<p>This one has infinitely many:</p>
<p><span class="math-container">\begin{array}{|c|c|c|}
\hline
&amp;A&amp;B\\\hline
A&amp;0,0&amp;0,0\\\hline
B&amp;0,0&amp;0,0\\\hline
\end{array}</span></p>
<p>A less trivial example (again with a continuum of NEs, h/t M.Greinecker):</p>
<p><span class="math-container">\begin{array}{|c|c|c|c|c|}
\hline
&amp;A&amp;B&amp;C&amp;D\\\hline
A&amp;1,-1&amp;-1,1&amp;0,0&amp;0,0\\\hline
B&amp;-1,1&amp;1,-1&amp;0,0&amp;0,0\\\hline
C&amp;0,0&amp;0,0&amp;1,-1&amp;-1,1\\\hline
D&amp;0,0&amp;0,0&amp;-1,1&amp;1,-1\\\hline
\end{array}</span></p>

CC BY-SA 4.0
2022-04-13T07:46:13.227Z
There should be more than four equilibria. Note that the set of strategies of a player guaranteeing the minmax and the set of strategies of the other player guaranteeing the maxmin are both closed under mixtures. By the Minimax theorem, both players playing such a mixture will result in a Nash equilibrium. So there should be a continuum of Nash equilibria.
Michael Greinecker
CC BY-SA 4.0
2022-04-13T14:38:06.900Z
@MichaelGreinecker: Thanks. You're right indeed.
Herr K.
CC BY-SA 4.0
2022-04-13T01:55:30.590Z
2022-04-13T14:37:22.570Z
2022-04-13T14:37:22.570Z
Herr K.
<p>Is there any example of a zero-sum game with multiple equilibria?</p>

<game-theory>
Multiple equilibria for zero sum game?
emma patterson
# Multiple equilibria for zero sum game?

(Asked by: username_0 on 2022-04-13)

Is there any example of a zero-sum game with multiple equilibria?

**Tags:** game-theory

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## 1 answers

### Answer by username_1 on 2022-04-13. Score: 6

This one has infinitely many:

\begin{array}{|c|c|c|}
\hline
&A&B\\\hline
A&0,0&0,0\\\hline
B&0,0&0,0\\\hline
\end{array}

A less trivial example (again with a continuum of NEs, h/t M.Greinecker):

\begin{array}{|c|c|c|c|c|}
\hline
&A&B&C&D\\\hline
A&1,-1&-1,1&0,0&0,0\\\hline
B&-1,1&1,-1&0,0&0,0\\\hline
C&0,0&0,0&1,-1&-1,1\\\hline
D&0,0&0,0&-1,1&1,-1\\\hline
\end{array}

> **Comments:**
> **username_2**: There should be more than four equilibria. Note that the set of strategies of a player guaranteeing the minmax and the set of strategies of the other player guaranteeing the maxmin are both closed under mixtures. By the Minimax theorem, both players playing such a mixture will result in a Nash equilibrium. So there should be a continuum of Nash equilibria.  
> **username_1**: @MichaelGreinecker: Thanks. You're right indeed.  

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