SOLVING LP MODEL GAMES GRAPHICALLY USING COMPUTER - Quantitative Techniques for management

Example : Solve the following game shown in table, by linear programming.

Game Problem

Game Problem

The linear programming formulation is given by,
For player A,
Maximize, z = v
Subject to the constraints,
v – 4x1 + 3x2< 0 ......................(i)
v + x1 - 4x2< 0 ......................(ii)
v + 4x1 + x2< 0 ......................(iii)
x1 + x2 = 1 ......................(iv)
where, x1, x2> 0
v is unrestricted.

For Player B,
Minimize, z = v
Subject to constraints,
v – 4y1 + y2 + 4y3> 0 ......................(v)
v + 3y1– 4y2 + y3> 0 ......................(vi)
y1 + y2 + y3 = 1 .....................(vii)
where, y1, y2, y3> 0
v is unrestricted.

The problem can be solved by using linear programming. This can also be solved by using two-person zero-sum game. The output result is given in Figure below:

Two-person Zero-sum Game, Output Result Using TORA

Two-person Zero-sum Game, Output Result Using TORA

The optimal strategies are,
A1 A2
Player A: 0.11 0.89
B1 B2 B3
Player B: 0.22 0 0.78
Value of the game, v = – 2.22


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