| Lester Dubins and Leonard Savage in their book, How to Gamble if You Must, defined a strategy, (sigma), as a sequence (sigma)(,0), (sigma)(,1),..., where (sigma)(,0) is a probability measure on a space, F, called the state space, and for each positive integer n, (sigma)(,n) is a map from F('n) to the space of probability measures on F. If the maps (sigma)(,n) are measurable then (sigma) is a measurable strategy. Measurable strategies have the property that they induce countably additive probability measures upon the space of histories, while arbitrary strategies may fail to do so. Measurable strategies are therefore more advantageous to work with than nonmeasurable ones. A longstanding open question asks, then, whether the same optimal return can be obtained if only measurable strategies are used. In a key theorem of this paper we prove that measurable strategies are adequate for approximating optimal return if and only if the optimal return function is measurable. Utilizing this result, and by characterizing the relationship between the sets of gambles available at different points in the state space, we are able to prove in several cases, such as, where the set of gambles available at each point in the state space is a subset of some given countable set of gambles, that measurable strategies are adequate for approximating optimal return.; We next consider the problem of limiting our selection of strategies to the class of stationary strategies. A strategy defined by the sequence (sigma)(,0), (sigma)(,1),... is stationary if there exists a function (gamma) defined on the state space, F, to the space of probability measures on F, such that for each n and (x(,1),...,x(,n)) (ELEM) F('n), (sigma)(,n)(x(,1),...,x(,n)) = (gamma)(x(,n)). We show that stationary strategies are adequate for approximating optimal return with respect to any utility function belonging to the class of functions which we call nearly-leavable shift-invariant utility functions. This class of functions includes many of the common utility functions used in gambling theory, such as the limsup and liminf payoff functions. |
