Family Of Chaotic Maps From Game Theory

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,n ->infinity(lim) 1/n(k=0)Sigma(n-1)f(k) (x) = bfor every x is an element of (0, 1). This solves a known open question whether there exists a 'natural' nontrivial smooth map other than x bar right arrow axe(-x) with centres of mass of all periodic orbits coinciding. We also study the dependence of the dynamics on the two parameters.