TURING GAMES: A STUDY OF EQUIVALENCE BETWEEN GAMES AND COMPUTATION DEVANG THAKKAR DEPARTMENT OF COMPUTER SCIENCE, COLUMBIA UNIVERSITY, NEW YORK, NY,10025
Abstract. In this paper I try to explore the equivalence of Turing Machines and games. The main idea I am trying to achieve is that every computation is a game and vice versa. Now the basic model that computer science proposes for computation is the model of a Turing machine where in the machine has a state transition function which is a set of rules that define what state you should be in and an input tape which consists of symbols based on which future decisions are made. The model that I propose tries to achieve a smooth transition between a game and a Turing Machine and hence allows application of mathematical techniques of either field to be applied to the other. I provide the basis and examples of two transforms; one that converts a Turing machine into a game and second that converts a game to a Turing machine. The main question that might be arising in your mind right now is what purpose does this serve? The reasoning here is if I am able to generalize my transforms and make these reductions possible then I can use the relation between language strings and paths to equilibrium that I prove in this paper to provide useful reductions from hard problems arising in the fields of both computational complexity and game theory to simpler ones in the other field. Finally I provide various open questions for future work in this area and also some applications of the results proposed here.
Introduction Game Theory  and Computer Science are both rich fields of mathematics which have evolved over the last five decades. The goals of both these fields are somewhat similar - to solve problems for humans. Game Theory studies the interaction of rational decision makers and gives a bunch of tools that would help in improving decisions. Computer Science on the other hand studies the art of solving problems using a computing machine. Both fields have been running in parallel since their induction (by the likes of von Neumann, Godel, Turing and others) and now they fields seem to be coming closer than before as computer scientists start taking a more rational approach to things(pun intended). Yanakakis and Papadimitriou explored the similarities between bounded rationality and computational complexity , Kearns et al  studied the use of properties of graph to give simpler algorithms to find important Game Theoretic parameters and there are many more such works. Standing on the shoulders of these giants the idea that I try to explore here is that of the analogies between a computation and a game and whether it is possible to our expertise in one field to simplify problems in the other. In section 1 I provide Key words and phrases. game theory, turing machine, complexity. 1
DEVANG THAKKAR DEPARTMENT OF COMPUTER SCIENCE, COLUMBIA UNIVERSITY, NEW YORK, NY,10025 2
the background material and survey of related results. In section 2 I provide the intuition for this work and some toy examples. 1. Background This section provides the necessary background material. 1.1. Machine Games. The line between the concepts of computational power and decision making power is a very thin one and a key step in relating computer science and game theory. The theory of games is sometimes considered flawed predicting rational behavior. Prisoners Dilemma is an example of this. This is where computer scientists step in. We can give a better view of strategy selection by attaching a complexity parameter to it. This was first pointed out by Papadimitriou. The main ideas of these works is to bound the strategy representing capacity of players and hence induce co-operation. They achieve this bound by representing players as finite automaton and then restricting the number of states the automata can have. Thus in general we could say we have machine playing games for players and we restrict the computational power of there machines and hence the decision making power of the players. Rubinstein and Osborne  give the following definitions for a machine and machine game. Definition 1.1.1. Let G = h(1, 2), (Ai ), (ui )i be a game A Machine of player i is defined as a four-tuple hQi , qi0 , fi , τi i in which - Qi is a set of states - qi0 ∈ Qi is the initial state - fi : Qi → Ai is the output function - τi : Qi × Ai → Qi is the transition function. Definition 1.1.2. A Machine Game of the δ-discounted infinitely repeated game h(1, 2), (Mi ), (i )i in which for each player i 1) Mi is the set of all finite machines for player i in the infinitely repeated game. 2) i is a preference ordering that is increasing in player i’s payoff in the repeated game and decreasing in the complexity of his machine: (M1 , M2 ) i (M10 , M20 ) whenever either Ui (M1 , M2 ) > Ui (M10 , M20 ) and c(Mi ) = c(Mi0 ) or Ui (M1 , M2 ) > Ui (M10 , M20 ) and c(Mi ) < c(Mi0 ). Where U is the utility function which in case of δ-discounted game is Ui (M1 , M2 ) = (t−1) (1 − δ)Σ∞ ui (at (M1 , M2 )) where at (M1 , M2 ) is the sequence induced by the t=1 δ machine game and c(Mi ) denotes the complexity of machine i. In order to give complexity criterion for games Papadimitriou and Yanakakis  represent a game as a polynomial computable function g(z,x,y) with 3 parameters: z is the string that encodes the game, and x and y encode the strategies. The value of g is a pair of integers - the payoffs. They prove the following interesting results. Theorem 1.1.3. (a)The class of all languages of the form (z: there is a strategy y for II with payoff ≥ 0 against strategy x=e of I) is precisely NP. (b)The class of languages of the form (z:there is a pure equilibrium in game z) is precisely Σp2 . (c)The class of all languages of the form (z:zero-sum game z has an equilibrium with payoff to I at least 0) is precisely EXP.
Figure 1. Tit-for-Tat (d)The class of languages of the form (z: game z has an equilibrium with payoff at least 0 to both I and II) is precisely NEXP. For a detailed proof of the above theorem refer  1.2. Equilibria in Machine Games. In this section I provide a survey of results obtained in the quest of finding an equilibria in repeated games. The importance of using automata to analyze repeated games is emphasized in the previous section. The question is how easy or hard is it to find equilibria in such repeated games. The ideas used in the study of equilibria in repeated games are termed as ”folk theorems” and they make use of trigger strategies to take punitive actions against players who deviate. The idea is that the notion of threats can stabilize a wide range of payoff profiles in repeated games. For infinitely repeated games Aumann and Hart  provide a general definition for The Folk Theorem Definition 1.2.1. The Folk Theorem In the infinitely repeated game all points in the mixed individually rational region are equlibria. Building on this Papadimitriou and Yanakakis provide a definition for The Folk Theorem for Automata where automata is used to play games. Definition 1.2.2. Folk Theorem for Automata The following are equivalent: a)The infinitely repeated game has a pure equilibrium in automata. b)The infinitely repeated game has a finite mixed equilibrium in automata. c)The pure individually rational region is non-empty. Based on the ideas of the folk theorem Littman and Stone  provide an polynomial time algorithm for finding a Nash Equilibrium in a repeated game with average payoff criterion.
DEVANG THAKKAR DEPARTMENT OF COMPUTER SCIENCE, COLUMBIA UNIVERSITY, NEW YORK, NY,10025 4
Figure 2. Main Idea 2. Turing Games 2.1. Introduction. As seen in the previous sections most work done so far deals with the use of automata to understand, represent and analyze repeated games. But is it possible to go the other way round and harness the power of game theory to solve problems in automata. In this section I survey certain results in this area and give my perspective on the idea. I begin by giving a formal definition of the Turing machine introduced by A. Turing in his 1936 paper . A modified definition is as follows Definition 2.1.1. Turing Machine We can describe a Turing Machine(TM) by a 7-tuple M = (Q, Σ, Γ, δ, q0 , B, F ) whose components have the following meanings: Q: The finite set of states of the finite control. Σ: The finite set of input symbols. Γ: The complete set of tape symbols,Σ is always a subset of Γ. δ: The transition function. The arguments of δ(q, X) are a state q and a tape symbol X. The value of δ(q, X), if it is defined, is a triple (p,Y,D), where : 1. p is the next state in Q. 2. Y is the symbol, in Γ, written in the cell being scanned, replacing whatever symbol was there. 3. D is a direction, either L or R, standing for ”left” or ”right”,respectively, and telling us the direction in which the head moves.
Figure 3. Automata Game q0 : The start state, a member of Q, in which the finite control is found initially. B: The blank symbol. This symbol is in Γ but not in Σ; i.e., it is not an input symbol. The blank appears initially in all but the finite number of initial cells that hold input symbols. F: The set of final or accepting states, a subset of Q. 2.2. Automata and Games. We can consider automata to be game between two players A and B such that A selects the states of the automata and B selects the input symbols. Thus A is nothing but the transition function of the automata. A strategy can be called a winning strategy for the automata if it is in one of the accepting states. Here we take into account the fact that halting problem is undecidable. The game can go on forever and we might never be able to say for a particular game whether there is a winning strategy for the automata. Thus we have set up a game between the transition function of the Turing machine say A and the input structure (or tape in case of a TM) M and we denote the game as G(A,M). For the example shown in figure 3 above Q0 is the initial state and the sample run of the game is (Q0 , a, Q1 , b, Q0 , a, Q1 , c, Q2 ) at the end of which the automata is in an accepting state and hence it is a winning strategy for the automata. Similar ideas are introduced by Gurevich and Harrington  where they use game techniques to analyze the Rabin tree automata and Buchi automata.
2.3. Transforms. In this section I provide the sketch of the requirements of the transforms.
DEVANG THAKKAR DEPARTMENT OF COMPUTER SCIENCE, COLUMBIA UNIVERSITY, NEW YORK, NY,10025 6
Proposition 2.3.1. Game to Turing Machine As discussed in the preceding sections a repeated game can be easily represented as an automata by taking into account the threat points in the game. A turing machine which is nothing but automata can thus be a useful tool in game analysis. The requirements of this transform would be two functions 1 - which maps players actions to states. 2 - which maps opponents actions to the transition function based on threats. Remark 2.3.2. Littman and Stone  give an alternate efficient representation for a repeated game by introducing the concept of counting nodes which allows representation of strategies of the form 2n tit-for-tat, where n is the number of states to wait to activate a threat. Proposition 2.3.3. Turing Machine to Game In order to represent a turing machine as a game we need three functions 1- which maps states of the turing machine to actions of player A. 2- which maps the transition function to actions of player B. 3- which maps action pairs to utilities for each player. Remark 2.3.4. The third function in the above proposition would give not the exact payoffs but a relation between payoffs based on which we can classify games. The following are the basic games 1. (a,b) (c,c) Here a > b > c and (a,b) and (b,a) are the two NE. (c,c) (b,a) 2. (a,a) (c,c) Here a > b > c and (a,a) and (b,b) are the two NE with (a,a) pref(c,c) (b,b) ered. 3. (a,a) (c,d) Here d > a > b > c and (b,b) is the NE . (d,c) (b,b) 4. (a,a) (c,d) Here d > a > c > b and (d,c),(c,d) are the NE . (d,c) (b,b) 5. (a,b) (b,a)
(b,a) Here a = −b there is no NE . (a,b)
The difficult part in transforming a turing machine to a game is calculating the utilities of players. In order to simplify the problem I assume that the Turing machine in question represents the strategies of both the players. In other words both players are using the same Turing machine to make their strategies. Now to calculate the payoff we need to predict what could be threat points. This could be a hard problem. A possible solution is to make generic structures for the standard games and look for their occurance in automata.
3. Conclusion The ideas suggested in the preceding sections point towards a game theoretic model for computation. The interesting areas to explore are the payoff calculations and better representations for games using automata. We suggest that further research in this direction is likely going to reveal additional equivalences between computation and games. Acknowledgment. The results of this paper were obtained during my M.S. studies at Columbia University and are also contained in my thesis . I would like to express deep gratitude to my supervisor Tal Malkin whose guidance and support were crucial for the successful completion of this project. I would also like to thank Prof. Yanakakis for useful discussions during the course of my research. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
M. Littman M. Kearns, S. Singh, Graphical models for games, UAI’01. C.Papadimitriou M. Yanakakis, On complexity as bounded rationality, STOC’94, 1994. A. Rubinstein M.J.Osborne, A course in game theory, The MIT Press, Cambridge, 1994. P. Stone M.L.Littman, A polynomial-time nash equilibrium algorithm for repeated games, EC’03 (2003). C. Papadimitriou, On players with bounded number of states, games and economic behaviour ed., 1992. S. Hart R.J.Aumann (ed.), Handbook of game theory with economic applications, vol1 ed., North Holland, 1982. Devang Thakkar, Game theoretic models of computation, Masters thesis, Columbia University, New York, Expected Dec. 2004. A.M. Turing, On computable numbers, with an application to the entscheidungsproblem, (1936). von Neumann and Oskar Morgensten, Theory of games and economic behaviour, New York:John Wiely and Sons, New York, 1944. Leo Harrington Yuri Gurevich, Trees, automata and games, ACM (1982).