Gambling Problems with a Limit Inferior Payoff

In the gambling theory of Dubins and Savage, the payoff from a sequence of states is the limit superior of their utilities. Here the limit inferior is used instead. For example, if the utility function is the indicator of a set G, then the Dubins and Savage payoff is 1 when infinitely many states are in G, while the payoff here is 1 when all but finitely many states are in G. There are some resulting changes in the theory, but it remains true that optimal stationary plans exist for finite problems.