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

In the 60ʼs Shapley provided an example of a two player fictitious play which generates periodic behaviour. In this game, player A prefers to copy Bʼs 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 Shapleyʼs 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. ► There exist infinitely many periodic orbits. ► Chaotic behaviour with players dithering between different strategies. ► There exists a periodic orbit (consisting of mixed strategies) of ‘jitter type’. ► Nearby, orbits jitter closer and further away as in a random walk motion. ► All this behaviour holds for an open set of games.