Not Every Two-Person Zero-Sum Game has Saddle Point

Theorem

Not every two-person zero-sum game has a saddle point.

Proof

Consider the game of Matching Pennies.

Recall its payoff table:

		$\text B$
$\text A$		$\begin{array}{r {{|}} c {{|}} } & \text{H} & \text{T} \\ \hline \text{H} & 1, -1 & -1, 1 \\ \hline \text{T} & -1, 1 & 1, -1 \\ \hline \end{array}$

Trivially, by inspection, this has no entry which is the smallest entry in its row and the largest entry in its column.

$\blacksquare$

Sources

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