Zero-Sum Games

# Two-Person Zero-Sum and The Gambler’sGames with Applications

This site is a part of the JavaScript E-labs learning objects for decision making. Other JavaScript in this series are categorized under different areas of applications in the MENU section on this page.

Game theory describes the situations involving conflict in which the payoff is affected by the actions and counter-actions of intelligent opponents. The following JavaScript is designed for two-person zero-sum games. Each player has at most strategies (i.e., choices) from which to select.

Enter the payoff matrix starting at the upper-right corner of the table. Then click on the Calculate button.

Notice that the payoff matrix is oriented for player I. Therefore, a positive payoff is a gain for player I and a loss for player II and a negative payoff is a gain for player II and a loss for player I. Clearly, the player's II objective is to minimize the payoff to player I. Note that one may multiply all elements of the payoff matrix to one player in order to obtain the payoff matrix to the other player.

By now you may have realized that the above game is not a pure random decision problem. Switching to different strategy with specific frequencies obtained by optimal solution is aimed at confusing the other player. The following game is an example of pure random decision-making problem.

Ruin Probability: The second JavaScript is for sensitivity analysis of the winning the target dollar amount or losing it all (i.e., ruin) game.

Let R = the amount of money you bring to the table, T = the targeted-winning amount, U = the size of each bet, and p = The probability of winnig any bet, then the probability of probability (W) of reaching the target, i.e., leaving with \$(R+T ) is:

W = (A -1) / (B -1)

where,

A = [ (1 - p) / p ] R / U

and

B = [ (1 - p) / p ] (T+R) / U

Therefore, the Ruin Probability, i.e., the probability of losing it all \$R is: (1 – W).

Notice: This results are subject to the condition that the targeted-winning amount (\$T) must be much less than amount of money you bring to the table (\$R). That is (\$T) must be a fraction (f) of (\$R).

Remember that if you bet too much you will leave a loser, while if you bet too little, your capital will grow too slowly. You may ask what fraction (f) of R you should bet always. Let V be the amount that you win for every dollar that you risk, then the optimal fraction is:

f = p - (1 - p) / V

For example for p = 0.5, and v = 2, the optimal decision value for f is 25% of your capital R. The above result, recommend that fraction (f) of R you should bet always, must never exceed p.

Payoff Matrix to Player I
 Player II 1 2 3 4 5 P l a y e r   I 1 2 3 4 5 Probability of a Targeted-Winning Amount or Losing It All Game

Amount of money you bring to the table (\$R):
The targeted-winning amount (\$T):
The size of each bet (\$U) :
The probability of winnig any bet (p) :
Probability of reaching the target, R+T (\$) :
Probability of losing it all, -R (\$) :

For Technical Details, Back to:
Two-Person Zero-Sum and The Gambler’s Games with Applications

Professor Hossein Arsham