant of a given type manages to take over the whole population, is usually smaller than 1. Rather, evolutionary stability of a particular type typically does not imply that a population is completely immune to invasion and cannot be taken over by a mutant type. However, the criterion (1) appears as the central necessary condition in Scha¤er (1988) and is also a necessary condition in a stochastic dynamic theory (see Nowak et al. 2004, Taylor et al. 2004, or Nowak 2006, condition (7.16)).10 The criterion has a nice intuition that has been emphasized by Scha¤er (1988). Suppose that a mutation appears in a given generation, which otherwise consists of n 1 players of a homogeneous ‘old’type. All n players now interact, pairwise, as in our context. If the mutant has a higher expected material payo¤ than the average of all other players in this generation, it does not warrant …nal success, but it is a good start for the mutant type. Criterion (1) rules out that mutations with this kind of a good start exist. In order to have such a good start, a mutant’s material 10

The theory of evolutionary stability in …nite populations discusses several concepts that typically consist of several conditions. Scha¤er (1988) discusses further conditions that consider evolutionary success if the number of mutants grows beyond 1. For a di¤erent, but related approach see Nowak (2006, condition 7.16). Like the majority of the literature applying the stability concept, which we refer to in the introduction, we focus on the necessary condition which is used in both concepts and considers the performance of a single mutant.

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payo¤ does not need to be higher than the average material payo¤ that emerged for ‘old’ type players if they interact only with homogeneous players of the ‘old’ type. The mutant can still do well even if the mutant’s material payo¤ is lower than the material payo¤ ( ; ) which ‘old’types achieve in the absence of any mutant. This can happen whenever the average material payo¤ of the ‘old’types is su¢ ciently reduced if the population is invaded by a mutant. What matters is the comparison of the mutant’s material payo¤ and the material payo¤ of the ‘old’types in a population with a mutant.

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Evolutionarily stable strategies

The following proposition characterizes the evolutionarily stable strategies of players in their roles of buyer/proposer and seller/responder: Proposition 1 A set of evolutionarily stable strategies is characterized by b=

1 4m2 2 4m2

2m 2m

1 m

(2)

vS : 2m

(3)

if m > 1, b 2 [0; 1=2] if m = 1, and (vS ) = vS +

1

Hence, sellers/responders accept an o¤er b if and only if vS 2 [0; VS (b)], where VS (b) =

8 > < > :

0 2bm 1 2m 1

1

if if if

b< 1 2m

1 2m

b 1 : b>1

(4)

Proof. First, we search for the mutant seller/responder strategy that maximizes the mutant’s relative material payo¤ in (1) for a given o¤er b. Second, we search for the mutant buyer/proposer strategy that maximizes (1), given that all players in the role of seller/responder use the strategy characterized by (4). Step 1 : seller/responder strategy mutation: Consider a population with 2m 1 ESS players and one mutant. This mutant chooses the ESS o¤er b if he is assigned the

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role of buyer/proposer.11 He chooses VSM (b) 6= VS (b) if he is in the seller/responder role. Thus, if the mutant acts as the buyer/proposer, he chooses b similar to an ESS player and gets an expected material payo¤ equal to Z

VS (b)

(vB

b) dvS = VS (b) (1

b) :

0

If the mutant acts as a seller/responder and chooses the mutant threshold level VSM , his expected material payo¤ is Z

VSM (b)

bdvS +

Z

1

VSM (b)

0

vS dvS = VSM (b)b +

1 1 2

2

VSM (b)

;

as the mutant interacts with an ESS buyer/proposer. The mutant has the buyer role and the seller role with probability 1=2; accordingly, E0

M

1 = VS (b) (1 2

b) +

1 1 1 VSM (b)b + 2 2

VSM (b)

2

(5)

:

The mutant behavior also in‡uences the average expected material payo¤ of a non-mutant. A non-mutant may have the role of buyer/proposer or seller/responder, and may interact with a non-mutant or a mutant. The following matrix summarizes possible matching combinations, their frequencies and payo¤s: matched co-player is ESS player is n

buyer/proposer

seller/responder

non-mutant VS (b) (1

b)

mutant VSM (b) (1

VS (b)b +

1 2

1

b)

1 2m 1

2m 2 2m 1

(VS (b))2

2m 2 2m 1

VS (b)b +

1 2

1

(VS (b))2

1 2m 1

For instance, the upper-left payo¤ is the payo¤ of a non-mutant player who is assigned the role of buyer/proposer and meets another non-mutant player who must then be a seller/responder. There are 2m 1 ‘other’players, and 2m 2 of them are non-mutants. Note that the payo¤ entries for the left and right lower cells are identi11

Note that here and in the following we omit the superscript “ESS” at all ESS choices but only indicate mutations from ESS by the superscript “M.”

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cal: If the mutant player is a mutation along the seller-responder dimension then in the role of buyer/proposer the mutant acts like a non-mutant. For a seller/responder non-mutant it therefore does not matter if the co-player is a mutant or not. Taking into account that any given non-mutant has a probability of 1/2 of becoming a buyer/proposer or seller/responder, the expected payo¤ of a non-mutant at stage 0 is E0

M

=

2m 2 1 1 VSM (b) (1 b) + VS (b) (1 2 2m 1 2m 1 1 1 + VS (b)b + 1 (VS (b))2 : 2 2

Maximizing the di¤erence E0 M order condition 1 b VSM 2

M

E0

b)

with respect to VSM yields the …rst-

1 1 (1 2 2m 1

b) = 0

which is solved for VSM (b) = VS (b) as given in (4). This function determines the …tness-maximizing choice in the role of the seller/responder for any given b. Inverting (4) on the relevant interval yields (3). Step 2 : buyer/proposer strategy mutation: Now consider a population with 2m 1 ESS players and one mutant who chooses the ESS value VS (b) if he is assigned to the role of seller/responder but deviates from the ESS strategy to bM 6= b if he is assigned the role of buyer/proposer. Analogous to the reasoning in step 1, we get the material payo¤ of the buyer/proposer mutant as E0

M

1 = VS bM 2

1

bM +

1 1 VS (b)b + 1 2 2

(VS (b))2

:

Moreover, the average expected material payo¤ of a non-mutant at stage 0 becomes E0

M

=

1 VS (b) (1 b) 2 h + 12 2m1 1 VS (bM )bM + 2m + 2m

2 1

VS (b)b +

13

1 2

1 2

1

1

2

V S bM (VS (b))2

:

Maximizing the di¤erence E0 condition 1 0 M V b 2 S

1

bM

M

M

E0

with respect to bM yields the …rst-order

VS bM 1 1 V 0 bM bM + VS bM 2 2m 1 S

Using (4), we get VS0 bM =

8 > < > :

0 2m 2m 1

0

if if if

b< 1 2m

VS bM VS0 bM

= 0: (6)

1 2m

b 1 : b>1

It is straightforward to verify that the di¤erence E0 M E0 M is independent of b for all b 2 = [1= (2m) ; 1] since all o¤ers b outside this interval are either rejected (if b < 1= (2m)) or accepted (if b > 1). Whenever m > 1, there is a unique interior solution to the …rst-order condition (6) which is given by bM = b as in (2). In this case, we also get an interior solution for VS (b). If m = 1 then the di¤erence E0 M E0 M is independent of bM for all bM 2 [0; 1=2] and strictly decreasing in bM for all bM > 1= (2m) = 1=2. But VS = 0 if bM 1=2. Therefore, if m = 1, there is a continuum of payo¤-equivalent equilibria in which b 2 [0; 1=2] and hence VS = 0. Intuitively, if m = 1, a player who is assigned the role of buyer/proposer cannot do better than o¤ering b = 1=2; but since this o¤er is rejected with probability 1, he can also o¤er less without a¤ecting his or any other player’s material payo¤. Higher o¤ers b > 1=2, however, strictly reduce a player’s …tness since they leave a large surplus to a seller/responder with a low material valuation vS . Before we proceed we discuss the properties of the ESS in Proposition 1. Note that (vS ) > vS for all vS < 1. This means that some o¤ers are rejected even if the responder’s valuation vS is strictly lower than these o¤ers b. This corresponds to a general insight obtained in the context of evolutionary stability in small populations: The concern for relative material payo¤ makes spiteful strategies evolutionarily stable. To illustrate, consider a seller/responder who considers to accept an o¤er b = vS + ", for small but positive ". This seller gains a very small material payo¤ compared to not selling the resource. The buyer/proposer who makes such an o¤er receives a potentially much higher rent if the seller accepts his o¤er, as he gets 1 b in case

14

of acceptance, and zero otherwise. Thus, for small " the harm imposed on such a buyer/proposer by rejecting outweighs the own sacri…ce from rejecting. The threshold function (vS ) for acceptance in (3) also shows that toughness is a function of the di¤erence between the buyer/proposer’s material payo¤ in case of acceptance, which is equal to 1, and the seller/responder’s material valuation vS . The divergence between the threshold (vS ) and vS is maximal for very small values of vS . For vS ! 1, the responder’s threshold converges toward his own valuation of keeping the resource that could be traded. We can also identify the role of a small population, compared to an in…nitely large population. It holds that (vS ) ! vS if m ! 1; hence, all o¤ers with b vS are accepted if the size of the population is su¢ ciently large. This outcome is similar to other applications of the concept of evolutionary stability in …nite populations. The negative impact on the material payo¤ of the buyer which the seller has by rejecting the o¤er is less important if there is a large population with many players who are not directly a¤ected by the particular choice of a given buyer/proposer or a given seller/responder. The buyer/proposer o¤er in (2) is less straightforwardly in line with the general intuition for evolutionary stability in small groups. The buyer’s choice can be seen as an accommodating reaction to the reluctance of the seller/responder to accept low price o¤ers. As the seller requires a higher own material payo¤ and does not accept otherwise, the buyer o¤ers to pay a higher price b. But accommodating the seller’s tough reservation price only partially explains the buyer’s o¤er behavior. The o¤er b in (2) is also shaped by concerns for relative standing: The o¤er is strictly lower than the o¤er which would maximize the buyer/proposer’s absolute material payo¤ for a given ESS of the seller/responder as in (4). As is straightforward to con…rm, the best response to (4) that maximizes own material payo¤ would be ~b = 1 + 1 > b. 2 4m The buyer/proposer does not want to o¤er as much as ~b since such a high o¤er bene…ts the seller and thereby reduces the buyer’s evolutionary …tness. Finally, as for the seller/responder, the evolutionary e¤ect for the buyer/proposer o¤er vanishes if the population size increases, which is also a consequence of the change in the responder strategy: limm!1 b = 1=2 (see below). 15

Comparison with standard equilibrium concepts. The take-it-or-leave-it offer bargaining game has been studied extensively and its solution has been used in many contexts to describe bargaining with incomplete information. We may contrast the evolutionarily stable strategies with Bayesian Nash equilibrium and with perfect Bayesian equilibrium (PBE) for rational players who aim at maximizing their own material payo¤. The simultaneous choices of b by the buyer/proposer and of the acceptance rule by the seller/responder as for Bayesian Nash equilibrium may correspond to the consideration of one-step deviations in the analysis of ESS. However, we concentrate on perfect Bayesian equilibrium, as the set of Bayesian Nash equilibrium is large and does not address the sequential nature of decision-making.12 For the parametric case which we consider here, sequential rationality requires that the o¤er is accepted by the seller/responder if and only if vS b = VSN (b). Here and below, the superscript “N ” refers to variables relating to the PBE, to distinguish the PBE values from the equilibrium values of ESS that have no superscript, or a superscript “ESS,” where this may help for a distinction. Given the belief about the seller’s reservation price, the PBE o¤er made by the proposer is bN = 1=2. Before proceeding note that there is some possible confusion about the meaning of ‘type’ here. The seller’s reservation price is what is referred to as his ‘type’ in the context of standard concepts of Bayesian equilibrium, whereas in the context of evolutionary equilibrium, a player’s type is identi…ed with the player’s actual action (compare also the discussion in the introduction). Contrasting the outcome in the ESS and the PBE, we …rst note that lim bESS = bN and lim VSESS (b) = b = VSN (b) :

m!1

m!1

Figure 2 compares the PBE with the ESS for di¤erent population sizes, illustrating the divergence of the ESS o¤er bESS and the critical ESS valuation VSESS bESS , which determines acceptance of the ESS o¤er. The …gure o¤ers two insights. First, the vertical distance between the two curves bESS and VSESS represents the minimum rent which the responder requires for himself to be willing to accept the 12

The set of Bayesian Nash equilibria is rather large. For instance, the seller/responder may choose to reject any price o¤er b 6= b0 for some b0 , and to accept b0 if and only if vS b0 . The buyer/proposer’s optimal reply to this strategy is b0 for any b0 1. In turn, the fact that the buyer always proposes b = b0 makes rejecting all prices b 6= b0 a best reply for the seller. Most of these equilibria do not survive reasonable re…nements or sequential rationality. This is why we focus on the perfect Bayesian equilibrium.

16

Figure 2: Comparison of the evolutionarily stable equilibrium and the perfect Bayesian equilibrium. o¤er. This minimum rent is zero in the PBE since bN = VSN = 1=2 in this case. But acceptance of an o¤er also grants a rent to the proposer, and this matters for evolutionary stability since a positive rent of the proposer reduces the evolutionary …tness of the responder. Therefore, the ESS strategy VSESS requires a strictly positive rent in order to accept an o¤er, and this minimum rent is decreasing in the number of players. Second, the buyer’s o¤er interacts with the ESS strategy of the seller/responder. For m = 1, the ESS o¤er is equal to 1=2 and is thus equal to the o¤er in the PBE, but for any larger population size m, the evolutionarily stable strategies are characterized by an o¤er that is higher than 1=2.13 The relationship between the ESS o¤er b and m is non-monotonic; the ESS o¤er b is increasing for m < 2, has a maximum at m = 2 and is then decreasing in m, converging to 1=2 for very large m. Intuitively, one main force that drives b away from 1=2 is the strong deviation of the seller/responder from the PBE value. In the limit case of m = 1, even sellers with a very low vS require a high o¤er in order to accept; but it is evolutionary stable for the buyer not to o¤er such high amounts. If the population size increases, the seller/responder 13

Note that, if m = 1, then VS (b) = 0 and b is rejected with probability one. In fact, if m = 1 then the buyer/proposer is indi¤erent between all b 2 [0; 1=2]; see the proof of Proposition 1 for details.

17

claims a lower surplus which makes it “cheaper” for the buyer/proposer to ensure a given probability of trade. The o¤er bESS takes into account two countervailing forces: (i) own relative standing concerns which are strongest if m is small; and (ii) an accommodating reaction to the rejection behavior of the responder which is strongest if m is small. The sum of these two e¤ects need not be monotonic in m and this can cause the hump in Figure 2. Probability of trade and ex post e¢ ciency. Now we can analyze the e¢ ciency properties of the bargaining outcome in ESS and compare it with the PBE. Proposition 2 Let ! be the probability of trade for ESS and W be the expected total material payo¤ in the ESS for a single pair of players. It holds that !=

1 1 2 4m 3

1 2

and W =

1 2

1 (2m 2 (4m

(7)

2)2 2m + 3)2 4m

Both ! and W are increasing in m with limm!1 ! = WN.

2 : 3 1 2

= ! N and limm!1 W =

7 8

=

For the proof of Proposition 2 see the appendix. As is well-known from Schaffer (1988) and the applications of his de…nition in a number of contexts, in …nite populations it is not the absolute material payo¤ but a type’s increase in relative material payo¤ that matters. In the bargaining context, this makes the sellers/responders “tough,” that is, more reluctant to accept o¤ers. The evolutionarily stable buyer/proposer strategy accommodates the seller/responder behavior, although relative payo¤ concerns prevent the buyer/proposer from making “too high” o¤ers. Overall, for the probability of trade and for the total expected material payo¤, the accommodating e¤ect of the buyer/proposer is weaker than the increased toughness of the seller/responder. This widens the interval of material valuations for which trade does not take place, compared to the outcome in the PBE. In other words, the probability of trade ! and the aggregate resource rent W are strictly lower for ESS than in the PBE, but converge to the PBE values when m becomes very large. The analysis in this section focused on o¤ers made by the uninformed buyer, giving all bargaining power to the uninformed player. This is the more relevant case 18

for an e¢ ciency comparison between ESS and PBE because the case of uninformed proposers and privately informed responders is the standard case known to cause ine¢ cient failure to trade also in the PBE. However, it is an extreme case. The complementary case allocates all bargaining power to the informed player, which is the seller in our setup. We now turn to this problem.

4 4.1

Extensions Informed proposers

Suppose that, unlike in the previous sections, the informed seller is the player who proposes a price for the buyer to pay, and the uninformed buyer decides whether or not to accept it. Apart from this relocation of bargaining power, we keep all other building blocks of the formal framework unchanged. The seller has a valuation vS 2 [0; 1] which is drawn from a uniform distribution. The seller knows his own valuation, the buyer knows the distribution. The buyer has an exogenously given valuation vB = 1 which is common knowledge. Evolutionarily stable strategies. We consider the following set of possible evolutionary strategies. The seller/proposer sets a price p, which may depend on the seller’s true material valuation vS of keeping the good. Accordingly, strategies of the seller/proposer are de…ned as functions p(vS ) : [0; 1] ! [0; 1) : The strategies of the buyer/responder are de…ned by threshold values 2 [0; 1). The threshold is de…ned such that the buyer/responder accepts all prices for which p and rejects all prices for which p > .14 An evolutionary strategy (p (vS ) ; ) consists of a seller/proposer strategy and a buyer/responder strategy. This leads to the following characterization of the evolutionarily stable strategies: Proposition 3 There is a continuum of evolutionarily stable strategies (p (vS ) ; ) such that a buyer/responder accepts the o¤er price p if p and rejects the o¤er 14

We could allow for an acceptance strategy of the buyer/responder that is non-monotonic in p. This may potentially lead to further ESS, but this does not justify the increase in complexity of the analysis caused by the possibility of arbitrary, non-monotonic acceptance strategies.

19

otherwise and a seller/proposer o¤ers to sell for p = for p = 1 if vS > PS , where PS is de…ned as PS = and where

2m 2m 1

1 : 2m 1

8m2 8m + 1 : 8m2 6m

if vS

PS and o¤ers to sell

(8)

(9)

A proof of Proposition 3 is in the appendix. Some properties of evolutionarily stable strategies in Proposition 3 are intuitively straightforward. The buyer/responder does not know the true valuation vS of the seller/proposer but must choose whether to accept the price on the basis of the o¤er price p only. The evolutionarily stable acceptance strategy by ESS buyers/responders is characterized by a threshold price : Buyers buy if the price o¤ered does not exceed this threshold price. For any ESS, the threshold price is strictly smaller than the buyer’s material valuation vB = 1 (compare (9)). The buyer’s acceptance behavior, in turn, a¤ects the considerations for the seller. Taking into account that the buyer/responder has a threshold strategy , the optimal o¤er strategy of the seller/proposer can make use of two possible o¤er prices only: p 2 f ; 1g. Price o¤ers higher than all lead to rejection (and are all equally pro…table), and price o¤ers p < would be accepted, but are inferior to o¤ers of p= . Whether a seller/proposer o¤ers to sell for p = < 1 (in which case he can sell with probability 1) or for p = 1 (essentially deciding not to trade) depends on the bene…t from trade for himself and for the buyer/responder. From an evolutionary perspective, trade is worthwhile for the seller/proposer only if the own bene…t from trade is su¢ ciently high in comparison to the bene…t from trade at p = for the buyer/responder. Since the buyer’s material bene…t is independent of vS but the seller’s material bene…t from trade is lower the higher vS , the comparison of costs and bene…ts explains the threshold for the seller’s valuation in (8): From an evolutionary point of view, it is bene…cial for a seller/proposer to sell for a price p = only if the own valuation of no trade is su¢ ciently small: vS PS . Moreover, it is intuitive that the critical threshold PS for the seller’s valuation is an increasing function of the possible transaction price p = : A higher transaction price for which the seller/proposer can sell the resource increases the seller’s own material advantage 20

Figure 3: The set of evolutionarily stable equilibria in case of informed proposers. from selling compared to keeping the good, and it lowers the buyer’s material bene…t from this purchase. The ESS value of PS depends on ; in fact, for a given responder threshold , the seller’s o¤er function is uniquely determined. But the value of also depends on the ESS value of PS . This is the case because when is determined the buyer/responder solves a similar trade-o¤ as the seller/proposer as to whether to trade or not: For any price p < 1, trade o¤ers a material reward to the buyer/responder. But trade also o¤ers a material reward to the seller if p > vS , and this is bad for the buyer/responder from an evolutionary point of view. Closer inspection of the proof of Proposition 3 shows why there is a whole range of combinations (PS ; ) that characterize evolutionarily stable strategies. If is large, the material reward for the seller is large, and the material reward for the buyer is low. Hence, there is an upper bound on the evolutionarily stable ; for higher values of , mutations in the role of the buyer that lead to the breakdown of trade would be evolutionarily advantageous. This explains the upper bound on in (9). The set of evolutionarily stable strategies is a function of the size of the population n = 2m, as illustrated in Figure 3. The solid function max (m) in Figure 3 represents max condition (9); all (p (vS ) ; ) with 0 and the corresponding threshold PS for the seller’s valuation as in (8) are evolutionarily stable. Since vS is a draw from 21

the uniform distribution on the unit interval, the threshold value PS is equal to the probability of trade. The dashed function ! max (m) in Figure 3 maps the highest achievable probability of trade for ESS, that is, equation (8) evaluated at the upper bound on , which yields 4 (m 1) : (10) ! max = 4m 3 The upper bound ! max on the probability of trade is equal to zero for m = 1 and converges to one if m ! 1. Evolutionary considerations of the material payo¤ relative to the payo¤ of the matched partner are strongest in small populations. However, if there are many other players to compare with, it is optimal for players to disregard the co-player’s payo¤ and maximize absolute own material payo¤. In the latter case, trade with a probability close to 1 can be achieved. The vertical di¤erence between the functions max and ! max also illustrates the fact that PS < : There is a range of valuations vS 2 (PS ; ) for which trade would yield a material bene…t to the seller but for which the seller decides not to trade. Comparison with the perfect Bayesian equilibrium (PBE). In close analogy to the ESS, there is a set of Bayesian Nash equilibria15 and a perfect Bayesian equilibrium for players who maximize their absolute material payo¤. Sequential rationality requires that the buyer/responder accepts all price o¤ers for prices smaller than or equal to 1 and rejects all prices higher than 1. This leads to the PBE characterized by an o¤er pN = 1 and a choice by the buyer/responder to accept all price o¤ers smaller than or equal to 1. Trade always takes place and is e¢ cient in this PBE, given our assumptions about vS 2 [0; 1] and vB = 1. For …nite m, however, the set of ESS does not include the e¢ cient case of (PS ; ) = (1; 1). Intuitively, the buyer/responder gets a material payo¤ of zero if trade takes place at (1; 1), that is, just as much as if no trade takes place. However, the buyer can reduce the seller’s material payo¤ (who would get = 1 in case of trade) by choosing not to trade: In the “no trade” case the expected material payo¤ of the seller is E (vS ) = 1=2 < 1. 15

This set is, again, very large. Suppose, for instance, that the buyer/responder chooses to accept a price o¤er for a particular p0 2 [0; 1] but rejects any other price o¤er, and the seller/proposer o¤ers this price p0 if vS p0 and demands a price of 1 or higher otherwise. It is straightforward to con…rm that these strategies are optimal replies to each other, if chosen simultaneously. Note that the set of Bayesian Nash equilibria includes the PBE and that the PBE is the most e¢ cient equilibrium in this set.

22

4.2

More general distributions of valuations

The case of uniformly distributed valuations of the previous sections has the advantage that it allows for closed-form solutions of the evolutionarily stable strategies and a meaningful comparison with the perfect Bayesian equilibrium. We show in this section how the intuition behind the results obtained carries over to more general probability distributions of the seller’s valuation. Consider the setup from Section 2 where the buyer/proposer makes an o¤er b to the seller/responder who values the resource by vS . Suppose that vS is drawn from a continuous and twice di¤erentiable cumulative distribution function F (vS ) with full support on [0; 1]. As we show in more detail in Appendix A.3 the choice of the seller/responder remains as in Proposition 1: The sellers’evolutionarily stable threshold function for acceptance is such that sellers accept an o¤er b if and only if vS 2 [0; VS (b)] where VS (b) =

8 > < > :

0 2bm 1 2m 1

1

if if if

b< 1 2m

1 2m

b 1 : b>1

Intuitively, for the seller’s choice the distribution of valuations vS within the population does not matter since his choice only a¤ects his own material payo¤ and the material payo¤ of the buyer who he is matched with. Similarly, the acceptance threshold of the seller/responder in the perfect Bayesian equilibrium remains equal to VSN (b) = b. The evolutionarily stable choice in the role of the buyer/proposer depends on the shape of the distribution function F of valuations of the sellers within the population. Compared to the perfect Bayesian equilibrium (PBE), the analysis in Appendix A.3 con…rms two countervailing e¤ects on the buyer’s o¤er behavior in evolutionarily stable strategies. First, the buyer’s ESS o¤er is in‡uenced by concerns for relative standing: If the seller’s acceptance threshold were the same as in the PBE the buyer would choose a lower o¤er in ESS than in PBE. Second, the buyer’s ESS o¤er accommodates to the seller’s reluctance to accept low o¤ers in the ESS (where VS (b) < b). These two e¤ects caused by considerations of evolutionary stability have already been identi…ed in Section 3 for uniformly distributed valuations of the seller; in case of a general distribution of valuations it is, however, not clear in general which of the two

23

e¤ects dominates. While for uniformly distributed valuations the ESS o¤er is higher than the o¤er in the PBE (compare Figure 2), Appendix A.3 contains examples for which the opposite is true if the population size becomes very small. Overall, the assumption of uniformly distributed valuations has clear analytical advantages but is not crucial for the main economic intuitions for the results obtained.

4.3

Trade across populations

In the main part of the paper we allowed all individuals to belong to one population only. We now consider a slightly di¤erent setup. We may think of two populations on two separate islands, without geographical mobility. Ultimatum bargaining may take place between the two islands, that is, between members of these two non-overlapping populations, denoted by H and K. Each population has its own population dynamics. Material payo¤ of individuals in one population may be irrelevant for the population dynamics in the other population, and vice versa. Members of the two populations may trade with each other, but they do not compete for …tness with each other. Our de…nition of ESS in the main setup compares the mutant’s expected material payo¤ with the average material payo¤ of the population which he belongs to, and all players belong to the same population. The departure from material payo¤ maximization emerges in this setup because players have an incentive to harm their bargaining partner, as this partner is a member of their own population. If, instead, players interact with players who are not in their comparison group, the departures for ESS from material payo¤ maximization disappear, even if we stick as closely as possible to the main setup. We have to adjust notation to account for two disjoint non-overlapping populations of equal size, denoted by H and K. Each population has a constant number of n = 2m players in each generation. The state game consists of pairwise ultimatum bargaining. But unlike in Section 2, for each interaction one player of H is randomly paired with one player of K. Within each two-player subgroup, nature randomly assigns the roles of seller and buyer. Each player has the same probability of 1/2 of being assigned the role of seller or buyer. The existence of two populations, and the formation of buyer-seller pairs across, rather than within these populations is the key di¤erence compared to our main setup. All other aspects remain as in the framework of Section 2. In particular, the seller owns a resource which, if he keeps it, gives him material

24

value vS . This material value vS is an independent draw from a uniform distribution on the unit interval [0; 1] and is the seller’s private information. The buyer attributes a material value vB = 1 to obtaining possession of the resource; vB is commonly known. The two players who are paired interact in ultimatum bargaining where the buyer makes a take-it-or-leave-it o¤er to the seller. Evolutionary strategies consist of a vector (bH ; H (vS ); bK ; K (vS )) with four components. Here, bH and bK are real numbers and the functions G (vS )

: [0; 1] ! [0; 1) for G 2 fH; Kg

describe threshold strategies: A seller/responder from group G 2 fH; Kg with a material valuation vS of keeping the good accepts an o¤er b G by a buyer/proposer from population group G if and only if b G G (vS ). As in the main setup we restrict consideration to pure strategies and strictly increasing threshold functions G . We also de…ne the functions VGS ( ) as the inverse functions of G (vS ) where these exist, denoting the threshold for vS below which the seller/responder is willing to accept a given o¤er b of the buyer. De…ning evolutionary stability. For a de…nition of evolutionarily stable strategies (ESS) and one-step mutations, we have to account for the richer setup with two populations. Suppose that 2n 1 = 4m 1 players (i.e., all players in one population, and all but one player in the other population) behave according to the candidate ESS strategies described by H = (bH ; H ) and K = (bK ; K ). We consider mutations of a player M who is a member of population G 2 fH; Kg. As M may be assigned the role as proposer/buyer or as seller/responder, we distinguish between one-step proposer mutations and one-step responder mutations. Denote by M G the strategy of player M . This player may either choose the ESS G = (bG ; G ), or deviM ate and choose some M G = (bG ; G ) as a buyer mutant, or deviate and choose some M M M M G ) the strategy pro…le G = (bG ; G ) as a seller-mutant. We denote by ( G ; G ; M if individual M of population G chooses G but all remaining individuals choose the candidate ESS. Focusing on evolutionary …tness in …nite populations within the concept of Scha¤er (1988), we are interested in the players’ex ante expected material payo¤s. We de…ne M M by E0 M G ) the expected material payo¤ of player M from group G G( G; G ;

25

given the state of information at stage 0, that is, before subgroups and roles are assigned. Then, we can de…ne ESS in analogy with De…nition 1: De…nition 2 The strategy that is played by all individuals in both groups G = H; K is an evolutionarily stable strategy (“ESS”) if there is no one-step mutation M G from G such that E0

M M ; G(

M G ;

G)

E0

M M ; G (

M G ;

G)

> 0:

(11)

This condition de…nes E0 GM ( M ; GM ; G ) as the expected material payo¤ of those players other than M who are members of the same population G 2 fH; Kg as M . This says that the population dynamics is governed by the relative material payo¤s of members within the same population. Material payo¤s of players in group K do not govern evolutionary success of types inside population H, and vice versa. Characterization of ESS. The following proposition characterizes the evolutionarily stable strategies of players in their roles of buyer/proposer and seller/responder in the modi…ed setup: Proposition 4 A monomorphism of evolutionarily stable strategies exists that is characterized by b = 1=2 and (vS ) = vS . This is shown in the Appendix. Players’ESS in Proposition 4 does not deviate from the maximization of their material payo¤s. This is in contrast to the deviations observed in small groups in Section 3. The deviations from material payo¤ maximization were driven by the fact that a player could increase his evolutionary …tness by imposing harm on other players in his own population. This made players willing to choose strategies as seller or buyer by which they sacri…ced own material payo¤, provided that the deviation strategy caused enough harm on members of his own population. In the modi…ed setup of this section, all transactions take place between members of di¤erent populations. Harming members of a di¤erent population, however, does not increase own …tness. If a player cannot in‡uence the average material payo¤ of other members in his population, and his own …tness depends only on own material payo¤ compared to the material payo¤ of members of his own population, the maximization of own material payo¤ also maximizes evolutionary …tness.

26

The modi…ed setup illustrates three important aspects. First, consideration of interaction in small groups is not su¢ cient to cause evolutionarily stable behavior that deviates from maximization of own material payo¤. It is important that the deviating behavior hurts members of the own comparison group. Second, the result in Proposition 1 is not driven by the assumption that all players may be in the roles as buyers or sellers with equal probability of 50 percent. The same assumption applies in Proposition 4, but the outcome is very di¤erent. Third, we note that the proof of Proposition 4 does not make use of the assumption that precisely half of the population in each group is attributed the role as buyer/proposer and half of them is attributed the role as seller/responder. The result in Proposition 4 also holds if all players in population H are buyers/proposers and all players in K are sellers/responders, or for any other combination of shares.

5

Conclusions

The evolutionarily stable equilibrium of the ultimatum o¤er bargaining game with one-sided incomplete information is characterized by a low probability of trade, compared to perfect Bayesian equilibrium. The breakdown of trade is a well-known possible outcome in the perfect Bayesian equilibrium of ultimatum bargaining if the uninformed party makes an ultimatum o¤er to a privately informed responder. For evolutionarily stable strategies, the parameter range in which trade takes place is even more limited than for perfect Bayesian equilibrium if the uninformed player makes the o¤er. This result carries over to the case where the informed player makes the ultimatum o¤er. The main reason behind the low-trade result is that players reject trade o¤ers that would increase their own material payo¤, if this hurts the rest of their own population su¢ ciently strongly. In the state game of the main setup we study, each player is matched with another player in the same population, and trade within these two-player subgroups is assumed to be e¢ ciency-enhancing. In this context evolutionarily stable behavior leads to tough bargaining behavior and to the rejection of too low o¤ers. From the outside, this behavior may be interpreted as “revenge”or costly punishment for an o¤er that is perceived as unfair. But what is behind the rejection behavior is the straightforward evolutionary stability logic developed by Scha¤er (1988): Rejecting o¤ers makes a responder relatively better o¤ than trading if trade mainly bene…ts the proposer 27

and gives the responder a much lower rent. The rejection of such o¤ers reduces the responder’s payo¤, but it reduces the proposer’s material payo¤ by even more. If material payo¤ relative to that of other players matters, players are more reluctant to accept o¤ers that give them only a small own material payo¤, in particular if they expect that acceptance will give the other side a large own material payo¤. The rejection behavior, in turn, a¤ects the evolutionarily stable o¤er-making. In the role of proposer, players anticipate the responder’s reluctance to accept. This has implications for the proposer’s choice and tends to make the evolutionarily stable o¤er more generous. This is true both if the proposer is the informed player and if the proposer is the uninformed player. But in both cases the o¤er is also shaped by the evolutionary considerations of the proposer. Proposers are reluctant to make concessions and to leave a too high share of the surplus to the responder. Overall, in the framework considered the accommodating behavior of the proposer is not su¢ cient to compensate for the responder’s increased demand. Thus, the evolutionarily stable probability of trade may be lower than in the corresponding perfect Bayesian equilibrium, and this e¤ect is strongest in small populations.

A A.1

Appendix Proof of Proposition 2

Equation (4) characterizes the critical level of vS for which the ESS o¤er b is accepted. As vS is uniformly distributed on [0; 1], the probability of trade (7) is obtained directly from inserting b in (2) into (4). This probability is strictly increasing in m with limm!1 ! = 1=2. In comparison, in the PBE, the equilibrium o¤er is bN = 1=2 and this o¤er is accepted if vS 1=2, which has a probability equal to 1=2. The aggregate resource rent that emerges from the take-it-or-leave-it o¤er in the PBE is Z 1 7 N N W = Pr vS b vB + vS dvS = : 8 bN

28

The expected aggregate resource rent for a given pair of players in the ESS is 1 2

W = 1 = 2

1 1 2 4m 3 1 (2m 2 (4m

vB +

2)2 2m + 3)2 4m

Z

1 1 2

1 1 2 4m 3

vS dvS

2 : 3

For m ! 1 this term converges to W N = 7=8. For smaller m the aggregate rent is smaller. For instance, for m = 1 we have W = 1=2: As the probability for trade is zero in this case, the resource stays with the seller/responder who has an expected rent of 1=2. Since 4m 2 @W = > 0; @m (4m 3)3 we can also con…rm that the aggregate resource rent is monotonically increasing in m in the range m 2 f1; :::1g.

A.2

Proof of Proposition 3

We assume that 2m 1 players follow the candidate ESS strategy (PS ; ) and consider one-step mutations. First we search for the mutant buyer/responder strategy that maximizes the mutant’s relative material payo¤ as de…ned in (1) for a given ESS behavior of all other players. Then we search for the mutant seller/proposer strategy that maximizes (1), given that all other players use the ESS candidate strategy. Part 1 : buyer/responder strategy mutation: Consider a population with 2m 1 ESS players. Their seller/proposer behavior is characterized by a critical value PS such that p = for vS PS and p = 1 for vS > PS . Their buyer/responder strategy is described by a threshold value for the buyer/responder such that the price o¤er is accepted if and only if p . Let there be one mutant who chooses the ESS candidate proposer strategy in the role of seller/proposer, but chooses a possibly di¤erent threshold price M if the player is in the buyer/responder role. This mutant’s expected material payo¤ at stage 0 is E0

M

=

(

1 E 2

ESS + S 1 E 2

1 [PS (vB 2 ESS +0 S

)] if if

M M

> <

:

With probability 1=2, the mutant has the role of a seller/proposer, in which he does 29

not behave di¤erently from an ESS player. The term E ESS is de…ned as the expected S payo¤ of an ESS player in the role of seller/proposer if all players behave according to the ESS candidate strategy. This is what the mutant gets in the role of the seller/proposer. With the remaining probability of 1=2, the mutant has the role of a buyer/responder. In this role he chooses M , where potentially M 6= . The mutant’s material payo¤ in this role depends on the direction in which M deviates. A deviation to a higher acceptance threshold M does not a¤ect the mutant player’s own expected material payo¤ if all other players stick to the ESS candidate strategy (in the role of a buyer he still gets PS (vB ) where PS is the probability 16 that the seller chooses a price p = ). Moreover, for < M < 1, the material payo¤ of all other players is also una¤ected if they stick to their evolutionarily stable strategies. For M 1, the material payo¤ of all other players is increased. Thus, deviations to M > weakly reduce the mutant’s relative material payo¤. Second, a deviation to a lower acceptance threshold M < causes the breakdown of trade, as the seller proposes a price p 2 f ; 1g. This reduces the mutant’s expected material payo¤ by 1 1 PS (vB ) = PS (1 ); 2 2 compared to a choice of the ESS threshold . But it also a¤ects a non-mutant’s material payo¤ if he is in the group with the mutant (which happens to non-mutants with probability 1= (2m 1)) and the mutant is assigned the role of the buyer/responder (which happens with probability 1=2). Thus, the elimination of trade by a choice of M < reduces the non-mutants’expected material payo¤ by 1 1 2 2m 1

Z

0

PS

(

1 1 vS )dvS = 2 2m 1

PS

(PS )2 2

:

Altogether, a deviation to M < eliminates trade and, hence, reduces both the mutant’s expected material payo¤ and that of the average ESS player. Such deviations bene…t the mutant type if the expected average loss of ESS players exceeds the 16

Note that for M 1, the mutant’s own material payo¤ in the role of the buyer is equal to PS (vB ) + (1 PS ) (vB 1) = PS (vB ), even though the probability of trade is di¤erent in cases < M < 1 and M 1. In the cases in which trade occurs for M 1 but not for < M < 1, this trade occurs at a price p = 1 = vB .

30

expected loss of the mutant, that is, if 1 1 2 2m 1

(PS )2 2

PS

1 > PS (1 2

).

This comparison reduces to PS < 2

4m + 4m

and de…nes a range (PS ; ) in which a deviation from toward deviation from the ESS candidate strategy. In turn, if PS

2

4m + 4m ,

M

<

is a pro…table

(12)

no pro…table deviation from exists and the choice of the ESS threshold yields the maximum relative payo¤. In other words, there is a range of (PS ; ) for which a mutation that deviates from the ESS candidate buyer/responder strategy is not evolutionarily advantageous. Intuitively, (12) states that the price o¤er p = cannot be too high. Part 2 : seller/proposer strategy mutation: Consider a population with 2m 1 players who behave according to the ESS candidate strategy and one mutant who chooses the ESS candidate value of in the role of buyer/responder but deviates to a di¤erent function P M (vS ) in the role of seller/proposer. Analogous to the reasoning in Part 1, price o¤ers p below are all accepted by buyer/responders who follow the ESS candidate strategy. Compared to p = , such o¤ers lead to a lower material payo¤ of the mutant player and a higher average payo¤ of non-mutant players; hence, such deviations are evolutionarily disadvantageous. Price o¤ers p above are not accepted by the ESS responders and lead to a breakdown of trade. Thus, the optimal mutant seller/proposer strategy is a threshold strategy with p = for vS PSM , leading to trade at the largest achievable price in the range vS 2 [0; PSM ], and with p > for vS > PSM , leading to a breakdown of trade in the range vS 2 PSM ; 1 . We search for the optimal threshold PSM , which is determined as the critical valuation vS at which the relative bene…t from trade at p = equals the relative bene…t from no trade. As the mutant is in the role of a seller/proposer with probability 1=2, the mutant’s

31

expected material surplus from trade compared to no trade is 1 2

Z

0

PSM

(

1 vS )dvS = 2

PSM

(PSM )2 2

:

A mutation in PS a¤ects the expected payo¤ of the non-mutant player who is teamed up with the mutant (which happens to a non-mutant with probability 1= (2m 1)) in case the mutant is assigned the role of seller/proposer (which happens with probability 1=2). This non-mutant’s expected material surplus, if in the role of the buyer, is equal to PSM (1 ). Since all other non-mutant players are not a¤ected by the mutant’s choice PSM , the mutant’s expected additional relative payo¤ from trade compared to no-trade is (PSM )2 1 1 1 PSM P M (1 ); 2 2 2 2m 1 S which is maximized for PSM =

2m 2m 1

1 : 2m 1

Accordingly, a pro…table mutation in the role of a seller/proposer does not exist if PS ( ) =

2m 2m 1

1 : 2m 1

(13)

This de…nes the ESS choice in the role of the seller/proposer as a function of the buyers’ acceptance threshold such that no …tness-increasing deviation exists for mutants in the role of a seller.17 The conditions (12) and (13) jointly determine combinations of PS and such that no evolutionarily advantageous deviations exist in the role of the seller or the role of the buyer. Inserting equation (13) for the seller’s choice into the “no deviation” condition (12) for the buyer yields an upper bound for (as given in (9)) below which no pro…table deviations in the role of the buyer/responder are possible. The combinations (PS ; ) de…ne a set of strategies that ful…ll the criterion in De…nition 1 for an ESS, and there is a continuum of responder acceptance thresholds and corresponding o¤er functions p (vS ) that are evolutionarily stable. 17

Note that the set of equilibria characterized in Proposition 3 also includes equilibria in which trade never occurs. More precisely, PS ( ) < 0 if is small; the corresponding ESS choice in the role as seller/proposer is p (vS ) = 1 for all vS 2 [0; 1].

32

A.3

Details on the case of a general distribution of valuations

Consider the setup from Section 2 and suppose that vS is drawn from a continuous and twice di¤erentiable distribution function F (vS ) with full support on [0; 1]. We consider evolutionary strategies that consist of an o¤er b of the buyer and a threshold strategy (vS ) where the function VS (b) denotes the inverse of and characterizes the threshold for the valuation vS below which the seller accepts the o¤er. Choice in the role of the seller/responder. Suppose that 2m 1 players follow the ESS strategy and denote by ESS and ESS the material payo¤s of ESS S B players in the role of seller and buyer, respectively, if they are matched with another ESS player. Consider a mutant who chooses a threshold VSM (b) when being assigned the role of the seller and the ESS o¤er b when being assigned the role of the buyer. Then, the mutant’s expected material payo¤ at stage 0 is E0

M

1 = 2

Z

VSM (b)

bdF (vS ) +

Z

1

vS dF (vS )

VSM (b)

0

!

+

1 2

ESS B :

A non-mutant’s expected material payo¤ at stage 0 is M

E0 De…ning

M S

@ M S @VSM

=

1 2

:= E0

1 2

ESS S

+

M

E0

1 F VSM (b) (vB 2m 1 M

b) +

2m 2m

2 1

ESS B

:

and using vB = 1 we get

1 1 1 b VSM (b) F 0 VSM (b) F 0 VSM (b) (1 2 2 2m 1 1 0 M 2mb 1 F VS (b) VSM (b) : = 2 2m 1 =

b) (14)

Since (2mb 1) = (2m 1) 2 [0; 1] for b 2 [0; 1], (14) is strictly positive for VSM (b) < 2mb 1 1 and strictly negative for VSM (b) > 2mb for any b within the support of vS . 2m 1 2m 1 Therefore, the optimal acceptance threshold is

VS (b) =

8 > < > :

0 2bm 1 2m 1

1

if if if

33

b< 1 2m

1 2m

b 1 ; b>1

(15)

just as in (4). Choice in the role of the buyer/proposer. Suppose that 2m 1 players follow the ESS strategy and consider a mutant who chooses an o¤er bM when being in the role of the buyer, and chooses the ESS acceptance threshold VS (b) given in (15) when being in the role of the seller. The mutant’s expected material payo¤ is E0

M

=

1 2

ESS S

1 + F VS bM 2

1

bM :

A non-mutant’s expected material payo¤ is E0

M

" 1 1 = 2 2m 1 + M B

De…ning @ M B @bM

1 2

Z

VS (bM )

bM dF (vS ) +

Z

1

vS dF (vS )

VS (bM )

0

!

2m + 2m

2 1

ESS S

#

ESS : B

:= E0

M

E0

M

we get

1 0 F VS bM VS0 bM 1 bM F VS bM 2 1 1 F VS bM + bM F 0 VS bM VS0 bM 2 2m 1 VS bM F 0 VS bM VS0 bM 1 1 = F 0 VS bM VS0 bM (2m 1) 1 bM 2 2m 1 1 1 F VS bM ((2m 1) + 1) 2 2m 1

=

bM

VS bM (16)

The corresponding …rst-order condition for the optimal choice of bM is F 0 V S bM

VS0 bM

(2m

1) + VS bM

2mbM = F VS bM

2m;

which, using the ESS choice of the seller/responder as given in (15) and denoting bM = b, is equivalent to 4m2 4m (1 4m2 4m + 1

b) =

34

F VSESS (b) ; F 0 (VSESS (b))

(17)

in case of an interior solution with b 2 (1= (2m) ; 1).18 For m > 1, since the left-hand side of (17) is strictly decreasing in b, a su¢ cient condition for a unique solution to (17) is @ F (z) 0 for all z 2 (0; 1) : (18) @z F 0 (z) Comparison to the perfect Bayesian equilibrium. An analogous analysis shows that the seller’s optimal choice in the perfect Bayesian equilibrium (PBE) is to accept an o¤er b if and only if vS b. Using this threshold function, the …rst-order condition for the buyer’s optimal o¤er is given by 1

b=

F (b) F 0 (b)

(19)

and characterizes a unique maximum if (18) holds. The comparison of (17) and (19) reveals two e¤ects. First, taking the seller’s acceptance behavior as …xed, if the seller/responder were to follow the PBE strategy and choose VS (b) = b in the ESS then for a given o¤er b the right-hand side of these two …rst-order conditions would be the same. But the left-hand side of (17) is smaller than the left-hand side of (19). Therefore, in case of identical responder behavior, the ESS o¤er of the buyer/responder would be lower than the corresponding o¤er in the PBE. Second, with VS (b) < b in the ESS and (18), the right-hand side of (17) becomes lower in the ESS than in the PBE; thus, the ESS o¤er b must increase and accommodate to the tougher responder behavior in the ESS. Taking both e¤ects together, we can identify the same qualitative e¤ects of considerations of evolutionary stability as in the example of a uniform distribution. Whether the resulting ESS o¤er is higher or lower than the o¤er in the PBE depends on the shape of F as well as on the population size m. While for a uniform distribution the ESS o¤er is higher than the o¤er in the PBE for all m 1, the ESS o¤er is lower than the o¤er in the PBE if, for instance, m = 2 and F (vS ) = vS , > 3. 18

It is straightforward to verify that limbM #1=(2m) @ M and limbM "1 @ M < 0. B =@b

35

M M B =@b

0, with strict inequality for m > 1,

A.4

Proof of Proposition 4

First, we search for the mutant seller/responder strategy that maximizes the mutant’s relative material payo¤ in (11) given that all other players use the candidate ESS in Proposition 4. Second, we search for the mutant buyer/proposer strategy that maximizes (11), given that all other players use the candidate ESS. Step 1 : seller/responder strategy mutation: Consider two populations with a total of 4m 1 ESS players and one mutant M 2 G. This mutant chooses the ESS o¤er bH = bK = b in the role of buyer/proposer; we denote his expected material payo¤ in this case by ESS B . If the mutant acts as a seller/responder, he chooses the mutant M threshold level VGS . As the mutant interacts with an ESS buyer/proposer who o¤ers b, the mutant’s expected material payo¤ in the role of the seller is Z

M (b) VGS

bdvS +

Z

1

M (b) VGS

0

M vS dvS = VGS (b)b +

1 1 2

M VGS (b)

2

The mutant has the buyer role and the seller role with probabilities (1 ), respectively, which includes the case = 1=2: Accordingly, E0

M M ; G(

M G ;

G)

=

ESS B

M ) VGS (b)b +

+ (1

1 1 2

M VGS (b)

: 2 (0; 1) and 2

:

(20)

The mutant behavior also in‡uences the average expected material payo¤ of a nonmutant, but only that of members of the other group G. Any player from group G other than M is not a¤ected by M ’s mutant behavior, as these players do not interact with M . Accordingly, E0 GM ( M ; GM ; G ), which is the average expected material payo¤ of the other members of G, is constant with respect to the mutaM tion. Maximizing the di¤erence E0 M ; GM ; G ) E0 GM ( M ; GM ; G ) is G( M M and yields the the same as maximizing E0 M ; GM ; G ) with respect to VGS G( …rst-order condition M (1 ) b VGS =0 M This implies that the optimal seller/responder behavior is to choose VGS = b, that is, to accept a price o¤er b if and only if the own valuation is at least equal to b. Step 2 : buyer/proposer strategy mutation: Consider a mutant M from group G who chooses the ESS value VGS (b) = b in the role of seller/responder, in which case his expected material payo¤ is denoted by ESS . In the role of buyer/proposer, the S

36

mutant deviates from the ESS strategy to bM G . Analogous to step 1, we get the material payo¤ of the buyer/proposer mutant as E0

M M ; G(

M G ;

G)

= V

G;S

bM G

1

bM G + (1

)

ESS : S

Making use of VHS (b) = VKS (b) = b, this can be written as E0

M M ; G(

M G ;

G)

= bM G 1

bM G + (1

)

ESS : S

Again, the expected material payo¤ of any member of group G other than the mutant M is una¤ected by the mutation of M since the mutant trades with members of M ; GM ; G ) with respect to bM population G only. Maximizing E0 M G( G yields the optimal choice bM G = 1=2 = b.

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