Summary
This paper considers the computability of the Nash equilibria of a game, i.e. the possibility of an algorithm to play the game with respect to Nash equilibria. We consider a two-person game, in which both players have countably many feasible actions and their particular payoff functions are computable in the sense that there are algorithms to compute the values of ones for any given action profile. It is proved that there exists no algorithm to decide whether or not a given action profile is a Nash equilibrium. Moreover, we show that the set of the Nash equilibria is not empty, but there exists no algorithm to enumerate it allowing repetitions. These results mean that no Nash equilibrium of the game cannot be computed by any human being in practice, although the existence of the Nash equilibria can be proved in theory. Indeed, it is impossible to supply the players with algorithms regarding how they should decide whether or not a given action profile is a Nash equilibrium of the game or how they should enumerate all Nash equilibria. On the other hand, I show that if players’s payoff functions are any rational number valued polynomials bounded from the above with rational number coefficients, then the Nash equilibria of the game are computable.
The author is grateful to Mamoru Kaneko, Kotaro Suzumura, and especially to Takashi Nagashima. Furthermore, he would like to acknowledge Thomas Brenner, Christian Ewerhart, Peter Hammond, Kaori Hasegawa, Midori Hirokawa, Wiebe van der Hoek, Ryo Kashima, Luchuan A. Liu, Kin Chung Lo, Akihiko Matsui, Shigeo Muto, Mikio Nakayama, John Nash, Jr., Yongsheng Xu, and Itzhak Zilcha for their advice, comments, encouragement, and discussions. Of course, any possible errors and misunderstandings are due to the author.
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References
Binmore, K. (1987): Modeling Rational Players, Part I. Economics and Philosophy 2, 179–214
Davis, M. (1958): Computability and Unsolvability. McGraw-Hill, New York
Gödel, K. (1936): Über die Länge von Beweisen (On the Length of Proofs). In: Ergebnisse eines Mathematischen Kolloquiums, Heft 7, 23–24; reprinted with English translation in (1986): Feferman, S., Dawson, J. W. Jr., Kleene, S. C., Moore, G. H., Solovay, R. M., van Heijenoort, J. (Eds.): Kurt Gödel Collected Works, Volume I. Oxford University Press, New York, 396–398
Gödel, K. (1946): Remarks before the Princeton Bicentennial Conference on Problems in Mathematics (unpublished); reprinted in (1990): Feferman, S., Dawson, J. W. Jr., Kleene, S. C., Moore, G. H., Solovay, R. M., van Heijenoort, J. (Eds.): Kurt Gödel Collected Works, Volume II. Oxford University Press, New York, 150–153
Hennie, F. (1977): Introduction to Computability, Addison-Wesley, Reading
Kaneko, M., Nagashima, T. (1996): Game Logic and its Applications I. Studia Logica 57, 325–354
Kaneko, M., Nagashima, T. (1997): Game Logic and its Applications II. Studia Logica 58, 273–303
Kleene, S. C. (1952): Introduction to Metamathematics. North-Holland, Amsterdam
Odifreddi, P. (1989): Classical Recursion Theory. North-Holland, Amsterdam
Rabin, M. O. (1957): Effective Computability of Winning Strategies. In: Dresher, M., Tucker, A. W., Wolfe, P. (Eds.): Contributions to the Theory of Games, Volume III. Princeton University Press, Princeton, 147–157 This fact was suggested by Professor Itzhak Zilcha. Computability of Nash Equilibrium 357
Rogers, H. (1967): Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York
Tashiro, H. (1998): Computational Playability of Backward Induction Solutions II. In: The North American Summer Meeting of the Econometric Society (June, 1998 ). Montreal, Canada
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Tashiro, H. (2003). Computability of Nash Equilibrium. In: Petrosyan, L.A., Yeung, D.W.K. (eds) ICM Millennium Lectures on Games. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05219-8_22
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DOI: https://doi.org/10.1007/978-3-662-05219-8_22
Publisher Name: Springer, Berlin, Heidelberg
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