Two-Person Zero-Sum Game with Multiple Solutions

Theorem

There exists a two-person zero-sum game with more than one solution.

Proof

Consider the game defined by the following payoff table:

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

This has two solutions:

$(1): \quad A: \left({1/3, 2/3}\right), B: \left({0, 1, 0}\right)$

$(2): \quad A: \left({2/3, 1/3}\right), B: \left({0, 1, 0}\right)$

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It follows that the strategies:

$\left({\dfrac t 3, \dfrac {2 \left({1 - t}\right)} 3}\right), B: \left({0, 1, 0}\right)$

for all $0 \le t \le 1$ are also solutions.

This article, or a section of it, needs explaining. In particular: Again, the explanation of this needs to wait till more learning has been done. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Sources

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