Two-Person Zero-Sum Game with Multiple Solutions

Theorem
There exists a two-person zero-sum game with more than one solution.

Proof
Consider the game defined by the following payoff table:

This has two solutions:


 * $(1): \quad A: \left({1/3, 2/3}\right), B: \left({0, 1, 0}\right)$


 * $(2): \quad A: \left({2/3, 1/3}\right), B: \left({0, 1, 0}\right)$

It follows that the strategies:
 * $\left({\dfrac t 3, \dfrac {2 \left({1 - t}\right)} 3}\right), B: \left({0, 1, 0}\right)$

for all $0 \le t \le 1$ are also solutions.