Fictitious Play in $3\times 3$ games: chaos and dithering behaviour

TL;DR

This paper investigates complex dynamic behaviors, including chaos and dithering, in a family of 3x3 games that generalize Shapley's example, revealing the existence of jitter-type periodic orbits.

Contribution

It demonstrates the presence of chaotic and dithering behaviors in generalized 3x3 games, introducing the concept of jitter-type periodic orbits and proving their prevalence.

Findings

Existence of chaotic dynamics in 3x3 games.

Identification of jitter-type periodic orbits.

Chaotic behavior holds for an open set of games.

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 continue to study a family of games which generalize Shapley's example by introducing an external parameter, and prove that there exists an abundance of periodic and chaotic behavior 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 {\em `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.