Family of chaotic maps from game theory
Thiparat Chotibut, Fryderyk Falniowski, Michał Misiurewicz
and Georgios Piliouras
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,
limn→∞ (1/n)
∑ 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↦axe-x
with centers of mass of all periodic orbits coinciding. We also study
the dependence of the dynamics on the two parameters.