MIXED STRATEGIES: GAMES WITHOUT SADDLE POINT - Quantitative Techniques for management

Game theory without Saddle Point Example

For any given pay off matrix without saddle point the optimum mixed strategies are shown in Table

Mixed Strategies

Let p₁ and p₂ be the probability for Player A.
Let q₁ and q₂ be the probability for Player B.
Let the optimal strategy be S_A for player A and S_B for player B.
Then the optimal strategies are given in the following tables.

Optimum Strategies

p₁and p₂ are determined by using the formulae,

and p₂= 1-p₁

and q₂= 1-q₁
the value of the game w.r.t. player A is given by,

Example : Solve the pay-off given table matrix and determine the optimal strategies and the value of game.

Game Problem

Solution: Let the optimal strategies of S_A and S_B is as shown in tables.

Optimal Strategies

The given pay-off matrix is shown below in Table.

Pay-off Matrix or Maximin Procedure

Therefore, there is no saddle point and hence it has a mixed strategy. Applying the probability formula,

and p₂= 1-p₁
= 1-1/4= 3/4

and q₂= 1-q₁

=14/4

The optimum mixed strategies are shown in table below.

Optimum Mixed Strategies