The Game of Red and Black

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Java Applet Interactive red and black game

Java Applet Simulation of the red and black experiment

The Stopping Rule

In the basic gambling model, we showed that any strategy of the gambler is futile if the trials are unfair. Nonetheless, it does turn out that some strategies are better than others. From now on, we will assume that the gambler's stopping rule is a very simple and standard one: he will bet on the trials until he either loses his entire fortune and is ruined or reaches a fixed target fortune a:

N = min{n = 0, 1, 2, ...: Xn = 0 or Xn = a}

This particular game is known as red and black and is named after the casino game roulette.

If we want to, we can think of the difference between the target fortune and the initial fortune as the entire fortune of the house. With this interpretation, the player and the house play symmetric roles, but with complementary win probabilities: play continues until either the player is ruined or the house is ruined. Our main interest is in the final fortune XN of the gambler. Note that this variable takes just two values--0 and a.

Mathematical Exercise 1. Show that the mean and variance of the final fortune are given by

  1. E(XN) = aP(XN = a)
  2. var(XN) = a2 P(XN = a) [1 - P(XN = a)]

Strategies

From Exercise 1, the gambler would like to maximize the probability of reaching the target fortune. Is it better to bet small amounts or large amounts, or does it not matter? How does the optimal strategy, if there is one, depend on the initial fortune, the target fortune, and the trial win probability? We will analyze and compare two strategies that are in a sense opposites:

The strategy of timid play is also referred to as gambler's ruin, perhaps because, as we will see, it is a very bad strategy in real (unfair) gambling houses.

The Applets

In the red and black game applet, you are the player and the computer is the house. Thus, you can use any strategy you like. The picture box on the left shows the initial fortune, target fortune, and current forturne. You can vary the target fortune with a list box, and you can vary the initial fortune, trial win probability, and current bet with scroll bars. Click the Play button to play a trial and the New Game button to start a new game. As you play a game, the trial outcomes and current fortune are recorded in the first table. When the game is finished (that is, when you reach the target or 0), the outcome of the game and the number of trials are recorded in the second table. As you play games, the relative frequencies of successes and failures are recorded in the third table and in the second graph. Finally, the average number of trials per game is recorded in the last table.

In the red and black simulation applet, the computer is both the player and the house. You can choose either timid play or bold play with a list box. As before, the picture box on the left shows the initial fortune, target fortune, and current forturne. You can vary the target fortune with a list box, and you can vary the initial fortune and trial win probability with scroll bars. As you run the simulation the outcome of the games and the number of trials are recorded in the first table. The density function and the empirical density function of the game win indicator J are shown in the second graph and table. The mean number of trials and the sample mean number of trials are shown in the last graph and table.

Simulation Exercises

Simulation Exercise 2. In the red and black game set the initial fortune to 8, the target fortune to 16, and the win probability to 0.5. Play 10 games with each of the following strategies. Note the behavior of the final fortune and the number of trials, and in particular, note which strategy seems to work best.

  1. Timid Play
  2. Bold Play
  3. Bet $4 on each trial.

Simulation Exercise 3. Repeat Exercise 1 with p = 0.45.

Simulation Exercise 4. Repeat Exercise 1 with p = 0.55.

Red and Black

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