Family of chaotic maps from game theory

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 Cesaro attracting on the diagonal, that is, \[ \lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}f^k(x)=b \] for every $x$ in the minimal invariant interval. This solves a known open question whether there exists a nontrivial smooth map other than $x\mapsto axe^{-x}$ with centers of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.