Suppose a two player zero sum game has the following payoff matrix.
| A\B | \[B_1\] |
\[B_2\] |
| \[A_1\] |
2 | -11 |
| \[A_2\] |
-4 | 3 |
\[A_1, \: A_2\]
with probabilities \[p\]
and \[1-p\]
respectively, then A will expect to receive \[2p-4(1-p)=6p-4\]
for every game he plays. If B plays strategy \[B_2\]
, then A will expect to win \[-p+3(1-p)=3-4p\]
.A will maximise their winnings if he chooses
\[p\]
such that his minimum winnings are as high as possible. This is when \[6p-4=3-4p \rightarrow p=\frac{7}{10}\]
.We can perform the same analysis for player and B.
If A plays strategy 1, and B picks strategies
\[B_1, \: B_2\]
with prbabilities \[q\]
and \[1-q\]
respectively, then B will expect to receive \[-(2q-(1-q))=-(3q-1)\]
for every game he plays. If A plays strategy \[A_2\]
, then B will expect to win \[-(-4q+3(1-q))=-(3-7q)\]
.B will maximise their winnings if he chooses
\[p\]
such that his minimum winnings are as high as possible. This is when \[-(3q-1)=-(3-7q) \rightarrow q=\frac{4}{10}\]
.