Further insight into the structure of bold and timid policies

Abstract

A gambler repeatedly plays a game until either he becomes broke or his fortune becomes equal to or exceeds a target amount. The gambler is allowed to make multiple bets, i.e. stake integral amounts on different alternatives of the game, and more than one bet may win simultaneously. The objective is to determine a strategy that maximizes the probability of attaining the target amount. When all bets have the same gain and alternatives of betting exist such that the relevant plays are feasible, the following results are obtained. For unfavorable games, a bold type of policy is shown to be optimal. A timed type of policy is shown to be best within a restricted class of policies for favorable games. In general, optimal policies contain multiple bets. Based on a numerical example, this is established for roulette also.