Ismail, Mehmet (2014): Maximin equilibrium.

PDF
MPRA_paper_97401.pdf Download (360kB)  Preview 
Abstract
We introduce a new solution concept called maximin equilibrium which extends von Neumann's maximin strategy idea to nplayer noncooperative games by incorporating common knowledge of 'rationality' of the players. Our rationality assumption is, however, stronger than the one of maximin strategy and weaker than the one of Nash equilibrium. Maximin equilibrium, just like maximin strategies, is a method for evaluating the uncertainty that players are facing by playing the game. We show that maximin equilibrium is invariant under strictly increasing transformations of the payoff functions. Notably, every finite game possesses a maximin equilibrium in pure strategies. Considering the games in von Neumann and Morgenstern mixed extension, we show that maximin equilibrium is a generalization of Nash equilibrium. In addition, we demonstrate that maximin equilibria and Nash equilibria coincide in twoplayer zerosum games. We propose a refinement of maximin equilibrium called strong maximin equilibrium. Accordingly, we show that for every Nash equilibrium that is not a strong maximin equilibrium there exists a strong maximin equilibrium that Pareto dominates it. In addition, no strong maximin equilibrium is ever Pareto dominated by a Nash equilibrium. Finally, we discuss maximin equilibrium predictions in several games including the traveler's dilemma. (Submitted to MPRA for stable archival purposes. This is the first Maastricht University version.)
Item Type:  MPRA Paper 

Original Title:  Maximin equilibrium 
Language:  English 
Keywords:  Noncooperative games, maximin strategy, zerosum games 
Subjects:  C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C72  Noncooperative Games 
Item ID:  97401 
Depositing User:  Mehmet Ismail 
Date Deposited:  12 Dec 2019 01:58 
Last Modified:  12 Dec 2019 01:58 
References:  Aliprantis, C. and K. Border (1994). Infinite Dimensional Analysis: A Hitchhiker’s Guide. Aumann, R. J. and M. Maschler (1972). Some thoughts on the minimax principle. Management Science 18(5Part2), 54–63. Aumann, R. J. (1974). Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics 1(1), 67–96. Aumann, R. J. (1976). Agreeing to disagree. The Annals of Statistics 4(6), pp. 1236–1239. Aumann, R. (1990). Nash equilibria are not selfenforcing, in ‘‘Economic DecisionMaking: Games, Econometrics and Optimization’’(JJ Gabszewicz, J.F. Richard, and LA Wolsey, Eds.). Basu, K. (1994). The traveler’s dilemma: Paradoxes of rationality in game theory. The American Economic Review 84(2), 391–395. Berge, C. (1959). Espaces topologiques: Fonctions multivoques. Dunod. Bernheim, B. D. (1984). Rationalizable strategic behavior. Econometrica 52(4), pp. 1007–1028. Capra, C. M., J. K. Goeree, R. Gomez, and C. A. Holt (1999). Anomalous behavior in a traveler’s dilemma? American Economic Review 89 (3), 678–690. Fishburn, P. (1970). Utility theory for decision making. Publications in operations research. Wiley. Gilboa, I. and D. Schmeidler (1989). Maxmin expected utility with nonunique prior. Journal of Mathematical Economics 18 (2), 141–153. Goeree, J. K. and C. A. Holt (2001). Ten little treasures of game theory and ten intuitive contradictions. American Economic Review 91(5), 1402–1422. Harsanyi, J. C. and R. Selten (1988). A General Theory of Equilibrium Selection in Games. MIT Press. Harsanyi, J. C. (1966). A general theory of rational behavior in game situations. Econometrica 34(3), pp. 613–634. Lewis, D. (1969). Convention: a philosophical study. Harvard University Press. McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2016). Gambit: Software Tools for Game Theory, version 14.0.1. Nash, J. F. (1950). Equilibrium points in nperson games. Proceedings of the national academy of sciences, 36(1), 4849. Nash, J. F. (1951). Noncooperative games. Annals of Mathematics 54 (2), 286–295. Pearce, D. G. (1984). Rationalizable strategic behavior and the problem of perfection. Econometrica 52(4), pp. 1029–1050. Rosenthal, R. W. (1981). Games of perfect information, predatory pricing and the chainstore paradox. Journal of Economic theory, 25(1), 92100. Rubinstein, A. (2006). Dilemmas of an economic theorist. Econometrica, 74(4):865–883. Rubinstein, A. (2007). Instinctive and cognitive reasoning: A study of response times. The Economic Journal 117(523), 1243–1259. Savage, Leonard J. (1954). The Foundations of Statistics. Wiley Publications in Statistics. Selten, Reinhard (1975). Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory, 4(1):25–55. Selten, Reinhard (1965). Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit. Zeitschrift fur die gesamte Staatswissenschaft: ZgS. von Neumann, J. (1928). Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100, 295–320. von Neumann, J. and O. Morgenstern (1944). Theory of Games and Economic Behavior (1953, Third ed.). Princeton University Press. Wald, A. (1950). Statistical decision functions. Wiley publications in statistics. Wiley. Inc. Wolfram Research. Mathematica version 8.0. 2010. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/97401 