9.3 Linear programming and 2 x 2 games : A geometric approach

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9.3 Linear programming and 2 x 2 games : A geometric approach This section will introduce the method of solving a non-strictly determined matrix game without recessive rows or columns. All such games can be converted into linear programming problems. The method applies to a matrix game M that has all positive payoffs.

The method of this section will be illustrated by an example. For the payoff matrix M find the optimal strategies for the two players.

Minimize y subject to the given constraints: Continued …. Add 5 to each entry of M to make all values positive: Minimize y subject to the given constraints:

Continued… Substitute the values for a, b, c, d into the inequalities… .

Solve the linear programming problem Solution is (2/26, 3/26)

Value of the game, v and probability values The value of the game is given by the equation The value of the original matrix with negative values is 26/5 – 5 = 1/5 - This is the probability matrix for R:

Solve the second linear programming problem to find the probability matrix for the second player - -

Value of the game and probability matrix for second player - -