# Asymmetries in Equilibrium Utility

In this lecture, 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?

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 $$1$$.