Timid Play is Optimal When the Trials are Favorable

Java Applet Interactive red and black game

Java Applet Simulation of the red and black experiment

In this section, we will assume that p 1 / 2 so that the trials are favorable (or at least not unfair) to the gambler. We will show that if the house requires that all bets be multiples of a basic unit (which of course real gambling houses do), then an optimal strategy is for the gambler to play timidly, making the minimum bet on each trial until he must stop. On the other hand, if we unrealistically assume that our money is infinitely divisible and the house allows arbitrarily small bets, we will show that the gambler can reach the target with probability 1.

First, let us assume that all bets must be multiples of a basic unit, which we might as well assume is $1. Thus the sets of valid fortunes and bets are

A = {0, 1, ..., a}, B_x = {0, 1, ..., min{x, a - x}}

Let f denote the win probability function for timid play. To show that timid play is optimal, we just need to verify that the condition for optimality holds:

1. pf(x + y) + qf(x - y) f(x) for x in A, y in B_x.

2. Show that equality holds in (1) if p = 1 / 2.

3. If p > 1 / 2, show that the inequality in (1) is equivalent to

p(q / p)^{x + y} + q(q / p)^{x - y} (q / p)^x

4. Show that the inequality in Exercise 3 is equivalent to

pq^{2 y} + q p^{2 y} - q^y p^y 0

5. Show that the inequality in Exercise 4 is equivalent to

pq(p^y - q^y)(p^{y - 1} - q^{y - 1}) 0

and hence is true for p > 1 / 2.

We now know that timid play is optimal when the trials are favorable.

6. In the red and black game set a = 16, x = 8, and p = 0.48. Define the strategy of your choice and play 100 games. Compare your relative frequency of wins with the probability of winning with timid play.

We will now assume that the house allows arbitrarily small bets and that p > 1/2, so that the trials are strictly favorable. In this case it is natural to take the target as the monetary unit so that the sets of fortunes and bets becomes

A = [0, 1], B_x = [0, min{x, 1 - x}} for x in A.

We will show that V(x) = 1 for x in (0, 1]. Our results for timid play will play an important role in our analysis, so we will let f(j; a) denote the probability of reaching an integer target a, starting at the integer j in [0, a], with unit bets.

First, let us fix a rational initial fortune x = k / n in [0, 1].

7. Let m be a positive integer. Suppose that, starting at x, the gambler bets 1/mn on each trial. Show this is equivalent to timid play with target mn and initial fortune mk and that hence the probability of reaching the target 1 is f(mk; mn).

8. Show that

f(mk; mn) converges to 1 as m increases.

9. Use the results of Exercises 7 and 8 to show that V(x) = 1 if x in (0, 1] is rational.

10. Use the result of Exercise 8 and the fact that V is increasing to show that V(x) = 1 for all x in (0, 1].

Red and Black