Parity and Ruin in a Stochastic Game
E. Ben-Naim and P.L. Krapivsky
We study an elementary two-player card game where in each round
players compare cards and the holder of the smallest card wins. Using
the rate equations approach, we treat the stochastic version of the
game in which cards are drawn randomly. We obtain an exact solution
for arbitrary initial conditions. In general, the game approaches a
steady state where the card densities of the two players are
proportional to each other. The leading small size behavior of the
initial card densities determines the corresponding proportionality
constant, while the next correction governs the asymptotic
time dependence. The relaxation towards the steady state exhibits a
rich behavior, e.g., it may be algebraically slow or exponentially
fast. Moreover, in ruin situations where one player eventually wins
all cards, the game may even end in a finite time.
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