Parity and ruin in a stochastic game

Abstract:

We study an elementary two-player card game where in each round players compare cards and the holder of the card with the smaller value 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 value densities of the two players are proportional to each other. The leading small value behavior of the initial densities determines the corresponding proportionality constant, while the next correction governs the asymptotic time dependence. The relaxation toward 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.