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Two-Person Zero-Sum Game with Mixed Strategies

Consider the following game:

 

Player B

I           II

a11 a12

 

a21 a22

 

I

Player A

II

 

Let us suppose this game does not have any saddle points, we assume that both the players use mixed strategies.

If player A choose strategy I with the probability p and strategy II with the probability 1 – p. For player B, if he chooses strategy I, then expected gain to the player A will be given by,

a11p + a21 (1 – p)

When player B chooses strategy II, expected gain to the player A will be given by,

a12p + a22 (1 – p)

 

Considering the optimal plan for player A; he requires to have equal expected gain for each strategies adopted by player B. In this case, we get-

a11p + a21 (1 – p) = a12p + a22 (1 – p)

p = a22 – a21/ (a11 + a22) – (a12 + a21)

 

In the similar way, if player B chooses strategy I with the probability q along with strategy with the probability (1 – q), the expected losses will be given by,

For player B:

a11q + a21 (1 – q) and a12q + a22 (1 – q)

Equating the expected losses, we get-

q = a22 – a21/ (a11 + a22) – (a12 + a21)

 

Now, value of the game, v can be calculated by substituting the value of p in any one of the above equations. On simplification, we get-

 

v = a11a22 – a12a21/ (a11 + a22) – (a12 + a21)

 

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