Family of chaotic maps from game theory

Abstract

ABSTRACT From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point b corresponding to a Nash equilibrium of such map f is usually repelling, it is globally Cesàro attracting on the diagonal, that is, for every . This solves a known open question whether there exists a ‘natural’ nontrivial smooth map other than with centres of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.