Optimal gambling strategy and relative risk aversion

We consider the optimal strategy for a gambler who is faced with a finite sequence of two-outcome games, i.e., “win” vs. “lose,” and wants to maximize the expected utility of his final wealth. Unlike the celebrated Kelly strategy in the maximization of the expected logarithm of the final period wealth, the gambler with a general utility function has only “non-myopic” optimal strategy in the sense that the solution depends on the stage (number of games left to bet) and the state (current wealth) of the gambling process. In this paper, we show that, under fairly standard assumptions, the optimal gambling strategy is myopic, i.e., the optimal wager is always to bet a constant fraction of the wealth, if and only if the gambler's utility function exhibits constant relative risk aversion. A numerical example is given to validate the theoretical results.