Zero-Sum Games Two-Person Zero-Sum and The Gamblerís
Games with Applications
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.
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
B = [ (1 - p) / p ] (T+R) / U
Therefore, the Ruin Probability, i.e., the probability of losing it all $R is: (1 Ė W).
Notice: This results are subject to the condition that the targeted-winning amount ($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) / 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.