Two-Person Zero-Sum Game which is Not Completely Mixed

Example of Two-Person Zero-Sum Game which is not Completely Mixed

Consider the two-person zero-sum game with the following payoff table:

		$\text B$
$\text A$		$\begin{array} {r {{|}} c {{|}} c {{|}} c {{|}} } & B_1 & B_2 & B_3 \\ \hline A_1 & 4 & 1 & 3 \\ \hline A_2 & 2 & 3 & 4 \\ \hline \end{array}$

It is seen by inspection that:

$A$'s optimum strategy is $\left({1/4, 3/4}\right)$

$B$'s optimum strategy is $\left({1/2, 1/2, 0}\right)$.

It is also noted, in passing, that $B_2$ dominates $B_3$.

Hence the result, by definition of completely mixed game.

$\blacksquare$

Sources

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