Repeated zero-sum games with budget

When a zero-sum game is played once, a risk-neutral player will want to maximize his expected outcome in that single play. However, if that single play instead only determines how much one player must pay to the other, and the same game must be played again, until either player runs out of money, optimal play may differ. Optimal play may require using different strategies depending on how much money has been won or lost. Computing these strategies is rarely feasible, as the state space is often large. This can be addressed by playing the same strategy in all situations, though this will in general sacrifice optimality. Purely maximizing expectation for each round in this way can be arbitrarily bad. We therefore propose a new solution concept that has guaranteed performance bounds, and we provide an efficient algorithm for computing it. The solution concept is closely related to the Aumann-Serrano index of riskiness, that is used to evaluate different gambles against each other. The primary difference is that instead of being offered fixed gambles, the game is adversarial.