A Note on the Characterization of Optimal Return Functions and Optimal Strategies for Gambling Problems

Abstract

We consider finite state gambling problems with the Dubins and Savage payoff and with the $\lim\inf$ payoff. For these models we show that the optimal return function with respect to all stationary strategies can be characterized similarly to the optimal return function. This enables us then to characterize those stationary strategies which are optimal within the set of all stationary strategies in the same way as it was done for optimal strategies by Dubins and Savage.