Rare Nash Equilibria and the Price of Anarchy in Large Static Games

TL;DR

This paper analyzes the asymptotic behavior of Nash equilibria in large static anonymous games with random types, deriving laws of large numbers, large deviation principles, and probabilistic bounds on the price of anarchy.

Contribution

It introduces a comprehensive probabilistic framework for understanding rare equilibria and the price of anarchy in large games, extending beyond existing worst-case analyses.

Findings

Characterizes almost sure limits of Nash equilibria as the number of players grows.

Establishes large deviation principles for rare Nash equilibria.

Provides probabilistic bounds on the price of anarchy in congestion games.

Abstract

We study a static game played by a finite number of agents, in which agents are assigned independent and identically distributed random types and each agent minimizes its objective function by choosing from a set of admissible actions that depends on its type. The game is anonymous in the sense that the objective function of each agent depends on the actions of other agents only through the empirical distribution of their type-action pairs. We study the asymptotic behavior of Nash equilibria, as the number of agents tends to infinity, first by deriving laws of large numbers characterizes almost sure limit points of Nash equilibria in terms of so-called Cournot-Nash equilibria of an associated nonatomic game. Our main results are large deviation principles that characterize the probability of rare Nash equilibria and associated conditional limit theorems describing the behavior of equilibria conditioned on a rare event. The results cover situations when neither the finite-player game nor the associated nonatomic game has a unique equilibrium. In addition, we study the asymptotic behavior of the price of anarchy, complementing existing worst-case bounds with new probabilistic bounds in the context of congestion games, which are used to model traffic routing in networks.