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:

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

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: $3$