# Dominated Strategy may be Optimal

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## 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:

$\text B$ | ||

$\text A$ | $\begin{array} {r {{|}} c {{|}} }
& B_1 & B_2 \\ \hline A_1 & 1 & 2 \\ \hline A_2 & 1 & 3 \\ \hline \end{array}$ |

This has two solutions:

- $(1): \quad A: \tuple {1, 0}, B: \tuple {1, 0}$

- $(2): \quad A: \tuple {0, 1}, B: \tuple {1, 0}$

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

$\blacksquare$

## Sources

- 1956: Steven Vajda:
*The Theory of Games and Linear Programming*... (previous) ... (next): Chapter $\text{I}$: An Outline of the Theory of Games: $3$