# 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 SA for player A and SB for player B.
Then the optimal strategies are given in the following tables.

Optimum Strategies

p1and p2 are determined by using the formulae,

p1 =
a22-a21
(a11+a22)-(a12+a21

and p2= 1-p1

q1=
a22-a21
(a11+a22)-(a12+a21

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

Value of the game, v =
a11 a22– a12a21
(a11+a22)-(a12+a21

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 SA and SB 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,

p1 =
a22-a21
(a11+a22)-(a12+a21
=
4-3
(5+4)-(2+3)
=
1
9-5
=
1
4

and p2= 1-p1
= 1-1/4= 3/4

q1=
a22-a21
(a11+a22)-(a12+a21
4-2
(5+4)-(2+3)
=
2
9-5
=
1
2

and q2= 1-q1

Value of the game, v =
a11 a22– a12a21
(a11+a22)-(a12+a21

=14/4

The optimum mixed strategies are shown in table below.

Optimum Mixed Strategies