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In this paper we relate dynamics associated to zero-sum games (Fictitiousplay) to Hamiltonian dynamics. It turns out that the Hamiltonian dynamics whichis induced from fictitious play, has properties which are rather different fromthose found in more classically defined Hamiltonian dynamics. Although thevectorfield is piecewise constant (and so the flow $\phi_t$ piecewise atranslation), 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 thefollowing property. There exists a periodic orbit $\Gamma$ of the Hamiltonianflow in $H^{-1}(1)$ so that the first return map $F$ to a section $Z\subsetH^{-1}(1)$ 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 $\cupA_n\cup \{x\}$ 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 apoint $z\in Z$ so that $F^i(z)\in A_{n(i)}$ for all $i\ge 0$.

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