A new class of Hamiltonian flows with random-walk behavior originating from zero-sum games and Fictitious Play

Abstract

In this paper we relate dynamics associated to zero-sum games (Fictitious play) to Hamiltonian dynamics. It turns out that the Hamiltonian dynamics which is induced from fictitious play, has properties which are rather different from those found in more classically defined Hamiltonian dynamics. Although the vectorfield is piecewise constant (and so the flow $\phi_t$ piecewise a translation), the dynamics is rather rich. For example, there exists a Hamilton $H$ so that for each $t>0$ the level set $H^{-1}(t)$ is homeomorphic to $S^3$ (the level sets consist of pieces of hyperplanes in $\R^4$) and with the following property. There exists a periodic orbit $\Gamma$ of the Hamiltonian flow in $H^{-1}(1)$ so that the first return map $F$ to a section $Z\subset H^{-1}(1)$ transversal to $\Gamma$ at $x\in \Gamma$ acts as a random-walk: there exist a nested sequence of annuli $A_n$ in $Z$ (around $x$ so that $\cup A_n\cup \{x\}$ is a neighbourhood of $x$ in $Z$) shrinking geometrically to $x$ so that for each sequence $n(i)\ge 0$ with $|n(i+1)-n(i)|\le 1$ there exists a point $z\in Z$ so that $F^i(z)\in A_{n(i)}$ for all $i\ge 0$.