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Operations Management - Game Theory, Goal Programming & Queuing Theory

Problem of Graphical Solution Of A 2X2 Game With No Saddle Point

   Posted On :  25.06.2018 12:29 am

Solve the following game by graphical method.

Problem 
 
Solve the following game by graphical method.



Solution
 
First consider the row minima



Next, consider the column maxima


 
We see that Maximum { row minima}   Minimum { column maxima }
 
So, the game has no saddle point. It is a mixed game.
 
Let p be the probability that player A will use his first strategy.
 
Then the probability that A will use his second strategy is 1-p.
 
When B uses his first strategy

The expected value of pay-off to player A is given by



When B uses his second strategy The expected value of pay-off to player A is given by



Consider equations (1) and (2). For plotting the two equations on a graph sheet, get some points on them as follows:



Graphical solution
 
Take probability and expected value along two rectangular axes in a graph sheet. Draw two straight lines given by the two equations (1) and (2). Determine the point of intersection of the two straight lines in the graph. This will provide the common solution of the two equations (1) and (2). Thus we would get the value of the game.

Represent the two equations by the two straight lines AB and CD on the graph sheet. Take the point of intersection of AB and CD as T. For this point, we have p = 1/3 and E = -2. Therefore, the value V of the game is -2.



Tags : Operations Management - Game Theory, Goal Programming & Queuing Theory
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