Two-Person Zero-Sum and The Gambler’s
Games 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.

Professor Hossein Arsham   

---

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

Kindly email your comments to:
Professor Hossein Arsham

---

MENU

Decision Tools in Economics & Finance ABC Inventory Classification Autoregressive Time Series Beta and Covariance Computations Bivariate Discrete Distributions Break-Even Analysis and Forecasting Categorized Probabilistic, and Statistical Tools Detecting Trend & Autocrrelation Determination of the Outliers Forecasting by Smoothing Inventory Control Models Linear Optimization Solvers to Download Linear Optimization with Sensitivity Maths of Money: Compound Interest Analysis Matrix Algebra, and Markov Chains Mean, and Variance Estimations Measuring Forecast Accuracy Other Polynomial Regressions Optimal Age for Replacement Parametric System of Linear Equations Performance Measures for Portfolios Plot of a Time Series Predictions by Regression Proportion Estimation Quadratic Regression Regression Modeling Seasonal Index Single-period Inventory Analysis Summarize Your Data System of Equations, and Matrix Inversion Test for Random Fluctuations Test for Seasonality Test for Stationary Time Series Time Series' Statistics Probabilistic Modeling Bayesian Inference for the Mean Bayes' Revised Probability Bivariate Discrete Distributions Comparing Two Random Variables Decision Making Under Uncertainty Determination of Utility Function Making Risky Decisions Measure the Quality of Your Decision Multinomial Distributions Two-Person Zero-Sum Games

Statistics Analysis of Covariance ANOVA for Condensed Data Sets ANOVA for Dependent Populations ANOVA: Testing the Means Bayesian Statistical Inference Bivariate Sampling Statistics Chi-square Test for Relationship Compatibility of Multi-Counts Confidence Intervals for Two Populations Descriptive Statistics Determination of the Outliers Empirical Distribution Function Equality of Multi-variances Estimations With Confidence Goodness-of-Fit for Discrete Variables Identical Populations Testing Index Numbers with Applications K-S Test for Equality of Two Populations Lilliefors Test for Exponentially Multiple Regressions Percentage: Estimation & Testing Paired Proportion Test Polynomial Regressions Pooling Means, and Variances P-values for the Popular Distributions Quadratic Regression Sample Size Determination Revising the Mean and the Variance Scattered Diagram and the Outliers Simple Linear Regression Subjective Assessment of Estimates Subjectivity in Hypothesis Testing Test for Several Correlation Coefficients Test for Homogeneity of a Population Test for Normality Test for Uniform Distribution Testing Poisson Process Test for Randomness Testing Several Proportions Testing the Mean Testing the Medians Testing the Correlation Coefficient Testing Two Populations Testing the Variance The Before-and-After Test The Other Means Two-Way ANOVA Test Two-Way ANOVA with Replications

---

The Copyright Statement: The fair use, according to the 1996 Fair Use Guidelines for Educational Multimedia, of materials presented on this Web site is permitted for non-commercial and classroom purposes only.
This site may be translated and/or mirrored intact (including these notices), on any server with public access. All files are available at http://home.ubalt.edu/ntsbarsh/Business-stat for mirroring.

Kindly e-mail me your comments, suggestions, and concerns. Thank you.

Professor Hossein Arsham   

---

Back to:

Dr Arsham's Home Page

---

EOF: © 1994-2015.

---