Approximation and characterization of Nash equilibria of large games

We characterize Nash equilibria of games with a continuum of players in terms of approximate equilibria of large finite games. This characterization precisely describes the relationship between the equilibrium sets of the two classes of games. In particular, it yields several approximation results for Nash equilibria of games with a continuum of players, which roughly state that all finite-player games that are sufficiently close to a given game with a continuum of players have approximate equilibria that are close to a given Nash equilibrium of the non-atomic game.

The Master Equation and the Convergence Problem in Mean Field Games

This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. The book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. The book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

Introduction

This chapter provides a background on recent advances in the theory of mean field games (MFGs). MFGs has met an amazing success since pioneering works of more than ten years ago. It gives a self-contained study of the so-called master equation and an answer to the convergence problem. MFGs should be understood as games with a continuum of players, each of them interacting with the whole statistical distribution of the population. In this regard, they are expected to provide an asymptotic formulation for games with finitely many players with mean field interaction. This chapter focuses on the converse problem, which may be formulated by confirming whether the equilibria of the finite games converge to a solution of the corresponding MFG as the number of players becomes very large.

Learning from Others' Outcomes

I develop a simple model of social learning in which players observe others’ outcomes but not their actions. A continuum of players arrives continuously over time, and each player chooses once-and-for-all between a safe action (which succeeds with known probability) and a risky action (which succeeds with fixed but unknown probability, depending on the state of the world). The actions also differ in their costs. Before choosing, a player observes the outcomes of K earlier players. There is always an equilibrium in which success is more likely in the good state, and this alignment property holds whenever the initial generation of players is not well informed about the state. In the case of an outcome-improving innovation (where the risky action may yield a higher probability of success), players take the correct action as K → ∞. In the case of a cost-saving innovation (where the risky action involves saving a cost but accepting a lower probability of success), inefficiency persists as K → ∞ in any aligned equilibrium. Whether inefficiency takes the form of under-adoption or over-adoption also depends on the nature of the innovation. Convergence of the population to equilibrium may be nonmonotone. (JEL D81, D83, O32, Q12, Q16)

From Peer Pressure to Biased Norms

This paper studies a coordination game between a continuum of players with heterogeneous tastes who perceive peer pressure when behaving differently from each other. It characterizes the conditions under which a social norm—a mode of behavior followed by many—exists in equilibrium and the patterns of norm compliance. The emergent norm may be biased compared to the average taste in society, yet endogenously upheld by the population. Strikingly, a biased norm will, under some circumstances, be more sustainable than a non-biased norm, which may explain the bias of various social and religious norms. (JEL D11, Z12, Z13)