  We introduce a simple one-parameter game derived from a model describing the
properties of a directed polymer in a random medium. At his turn, each of the
two players picks a move among two alternatives in order to maximize his final
score, and minimize opponent's return. For a game of length $n$, we find that
the probability distribution of the final score $S_n$ develops a traveling wave
form, ${\rm Prob}(S_n=m)=f(m-v n)$, with the wave profile $f(z)$ unusually
decaying as a double exponential for large positive and negative $z$. In
addition, as the only parameter in the game is varied, we find a transition
where one player is able to get his maximum theoretical score. By extending
this model, we suggest that the front velocity $v$ is selected by the nonlinear
marginal stability mechanism arising in some traveling wave problems for which
the profile decays exponentially, and for which standard traveling wave theory
applies.
