Main Article Content
Suppose a game is played repeatedly by a finite collection of players. At every step, each player plays his optimal strategy given the observed probabilities of play for the strategies used by the other players. This generates a (time-dependent) map of the joint strategy space into itself known as ‘fictitious play’(FP). This map can be approximated by a discontinuous vector field. ‘Weak’ solutions for this dynamics are defined, and shown to exist and be unique under certain generic conditions. These weak solutions are also shown to be limits of the original discrete dynamics as the step size approaches zero. It is shown that this process lends itself to a reasonable interpretation of bounded rationality in the appropriate context.
Keywords: Games, Dynamic Systems, Complexity, Bounded Rationality.
Authors wishing to include figures, tables, or text passages that have already been published elsewhere are required to obtain permission from the copyright owner(s) for both the print and online format and to include evidence that such permission has been granted when submitting their papers. Any material received without such evidence will be assumed to originate from the authors.
Brown, G.W., “Iterative Solutions of Games by Fictitious Play” in Activity Analysis of Production and Allocation (T.C. Koopmans, Ed.) New York, Wiley, 1951.
Canning, David, “Rationality and Game Theory when Players are Turing Machines”, STICERD Disc. Paper TE/88/183, London School of Economics, October 1988.
Fudenberg, D. and D. Kreps, “Learning Mixed Equilibria”, Games and Economic Behaviour, 5, 1993, 320-367.
Gaunersdorfer, A. and J. Hofbauer, “Fictitious Play” Shapely Polygons, and the Replicator Equation”, Games and Economic Behaviour 11, 1995, 279-303.
Gottinger, H.W., “Complexity of Games and Bounded Rationality”, Optimization Vol. 21, 1990, pp. 991-1003.
Hofbauer, J., “Stability for Best Response Dynamics”, Inst. For Mathematik, Univ. Wien, 1995, mimeo.
Kalai,E.,“Bounded Rationality and Strategic Complexity in Repeated Games“, Center for Math. Studies in Economics and Management Science, Northwestern Univ., Evanston, Illinois, Disc.Paper 783, June 1988
Kreps, D., P. Milgrom, J. Robert, and R. Wilson, “Rational Cooperation in the Finitely Repeated Prinsoner’s Dilemma’ Journal of Economic Theory 27, 1982, 245-252.
Kreps, D., M., Game Theory and Economic Modelling, Oxford University Press, Oxford, 1990.
Matsui, A., “Best Response Dynamics and Socially Stable Strategies”, Journal of Economic Theory, 57, 1992, 343-362.
Megiddo,N., Remarks on Bounded Rationality, IBM Almaden Research Center, Research Paper, San Jose, Ca. 1986
Metrick, A. and B. Polack, “Fictitious Play in 2x2 games: a geometric proof of convergence, Economic Theory 4, 1994, 923-933.
Monderer, D., D. Samet and A. Sela, “Belief Affirming in Learning Processes”, Israel Inst. of Business Research. Fac. of Management, Tel Aviv Univ., Working Paper No. 15/96, June 1996.
Robinson, Julia, “An Iterative Method of Solving a Game”, Annals of Mathematics, Vol. 53 (1951), pp. 296-301.
Rosenmüller, J., “Über Periodizitätseigenschaften spieltheoretischer Lernprozesse”, Z. Wahrsch. verw. Gebiete 17, 1971, 259-308.
Rubinstein, A., “Finite Automata Play the Repeated Prisoner’s Dilemma”, Journal of Economic Theory, Vol. 39 (1986), pp. 83-96.
Shapley, L.S., “Some Topics in Two Person Games”, in Dresher, M., Shapley, L.S. and Tucker, A.W., eds., Advances in Game Theory, Princeton University Press, Princeton, 1964.
Smale, Steve, “Differentiable Dynamic Systems”, in The Mathematics of Time, Springer, New York, 1980a.
Smale, Steve, “The Prisoner’s Dilemma and Dynamical Systems Associated to Non-Cooperative Games”, Econometrica, Vol. 48, 1980b,1617-1634.