1. 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.

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

3. 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:

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

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

6. 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:

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

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