Games with Mixed Strategies
In some cases, no pure strategy solutions present for the game. In other words, saddle point does not present. In all these games, both players may accept an optimal mix of the strategies known as Mixed Strategy to 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 as Probabilistic games.
The probabilistic mixed strategy games without saddle points are generally solved by any of the below specified methods
2x2, mx2 and 2xn games
2x2, mx2, 2xn and mxn games
A 2 x 2 payoff matrix where there is no saddle point can be solved with the help of analytical method.
Given below the matrix
[ a11 a12 ]
Value of the game is
V= (a11a22 - a21a12) / (a11+a22) - (a12+a21)
With the coordinates
x1 = (a22 – a21)/(a11+a22)-(a12+a21) , x2 = a11- a12 / (a11+a22) – (a12+a21)
y1 = a22 - a12 / (a11+a22)-(a12+a21), y2 = a11- a21/ (a11+a22) – (a12+a21)
Alternative process to solve the strategy
The graphical method is required to solve the games whose payoff matrix has
Algorithm for solving 2 x n matrix games
Algorithm for solving m x 2 matrix games
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 p1, p2, p3 where p1 + p2 + p3 = 1 that maximizes his least expected gain.
Likewise B would select the set of strategies with the non-zero probabilities like q1, q2, q3 where q1 + q2 + q3 = 1 that minimizes his highest expected loss.
Determine the minimax and maximin value in the given matrix
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 = p1/V + p2/V + p3/V
& constraints ≥ 1
It can be written as
Min 1/V = x1 + x2 + x3
Likewise for B, we obtain the LPP as the dual of the above LPP
Max 1/V = Y1 + Y2 + Y3
& constraints ≤ 1
Where Y1 = q1/V, Y2 = q2/V, Y3 = q3/V
Work out the LPP with the use of simplex table and get the optimum strategy for the players..
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