Dominated Strategy may be Optimal

Theorem
A dominated strategy of a game may be the optimal strategy for a player of that game.

Proof
Consider the game defined by the following payoff table:

This has two solutions:


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


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

Thus both pure strategies for $A$ are optimal, but $A_1$ is dominated by $A_1$.