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## Games with Mixed Strategies

Games with Mixed StrategiesMixed Strategyto determine a saddle point. The optimal mix for each player may be determined by allocating each strategy a probability of it being selected. Therefore these mixed strategies are probabilistic combinations of accessible better strategies and these games consequently known asProbabilistic games.The probabilistic mixed strategy games without saddle points are generally solved by any of the below specified methods

Sl. No.MethodApplicable to1

Analytical Method

2x2 games

2

Graphical Method

2x2, mx2 and 2xn games

3

Simplex Method

2x2, mx2, 2xn and mxn games

Analytical MethodA 2 x 2 payoff matrix where there is no saddle point can be solved with the help of analytical method.

Given below the matrix

[ a

_{11}a_{12}]a

_{21}a_{22}Value of the game is

V= (a

_{11}a_{22}- a_{21}a_{12}) / (a_{11}+a_{22}) - (a_{12}+a_{21})With the coordinates

With the coordinates

x

_{1}= (a_{22}– a_{21})/(a_{11}+a_{22})-(a_{12}+a_{21}) , x_{2 = }a_{11}- a_{12}/ (a_{11}+a_{22}) – (a_{12}+a_{21})y

_{1 = }a_{22 }- a_{12 }/ (a_{11}+a_{22})-(a_{12}+a_{21}), y_{2 = }a_{11}- a_{21}/ (a_{11}+a_{22}) – (a_{12}+a_{21})Alternative process to solve the strategyGraphical method

The graphical method is required to solve the games whose payoff matrix has

Algorithm for solving 2 x n matrix games_{1}= 0, x_{1}= 1^{st}row in the payoff matrix on the vertical line x_{1}= 1 and the points of the II^{nd}row in the payoff matrix on the vertical line x_{1}= 0._{1j}on axis x_{1}= 1 is then connected to the point a_{2j}on the axis x_{1}= 0 to provide a straight line. Make 'n' straight lines for j=1, 2... n and find out the highest point of the lower envelope achieved. This point will be themaximin point.Algorithm for solving m x 2 matrix games_{1}=0, x_{1}= 1^{st}row in the payoff matrix on the vertical line x_{1}= 1 and the points of the II^{nd}row in the payoff matrix on the vertical line x_{1}= 0._{1j}on axis x_{1}= 1 is then connected to the point a_{2j}on the axis x_{1}= 0 to provide a straight line. Make ‘n’ straight lines for j=1, 2… n and find out the lowest point of the upper envelope achieved. This point will be theminimax point.Simplex Method

Assume the 3 x 3 matrix

According to the assumptions, A always tries to select the set of strategies with the non-zero probabilities like p

_{1}, p_{2}, p_{3}where p_{1}+ p_{2}+ p_{3}= 1 that maximizes his least expected gain.Likewise B would select the set of strategies with the non-zero probabilities like q

_{1}, q_{2}, q_{3}where q_{1}+ q_{2}+ q_{3}= 1 that minimizes his highest expected loss.Step 1Determine the minimax and maximin value in the given matrix

Step 2

The purpose of A is to maximize the value, which is equal to minimizing the value 1/V. The LPP can be written as

Min 1/V = p

_{1}/V + p_{2}/V + p_{3}/V& constraints ≥ 1

It can be written as

Min 1/V = x

_{1}+ x_{2}+ x_{3}& constraints ≥ 1

Likewise for B, we obtain the LPP as the dual of the above LPP

Max 1/V = Y

_{1}+ Y_{2}+ Y_{3}& constraints ≤ 1

Where Y

_{1}= q_{1}/V, Y_{2 }= q_{2}/V, Y_{3 }= q_{3}/VStep 3

Work out the LPP with the use of simplex table and get the optimum strategy for the players..