Optimal Stopping Strategies in Gambler’s Ruin Game

We consider a game-theoretic version of the gambler’s ruin problem. In each of the n steps, two players with different capitals compete over a unit of capital. The players’ chances in each step are equal. Accordingly, the capital of each player can increase or decrease by one unit with equal probability. The player wins if the opponent runs out of capital. In this case, the player gets 1 as payoff. If the game has not ended within the time interval n, then the players gain nothing. At each step, the players are required to pay a value of c. Two variants of the game are examined: one where one player’s capital is infinite, and the other where both players’ capitals are infinite. The player’s strategy is the stopping time in the game in order to maximize the expected payoff. The players’ optimal stopping strategies and payoffs are determined. The numerical results of payoff simulation for different n are reported.