Family of chaotic maps from game theory

TL;DR

This paper introduces a new family of chaotic maps derived from game theory, analyzing their complex dynamics, including periodic orbits and chaos, and addressing an open question about the centers of mass of periodic orbits.

Contribution

It presents a novel two-parameter family of maps from game theory, demonstrating chaotic behavior and solving an open problem regarding centers of mass of periodic orbits.

Findings

Existence of chaotic dynamics on the invariant diagonal.

Fixed point corresponding to Nash equilibrium is globally Cesaro attracting.

Addresses an open question about centers of mass of periodic orbits.

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.