On Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Limited Playing Times

On Nonoptimality of Bold Play for Subfair Red-And-Black with a Rational-Valued House Limit

In the subfair red-and-black gambling problem, a gambler can stake any amount in his possession, winning an amount equal to the stake with probability w and losing the stake with probability 1 − w, where 0 < w < ½. The gambler seeks to maximize the probability of reaching a fixed fortune (to be normalized to unity) by gambling repeatedly with suitably chosen stakes. In their classic work, Dubins and Savage (1965), (1976) showed that it is optimal to play boldly. When there is a house limit of l (0 < l < ½), so that the gambler can stake no more than l, Wilkins (1972) showed that bold play remains optimal provided that 1 / l is an integer. On the other hand, building on an earlier surprising result of Heath, Pruitt and Sudderth (1972), Schweinsberg (2005) recently showed that, for all irrational 0 < l < ½ and all 0 < w < ½, bold play is not optimal for some initial fortune. The purpose of the present paper is to present several results supporting the conjecture that, for all rational l with 1 / l not an integer and all 0 < w < ½, bold play is not optimal for some initial fortune. While most of these results are based on Schweinsberg's method, in a special case where his method is shown to be inapplicable, we argue that the conjecture can be verified with the help of symbolic-computation software.

The Action Gambler and Equal-Sized Wagering

A gambler with an initial bankroll is faced with a finite sequence of identical and independent bets. For each bet, he may wager up to his current bankroll, and will win this amount with probability p or lose it with probability 1-p. His problem is to devise a wagering strategy that will maximize his final expected utility with the side condition that the total amount wagered (i.e. the total ‘action’) be at least his initial bankroll. Our main result is an expression that characterizes when the strategy of placing equal-sized wagers on all bets is optimal. In particular, for a given bankroll B, utility function f (concave, increasing, differentiable), and n bets, we show that it is optimal to wager b/n on each bet if and only if the probability of winning each bet is less than or equal to some value p⋆∈[½,1] (where p⋆ is an explicit function of B, f, and n). We prove the result by using a basic nonlinear programming technique.

Strong Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Time-Dependent Parameters

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 − r)/r times the stake with probability w and losing the stake with probability 1 − w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune (the goal) by gambling repeatedly with suitably chosen stakes. This problem has recently been extended in a unifying framework to account for limited playing time as well as future discounting, under which bold play is known to be optimal provided that w ≤ ½ ≤ r. This paper is concerned with a further extension of the Dubins-Savage gambling problem involving time-dependent parameters, and shows that bold play not only maximizes the probability of reaching the goal, but also stochastically minimizes the number of plays needed to reach the goal. As a result, bold play also maximizes the expected utility, where the utility at the goal is only required to be monotone decreasing with respect to the number of plays needed to reach the goal. It is further noted that bold play remains optimal even when the time-dependent parameters are random.

Explicit Optimal Strategy for the Vardi Casino with Limited Playing Time

The Vardi casino with parameter 0 < c < 1 consists of infinitely many tables indexed by their odds, each of which returns the same (negative) expected winnings -c per dollar. A gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes and tables (odds). The optimal strategy is derived explicitly subject to the constraint that the gambler is allowed to play only a given finite number of times. Some properties of the optimal strategy are also discussed.