Pure strategy problem
Example : is solved using computer with TORA. From Main menu of TORA package select Zero-sum Games option. Click Go to Input Screen Enter the input values of the problem as shown in the figure.
Solving Pure Strategy Problem Using TORA (Input Screen)
Now, go to solve menu and click. Another screen appears with Solved Problem Select solve problem and click LP-based. Then select the output format screen and click Go to Output Screen. The following output screen is displayed, as shown in Figure.
Solving Pure Strategy Problem Using TORA (Output Screen)
The results of the problem can be read directly from the output screen.
Value of the Game to Player A = 1.00
Player A optimal strategies:
Strategies: A1 A2 A3
Probability: 0 1 0
Player B optimal strategies:
Strategies: B1 B2 B3
Probability: 1 0 0
The output also includes the linear programming formulation for Player A.
Mixed Strategy Problem
The output screen for the problem is shown in Figure.
Solving Mixed Strategy Problems Using TORA (Output Screen)
Here the players play both the strategies in what turns out to be a mixed strategy game.
A1 A2 B1 B2
Player A : 0.25 0.75 Player B : 0.5 0.5
Value of the game, v = 3.50.
Example : Solve the following 2 × 3 game given below in table graphically, using computer.
Solution: The game does not possess any saddle point and hence the solution has mixed strategies.
A’s expected payoffs against B’s pure moves are given by
Mixed Strategies Compared
The expected payoff equations are plotted as functions of p which show the payoffs of each column represented as points on two vertical axis. Strategy B1 is plotted by joining value 1 on axis 2 with the value 9 on axis 1. Similarly, other equations are drawn. The output using TORA is given in the figure below:
Graphical Solution of Game Using TORA (Output Screen)
Player A always wants to maximize his minimum expected payoff. Consider the highest point of intersection I on lower envelope of A’s expected payoff equation. The lines B2 and B3 passing through I, are the strategies that B needs to play. Therefore the given matrix is reduced to 2×2 matrix as shown in Table.
Reduced 2×2 Matrix
Solving the 2x2 matrix, the optimal strategies are obtained using the usual method
The value of the game, v = 4.40.
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