<p>In &quot;nonatomic games&quot; (I'm a bit informal here), players that are identical will generally have the same payoff in every equilibrium. However, the payoff can vary from equilibrium to equilibrium.</p>
<p>Let's take the space of players to be <span class="math-container">$[0,1]$</span>. All players can choose actions from the same space <span class="math-container">$A$</span>. There is a set <span class="math-container">$S$</span> that describes a summary statistic of an action profile, such as the population distribution on choices in <span class="math-container">$A$</span>, the &quot;average&quot; action, etc. Importantly, this summary does not depend on any single player; every player can change their behavior without changing this statistic. All players have the same payoff function <span class="math-container">$v:A\times S\to\mathbb{R}$</span>. If you have an equilibrium in such a game, it will come with some equilibrium summary statistic <span class="math-container">$s^*$</span> that no player can influence. Then every player will choose a best response to <span class="math-container">$s^*$</span> and receive the same payoff, namely <span class="math-container">$\max_A v(\cdot,s^*)$</span>.</p>
<p>But different equilibria can come with different summary statistics. Here is an example: Let <span class="math-container">$A=\{0,1\}$</span> and let each element of <span class="math-container">$S$</span> specify the fraction of players choosing action <span class="math-container">$1$</span>. The payoff function is given by <span class="math-container">$v(a,s)=a-2|a-s|$</span>. There is an equilibrium in which everyone plays <span class="math-container">$0$</span> and receives a payoff of <span class="math-container">$0$</span> and an equilibrium in which everyone plays <span class="math-container">$1$</span> and receives a payoff of <span class="math-container">$1$</span>.</p>

CC BY-SA 4.0
2022-12-26T00:22:00.690Z
2022-12-26T00:22:00.690Z
Michael Greinecker
<p>In <a href="https://youtu.be/JUc5BbotCVA?t=2571" rel="nofollow noreferrer">this lecture</a>, the professor says that all Nash Equilibria have the same utility in non-atomic selfish routing, whereas this is not guaranteed in atomic selfish routing.  It is unclear how general the statement is intended to be.</p>
<p>Does this hold for games in general?  If a game is played by a sufficiently large number of nearly-identical players, such that none of them has &quot;market power&quot; to meaningfully manipulate utility, does that guarantee that all Nash Equilibria have identical utility?  Are asymmetries in utility at Nash Equilibria only a product of at least one player having &quot;market impact&quot; on utility?  If so, how can one intuitively see that this fact actually holds?</p>

<game-theory><utility><nash-equilibrium><mechanism-design>
Asymmetries in Equilibrium Utility
user10478
# Asymmetries in Equilibrium Utility

(Asked by: username_0 on 2022-12-25)

In [this lecture](https://youtu.be/JUc5BbotCVA?t=2571), the professor says that all Nash Equilibria have the same utility in non-atomic selfish routing, whereas this is not guaranteed in atomic selfish routing. It is unclear how general the statement is intended to be.

Does this hold for games in general? If a game is played by a sufficiently large number of nearly-identical players, such that none of them has "market power" to meaningfully manipulate utility, does that guarantee that all Nash Equilibria have identical utility? Are asymmetries in utility at Nash Equilibria only a product of at least one player having "market impact" on utility? If so, how can one intuitively see that this fact actually holds?

**Tags:** game-theory, utility, nash-equilibrium, mechanism-design


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## 1 answers

### Answer by username_1 on 2022-12-26. Score: 3 | Accepted answer

In "nonatomic games" (I'm a bit informal here), players that are identical will generally have the same payoff in every equilibrium. However, the payoff can vary from equilibrium to equilibrium.

Let's take the space of players to be $[0,1]$. All players can choose actions from the same space $A$. There is a set $S$ that describes a summary statistic of an action profile, such as the population distribution on choices in $A$, the "average" action, etc. Importantly, this summary does not depend on any single player; every player can change their behavior without changing this statistic. All players have the same payoff function $v:A\times S\to\mathbb{R}$. If you have an equilibrium in such a game, it will come with some equilibrium summary statistic $s^\*$ that no player can influence. Then every player will choose a best response to $s^\*$ and receive the same payoff, namely $\max\_A v(\cdot,s^\*)$.

But different equilibria can come with different summary statistics. Here is an example: Let $A=\{0,1\}$ and let each element of $S$ specify the fraction of players choosing action $1$. The payoff function is given by $v(a,s)=a-2|a-s|$. There is an equilibrium in which everyone plays $0$ and receives a payoff of $0$ and an equilibrium in which everyone plays $1$ and receives a payoff of $1$.


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