  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.
