Efficiency of equilibria in games with random payoffs

TL;DR

This paper studies the behavior of pure equilibria in large random two-strategy games, showing that certain equilibrium-related payoffs converge to deterministic limits as the number of players grows.

Contribution

It provides a detailed analysis of the asymptotic properties of pure equilibria and social utility in large random games with i.i.d. payoffs, including convergence results.

Findings

Number of pure equilibria follows a Poisson(1) distribution without atoms.

Average social utility converges to deterministic limits as players increase.

Optimal and extremal pure Nash equilibria payoffs become predictable in large games.

Abstract

We consider normal-form games with $n$ players and two strategies for each player, where the payoffs are i.i.d. random variables with some distribution $F$ and we consider issues related to the pure equilibria in the game as the number of players diverges. It is well-known that, if the distribution $F$ has no atoms, the random number of pure equilibria is asymptotically Poisson$(1)$. In the presence of atoms, it diverges. For each strategy profile, we consider the (random) average payoff of the players, called Average Social Utility (ASU). In particular, we examine the asymptotic behavior of the optimum ASU and the one associated to the best and worst pure Nash equilibria and we show that, although these quantities are random, they converge, as $n\to\infty$ to some deterministic quantities.