Consider the following game:
Player B
I II
a_{11}  a_{12}

a_{21}  a_{22}

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,
a_{11}p + a_{21} (1 – p)
When player B chooses strategy II, expected gain to the player A will be given by,
a_{12}p + a_{22} (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
a_{11}p + a_{21} (1 – p) = a_{12}p + a_{22} (1 – p)
p = a_{22} – a_{21}/ (a_{11} + a_{22}) – (a_{12} + a_{21})
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:
a_{11}q + a_{21} (1 – q) and a_{12}q + a_{22} (1 – q)
Equating the expected losses, we get
q = a_{22} – a_{21}/ (a_{11} + a_{22}) – (a_{12} + a_{21})
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 = a_{11}a_{22} – a_{12}a_{21}/ (a_{11} + a_{22}) – (a_{12} + a_{21})
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