<ol>
<li>There is no guarantee that symmetric games have symmetric equilibria.
See <a href="https://www.rochester.edu/college/faculty/markfey/papers/SymmGame3.pdf" rel="noreferrer">this paper</a> for concrete examples. </li>
<li><p>There is also no guarantee that symmetric games have pure-strategy equilibria. For example,  the following game is symmetric and has no pure-strategy equilibria 
<span class="math-container">$$
\begin{array}{cccc}
&amp; \mathrm{a} &amp; \mathrm{b} &amp; \mathrm{c}\\
\mathrm{a} &amp; 1,1 &amp; 0,2 &amp; 5,0\\
\mathrm{b} &amp; 2,0 &amp; 3,3 &amp; 0,4\\
\mathrm{c} &amp; 0,5 &amp; 4,0 &amp; 1,1\\
\end{array}
$$</span></p></li>
<li><p>There are two special kinds of games that are known to always have pure strategy equilibria: <a href="https://web.stanford.edu/~jdlevin/Econ%20286/Supermodular%20Games.pdf" rel="noreferrer">supermodular games</a> and <a href="https://en.wikipedia.org/wiki/Potential_game" rel="noreferrer">potential games</a>. </p></li>
</ol>

CC BY-SA 4.0
2020-06-03T20:23:19.340Z
The first has a symmetric equilibrium. Just not in pure strategies.
Walrasian Auctioneer
CC BY-SA 4.0
2020-06-03T20:31:32.180Z
Thank you, I will change the example.
brunosalcedo
CC BY-SA 4.0
2020-06-03T20:16:33.447Z
2020-06-03T20:32:09.990Z
2020-06-03T20:32:09.990Z
brunosalcedo
<p>I have 2 symmetric players <span class="math-container">$A$</span> and <span class="math-container">$B$</span>. </p>

<p>Each of them has 2 decision variables <span class="math-container">$x_i\in[0, \beta]$</span> and <span class="math-container">$y_i\in[0,1]$</span>, where <span class="math-container">$i\in\{A,B\}$</span>. </p>

<p>Their payoff functions are symmetric, i.e., if you swap the label <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, you get the other player's payoff function. And the payoff function is continuous in both <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. </p>

<p>My question is: for this type of game, does it always have symmetric pure strategy equilibrium? In general, what are the conditions to guarantee the existence of symmetric pure strategy equilibrium? </p>

CC BY-SA 4.0
2020-06-03T19:54:03.907Z
In general it does not even have a pure strategy equilibrium. I believe there are some results of existence of such equilibrium in supermodular games.
Walrasian Auctioneer
<game-theory><nash-equilibrium>
Existence of Symmetric Pure Strategy Equilibrium
KevinKim
# Existence of Symmetric Pure Strategy Equilibrium

(Asked by: username_0 on 2020-06-03)

I have 2 symmetric players $A$ and $B$.

Each of them has 2 decision variables $x\_i\in[0, \beta]$ and $y\_i\in[0,1]$, where $i\in\{A,B\}$.

Their payoff functions are symmetric, i.e., if you swap the label $A$ and $B$, you get the other player's payoff function. And the payoff function is continuous in both $x$ and $y$.

My question is: for this type of game, does it always have symmetric pure strategy equilibrium? In general, what are the conditions to guarantee the existence of symmetric pure strategy equilibrium?

**Tags:** game-theory, nash-equilibrium
> **Comments:**
> **username_1**: In general it does not even have a pure strategy equilibrium. I believe there are some results of existence of such equilibrium in supermodular games.  

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## 1 answers

### Answer by username_2 on 2020-06-03. Score: 6 | Accepted answer

1. There is no guarantee that symmetric games have symmetric equilibria.
   See [this paper](https://www.rochester.edu/college/faculty/markfey/papers/SymmGame3.pdf) for concrete examples.
2. There is also no guarantee that symmetric games have pure-strategy equilibria. For example, the following game is symmetric and has no pure-strategy equilibria
   $$
   \begin{array}{cccc}
   & \mathrm{a} & \mathrm{b} & \mathrm{c}\\
   \mathrm{a} & 1,1 & 0,2 & 5,0\\
   \mathrm{b} & 2,0 & 3,3 & 0,4\\
   \mathrm{c} & 0,5 & 4,0 & 1,1\\
   \end{array}
   $$
3. There are two special kinds of games that are known to always have pure strategy equilibria: [supermodular games](https://web.stanford.edu/~jdlevin/Econ%20286/Supermodular%20Games.pdf) and [potential games](https://en.wikipedia.org/wiki/Potential_game).

> **Comments:**
> **username_1**: The first has a symmetric equilibrium. Just not in pure strategies.  
> **username_2**: Thank you, I will change the example.  

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