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Two-person Zero-sum Game with Mixed Strategies

Consider the following game :

If this game does not have saddle point, then we assume that both players use mixed

strategies.

 

Let player A select strategy I with probability p and strategy II with probability 1 – p Suppose player B select strategy I, then the expected gain to player A is given by a11 p + a21 (1 - p).

 

If player B select strategy II, then the expected gain to player A is given by a12 p + a22 (1 - p).

 

The optimal plan for player A requires that its expected gain to be equal for each strategies of player B. Thus we obtain

 

all P + a21 (1 - p) = a l2 P + a22 (1 - p)

 

p = a22-a21/(a11+a22) – (a12 +a21)

 

Similarly, let player B selects strategy I with probability q and strategy II with probability

1-q . The expected loss to player B with respect to the strategies of player A are

 

a11 q + a12 (1 - q) and a21 q + a21 (1 - q).

 

By equating the expected losses of player B we obtain

q = a22-a12/(a11+a22) – (a12 +a21)

 

The value of game v is found by substituting the value of p in one of the equations for the expected gain of A and on simplification, we obtain

 

v = a22 a22-a12­ a21 /(a11+a22) – (a12 +a21)

 

Example 3. Determine the solution of the following game :

Solution. Clearly the given game has no saddle point. So the players have to use mixed Strategies

Let the mixed strategies for A as SA   =

 

where                                                     P2   =

and the mixed strategies for B as SB

 

q2    =

(A1 A2

P1 P2 )

1 - P 1

(B1 B2

ql q2 )

1 - ql

 

P1 = 4-2/(3 +4) - (1 + 2) = 2/4 = 1/2,  p2 = 1 - p1 = 1/2

ql = 4 – 1/(3 + 4) - (1 + 2)= ¾, q2 = 1-q1 = 1/4

 

Thus the optimal strategy for A is SA = A1   A2

½       ½

and for B is                                         SB =  ( B1    B2

¾    ¾ )

and the value of the game is 5/2.