Fictitious play in 3x3 games: Chaos and dithering behaviour
In the 60's Shapley provided an example of a two player fictitious play which generates periodic behaviour. In this game, player A prefers to copy B's behaviour and player B prefers to play one strategy ahead of player A. In this paper we continue to study a family of games which generalise Shapley's example by introducing an external parameter, and prove that there exists an abundance of periodic and chaotic behaviour with players dithering between different strategies. The reason for all this, is that there exists a periodic orbit (consisting of playing mixed strategies) which is of 'jitter type': such an orbit is neither attracting, repelling or of saddle type as nearby orbits jitter closer and further away from it in a manner which is reminiscent of a random walk motion. We prove that this behaviour holds for an open set of games.
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- Drew Fudenberg & David K. Levine, 1996.
"The Theory of Learning in Games,"
Levine's Working Paper Archive
624, David K. Levine.
- A. Gaunersdorfer & J. Hofbauer, 2010.
"Fictitious Play, Shapley Polygons and the Replicator Equation,"
Levine's Working Paper Archive
438, David K. Levine.
- Gaunersdorfer Andrea & Hofbauer Josef, 1995. "Fictitious Play, Shapley Polygons, and the Replicator Equation," Games and Economic Behavior, Elsevier, vol. 11(2), pages 279-303, November.
- Matsui, Akihiko, 1992. "Best response dynamics and socially stable strategies," Journal of Economic Theory, Elsevier, vol. 57(2), pages 343-362, August.
- Berger, Ulrich, 2007. "Two more classes of games with the continuous-time fictitious play property," Games and Economic Behavior, Elsevier, vol. 60(2), pages 247-261, August.
- Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions; Part II: Applications," Working Papers hal-00242974, HAL.
- Viossat, Yannick & Sorin, Sylvain & Hofbauer, Josef, 2009.
"Time average replicator and best reply dynamics,"
Economics Papers from University Paris Dauphine
123456789/1014, Paris Dauphine University.
- Jordan J. S., 1993. "Three Problems in Learning Mixed-Strategy Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 368-386, July.
- Michel Benaim & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions II: Applications," Levine's Bibliography 784828000000000098, UCLA Department of Economics.
- Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
- Milgrom, Paul & Roberts, John, 1991. "Adaptive and sophisticated learning in normal form games," Games and Economic Behavior, Elsevier, vol. 3(1), pages 82-100, February.
- Monderer, Dov & Shapley, Lloyd S., 1996. "Fictitious Play Property for Games with Identical Interests," Journal of Economic Theory, Elsevier, vol. 68(1), pages 258-265, January.
- Berger, Ulrich, 2008. "Learning in games with strategic complementarities revisited," Journal of Economic Theory, Elsevier, vol. 143(1), pages 292-301, November.
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