Deciding when to quit the gambler's ruin game with unknown probabilities

In the standard definition of the classical gambler's ruin game, a persistent player enters in a stochastic process with an initial budget b 0 , which is, round after round, either increased by 1 with probability p, or decreased by 1 with probability 1 − p. The player wins the game if the budget reaches a given objective value g, and loses the game if the budget drops to zero (the gambler is ruined). This article introduces the decisional gambling process, where the parameter p is hidden, and the player has the possibility to stop the game at any round keeping earnings. In this case, the best a player can do is to maintain an estimate of p based on the observed outcomes, and use it to decide whether is better to stay or quit the game. The main contribution of this article is to bring the question of finding the optimal stopping time to the gambler's ruin game. Different heuristics are analyzed and evaluated according to their performance in maximizing the gambler's expected final budget.