The Basic Gambling Model

In this module, we will study one of the simplest gambling models. Yet in spite of its simplicity, the mathematical analysis leads to some beautiful and sometimes surprising results that have importance and application well beyond gambling.

Assumptions

Here is the basic situation: The gambler starts with an initial (non-random) sum of money. He may bet on a simple trial with two outcomes—win or lose. If he wins a trial, he receives the amount of the bet; if he loses a trial, he must pay the amount of the bet. Thus, the gambler plays at even stakes.

Let us try to formulate the gambling experiment mathematically and decide on some assumptions for the basic random variables. First, we assume that the trials are independent and that the probabilities of winning and losing remain constant from trial to trial. Thus, we have a sequence of Bernoulli trials:

1. I₁, I₂, ... where I_j is the outcome of trial j (1 win or 0 lose)
2. I₁, I₂, ... are independent and P(I_j = 1) = p, P(I_j = 0) = q = 1 - p.

If p = 0, then the gambler always loses and if p = 1 then the gambler always wins. These trivial cases are not interesting so we will usually assume that 0 < p < 1. In real gambling houses, of course, p < 1 / 2 (that is, the trials are unfair to the player), so we will be particularly interested in this case.

Random Processes

The gambler's fortune over time is the basic random process of interest: Let

X₀ = the initial fortune, X_i = the fortune after i trials.

The gambler's strategy consists of the decisions of how much to bet on the various trials and when to quit. Let

Y_i = the amount of the i'th bet.

and let N denote the number of trials played by the gambler. If we want to, we can always assume that the trials go on forever, but with the assumption that the gambler bets 0 on all trials after N. With this understanding, the trial outcome, fortune, and bet processes are defined for all i.

Note that the fortune process is related to the wager process as follows: for j = 1, 2, ...

X_j = X_{j - 1} + Y_j if I_j = 1,
X_j = X_{j - 1} - Y_j if I_J = 0

1. Show that

X_j = X_{j - 1} + (2I_j - 1)Y_j for j = 1, 2, ...

Strategies

The gambler's strategy can be very complicated. For example, the gambler's bet on trial n (Y_n) or his decision to stop after n - 1 trials ({N = n - 1}) could be based on the entire past history of the game, up to time n:

H_n = (X₀, Y₁, I₁, Y₂, I₂, ..., Y_{n - 1}, I_{n - 1})

Moreover, they could have additional sources of randomness. For example a gambler playing roulette could partly base his bets on the roll of a lucky die that he keeps in his pocket. However, the gambler cannot see into the future (unfortunately from his point of view), so we can at least assume that

Y_n and {N = n - 1} are independent of I_n, I_{n + 1}, I_{n + 2} ...

We will now show that, at least in terms of expected value, any gambling strategy is futile if the trials are unfair.

2. Use the result of Exercise 1 and the assumption of no prescience to show that

E(X_i) = E(X_{i - 1}) + (2p - 1)E(Y_i ) for i = 1, 2, ...

3. Suppose that the gambler has a positive probability of making a real bet on trial i. Use the result of Exercise 2 to show that

1. E(X_i) < E(X_{i - 1}) if p < 1 / 2
2. E(X_i) = E(X_{i - 1}) if p > 1 / 2
3. E(X_i) = E(X_{i - 1}) if p = 1 / 2

Exercise 3 shows that on any trial in which the gambler makes a bet, his expected fortune strictly decreases if the trials are unfair; his expected fortune remains the same if the trials are fair; and his expected fortune strictly increases if the trials are favorable.

Stationary Deterministic Strategies

As we noted earlier, a general strategy can depend on the past history and can be randomized. However, since the underlying Bernoulli trials are independent, one might guess that these complicated strategies are no better than simple strategies in which the amount of the bet and the decision to stop are based only on the gambler's current fortune. These simple strategies do indeed play a fundamental role and are referred to as stationary, deterministic strategies. Such a strategy can be described by a function S from the space of fortunes to the space of allowable bets, so that S(x) is the amount that the gambler bets when his current fortune is x.

Red and Black