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: $3$