Two Persons Zero Sum Games

8/13/2019 Two Persons Zero Sum Games

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Two-Person Zero-Sum and The GamblersGames with Applications

This site is a part of the JavaScriptE-labslearning objects

for decision making. Other JavaScript in this series arecategorized under different areas of applications in the

MENUsection on this page.Professor Hossein Arsham

Game theory describes the situations involving conflict inwhich the payoff is affected by the actions and counter-

actions of intelligent opponents. The following JavaScript isdesigned 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 ofthe 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 lossfor 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 tominimize the payoff to player I. Note that one may multiplyall 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 apure random decision problem. Switching to differentstrategy with specific frequencies obtained by optimal

solution is aimed at confusing the other player. The followinggame is an example of pure random decision-making

problem.

Ruin Probability:The second JavaScript is for sensitivityanalysis 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 = thetargeted-winning amount, U = the size of each bet, and p =

The probability of winnig any bet, then the probability ofprobability (W) of reaching the target, i.e., leaving with

$(R+T ) is:
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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 losingit all $R is: (1 W).

Notice:This results are subject to the condition that thetargeted-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) / VFor 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

1

2

3


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I 4

5

Clear

Probability of a Targeted-Winning Amount or Losing ItAll Game

Amount of money you bring to the table ($R): 90

The targeted-winning amount ($T): 10

The size of each bet ($U): 1

The probability of winnig any bet (p): 0.49

Clear

Probability of reaching the target,R+T ($):

Probability of losing it all, -R ($):

For Technical Details, Back to:

Two-Person Zero-Sum and The Gamblers Games with

Applications

Kindlyemailyour comments to:Professor Hossein Arsham

MENU
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