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 a_{11} p + a_{21} (1 - p).

If player B select strategy II, then the expected gain to player A is given by a_{12} p + a_{22} (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

a_{ll} P + a_{21} (1 - p) = a_{ l2} P + a_{22} (1 - p)

p = a_{22}-a_{21}/(a_{11}+a_{22}) – (a_{12} +a_{21})

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

a_{11} q + a_{12} (1 - q) and a_{21} q + a_{21} (1 - q).

By equating the expected losses of player B we obtain

q = a_{22}-a1_{2}/(a_{11}+a_{22}) – (a_{12} +a_{21})

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 = a_{22 }a_{22}-a_{12} a_{21 }/(a_{11}+a_{22}) – (a_{12} +a_{21})

** **

**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 S_{A =}
where P2 = and the mixed strategies for B as S
q |
(A1 A2
P 1 - P (B1 B2 q 1 - ql |

P_{1} = 4-2/(3 +4) - (1 + 2) = 2/4 = 1/2, p2 = 1 - p_{1} = 1/2

q_{l} = 4 – 1/(3 + 4) - (1 + 2)= ¾, q_{2} = 1-q_{1} = 1/4

Thus the optimal strategy for A is S_{A} = A1 A2

½ ½

and for B is S_{B} = ( B1 B2

¾ ¾ )

and the value of the game is 5/2.