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We introduce a simple one-parameter game derived from a model describing theproperties of a directed polymer in a random medium. At his turn, each of thetwo players picks a move among two alternatives in order to maximize his finalscore, and minimize opponent's return. For a game of length $n$, we find thatthe probability distribution of the final score $S_n$ develops a traveling waveform, ${\rm Prob}(S_n=m)=f(m-v n)$, with the wave profile $f(z)$ unusuallydecaying as a double exponential for large positive and negative $z$. Inaddition, as the only parameter in the game is varied, we find a transitionwhere one player is able to get his maximum theoretical score. By extendingthis model, we suggest that the front velocity $v$ is selected by the nonlinearmarginal stability mechanism arising in some traveling wave problems for whichthe profile decays exponentially, and for which standard traveling wave theoryapplies.

Contest based on a directed polymer in a random medium