GAMBLERS_RUIN_SIMULATION
Simulation of Gambler's Ruin

GAMBLERS_RUIN_SIMULATION is a MATLAB program which simulates the game of gambler's ruin.

In the game of gambler's ruin, two gamblers, A and B, start with $A_STAKES and $B_STAKES, respectively, and play until one of them is bankrupt. They repeatedly flip a coin. On heads, A wins one dollar from B, and on tails, B wins one dollar from A.

The game is easy to simulate, and can also be analyzed. It is worth study in part because the analysis or simulation of the game reveals some surprising features about the chances of winning, the typical length of a game, and the number of times each player will be in the lead.

In particular, most people will be surprised to find that if A starts with $1 and B with $100, the "average" game will take 100 coin tosses.

It is known that, if A and B start the game with $A_STAKES and $B_STAKES,

• the expected number of coin tosses required to complete the game is A_STAKES * B_STAKES.
• the probability that A will win the game is A_STAKES / ( A_STAKES+B_STAKES).
• the lead may change many times in one game, but, if a large number of games are played, the most common case is that the lead never changes.

Note that the game of gambler's ruin can be regarded as a case of a 1D random walk in a finite region, in which the "particle" starts A_STAKES steps from one wall and B_STAKES steps from the other, and in which the walk ends when the particle strikes a wall.

The program GAMBLERS_RUIN_PLOT displays the "score" for player A at each step of the game.

The program GAMBLERS_RUIN_SIMULATION plays the game a number of times, and computes statistics of the number of steps and the probability of winning, as well as histograms of the number of steps and the number of times the lead was reversed.

Usage:

gamblers_ruin_plot ( a_stakes, b_stakes )

• a_stakes and b_stakes are the amounts of money with which A and B start the game.

gamblers_ruin_simulation ( a_stakes, b_stakes, game_num )

• a_stakes and b_stakes are the amounts of money with which A and B start the game.
• game_num is the number of games to play.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

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Reference:

1. Martin Gardner,
   The Mathematical Circus,
   Mathematics Association of America, 1996,
   ISBN13: 978-0883855065,
   LC: QA95.G287.
2. Ian Stewart,
   "Repealing the Law of Averages",
   Scientific American,
   Volume 278, Number 4, April 1998, pages 102-104.

Source Code:

• gamblers_ruin_simulation.m, plays the game many times, print some statistics and displays some histograms.
• gamblers_ruin_plot.m, plays the game once, and displays the status of A's winnings.

Examples and Tests:

• flips_70_70_hist.png, a histogram of the number of times the lead changes when A and B both start the game with $70. The game was run 1,000 times.
• game_5_7_plot.png, a plot of A's wininings in a single game in which A began with $5 and B with $7.
• steps_70_70_hist.png, a histogram of the number of coin tosses required to complete then game when A and B both start the game with $70. The game was run 1,000 times.

You can go up one level to the MATLAB source codes.