Gambler's Ruin Problem in Several Dimensions

We give an explicit solution of the Gambler s Ruin Problem when the players use two or more currencies We also determine the asymptotics of the expected duration of the game when both players have equal amounts of each currency Introduction In the one dimensional Gambler s Ruin Problem two players start out with i and N i dollars respectively At each step they toss a fair coin to decide who wins a dollar from the opponent The game is over when one of them goes bankrupt It is well known that the expected duration of the game is i N i see or almost any other textbook on probability In the two dimensional variant the players use two di erent currencies say dollars and euros They start out with i dollars j euros and N i dollars M j euros respectively At each step they toss fair coins to decide the currency and the winner The game is over when one of them runs out of either currency What is the expected duration of the game According to no closed form solution of this problem is known to exist and probably doesn t exist Denote by game i j the game with the rst player s initial assets equal to i j Assume that i N and j M Then after the rst step game i j turns into one of game i j game i j game i j or game i j each with probability It follows that the expected duration ai j of game i j satis es the recurrence equation ai j ai j ai j ai j ai j i N j M Supported in part by MZT RS under grant J and the boundary conditions a j aN j ai ai M i N j M The unknown ai j i N j M can be obtained from by straightfor ward linear algebra Instead of solving this linear system of N M equations Orr and Zeilberger have shown how to obtain the values a j aN j j M and ai ai M i N from a system containing O N M equations only The remaining ai j can then be computed directly from the recurrence ai j ai j ai j ai j ai j i N j M Because of the obvious symmetries ai j aN i j ai M j aN i M j it su ces to com pute a quarter of these numbers only Note however that in oating point arithmetic this computation is numerically unstable due to the coe cient in front of ai j in In this paper we give an explicit solution of the two dimensional Gambler s Ruin Problem Sec as well as of its obvious generalization to larger numbers of currencies Sec Although our solution is not in closed form because it contains a double sum it provides a direct way to compute the expected duration of the game without having to solve linear systems or use recursive computations In Section we express the middle element aN N as a single sum when N is a power of two In Section we determine the asymptotics of the middle element aN N N when N is even for any number of currencies Solution of the dimensional problem For the sake of simplicity we henceforth assume that M N but it is straightforward to generalize our results to the case M N Let A ai j N i j be the matrix of unknown values ai j Writing in the form ai j ai j ai j ai j ai j ai j i j N and using we see that A satis es the matrix equation AD DA J where