Linear programming technique - Linear Programming Approach To Game Theory

When there is neither saddle point nor dominance in a problem of game theory and the payoff matrix is of order 3x3 or higher, the probability and graphical methods cannot be employed. In such a case, linear programming approach may be followed to solve the game.

Linear programming technique
 
A general approach to solve a game by linear programming technique is presented below. Consider the following game:

Statement of the problem

We can use simplex method to solve the above problem. For this purpose, we have to introduce non-negative slack variables s₁ , s₂ ,..., s_m to each of the inequalities. So the problem can be restated as follows: Restatement of the problem:

In a similar manner we can determine the optimal strategy for player A.
 
Application
 
We illustrate the method for a 2X2 zero sum game.