Fictitious Play in 3x3 Games: the transition between periodic and chaotic behaviour

TL;DR

This paper explores how the dynamics of a 3x3 fictitious play game transition from periodic to chaotic behavior as a parameter varies, using a geometric dynamical systems approach.

Contribution

It generalizes Shapley's 2x2 example by introducing a parameter, analyzing bifurcations, and studying the transition from periodic to chaotic dynamics in 3x3 games.

Findings

Periodic behavior disintegrates into chaos at a critical parameter.

Dynamics become periodic again at a further parameter value.

The study uses a geometric approach to analyze bifurcations.

Abstract

In the 60's Shapley provided an example of a two player fictitious game with periodic behaviour. In this game, player $A$ aims to copy $B$'s behaviour and player $B$ aims to play one ahead of player $A$. In this paper we generalize Shapley's example by introducing an external parameter. We show that the periodic behaviour in Shapley's example at some critical parameter value disintegrates into unpredictable (chaotic) behaviour, with players dithering a huge number of times between different strategies. At a further critical parameter the dynamics becomes periodic again and both players aim to play one ahead of the other. We study the dynamics of a two player continuous time bimatrix fictitious play with the dynamics of a one-parameter family of $3 \times 3$ games that includes a well-known example of Shapley's as a special case. In this paper we adopt a geometric (dynamical systems) approach and study the bifurcations of simple periodic orbits. Here we concentrate on the periodic behaviour, while in a sequel we shall describe the chaotic behaviour.