On the Approximation Performance of Fictitious Play in Finite Games
We study the performance of Fictitious Play, when used as a heuristic for finding an approximate Nash equilibrium of a two-player game. We exhibit a class of two-player games having payoffs in the range [0,1] that show that Fictitious Play fails to find a solution having an additive approximation guarantee significantly better than 1/2. Our construction shows that for n×n games, in the worst case both players may perpetually have mixed strategies whose payoffs fall short of the best response by an additive quantity 1/2 − O(1/n 1 − δ ) for arbitrarily small δ. We also show an essentially matching upper bound of 1/2 − O(1/n).
KeywordsNash Equilibrium Mixed Strategy Approximation Performance Pure Strategy Adjacent Block
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- 1.Brandt, F., Fischer, F., Harrenstein, P.: On the Rate of Convergence of Fictitious Play. In: 3rd Symposium on Algorithmic Game Theory, pp. 102–113 (2010)Google Scholar
- 3.Bosse, H., Byrka, J., Markakis, E.: New Algorithms for Approximate Nash Equilibria in Bimatrix Games. In: Proceedings of the 3rd International Workshop on Internet and Network Economics, pp. 17–29 (2007)Google Scholar
- 5.Conitzer, V.: Approximation Guarantees for Fictitious Play. In: Procs of 47th Annual Allerton Conference on Communication, Control, and Computing, pp. 636–643 (2009)Google Scholar
- 11.Shapley, L.: Some topics in two-person games. In: Advances in Game Theory. Annals of Mathematics Studies, vol. 52, Princeton University Press, Princeton (1964)Google Scholar