Fictitious play in 3 x 3 games: Chaos and dithering behaviour

In the 60's Shapley provided an example of a two player fictitious play which generated periodic behaviour. In this game, player A prefers to copy B's behaviour and player prefers to play one strategy ahead of player A. In this paper we continue to study 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 player. dithering between different strategies. The reason for all this, is that there exists a period 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 in a manner which is reminiscent of a random walk motion. We prove that this behavior holds for an open set of games. 2010 Elsevier Inc. All rights reserved.