Mean Field Equilibrium: Uniqueness, Existence, and Comparative Statics

Abstract

The standard solution concept for stochastic games is Markov perfect equilibrium (MPE); however, its computation becomes intractable as the number of players increases. Instead, we consider mean field equilibrium (MFE) that has been popularized in the recent literature. MFE takes advantage of averaging effects in models with a large number of players. We make three main contributions. First, our main result provides conditions that ensure the uniqueness of an MFE. We believe this uniqueness result is the first of its nature in the class of models we study. Second, we generalize previous MFE existence results.Third, we provide general comparative statics results. We apply our results to dynamic oligopoly models and to heterogeneous agent macroeconomic models commonly used in previous work in economics and operations.

Markov Distributional Equilibrium Dynamics in Games with Complementarities and No Aggregate Risk

“…9 Our approach delivers a new collection of computable equilibrium comparative statics/dynamics results. Thus, we complement and extend the recent stationary equilibrium comparative statics results from a class of mean field games and oblivious equilibria to distributional games and dynamic equilibrium (see Acemoglu and Jensen, 2015;Light and Weintraub, 2019).…”

“…The literature is extensive and we refer the reader to Acemoglu and Jensen (2015) and Light and Weintraub (2019) for an excellent discussion and citations. In particular, our results provide a foundation for a theory of equilibrium monotone comparative transitional dynamics relative to ordered perturbations of the space of games/economies.…”

“…A second approach (often used in conjunction with aggregation and stationarity) is to simplify interaction in the equilibrium by imposing some behavioral features in its definition. Such approaches include notions of oblivious equilibria (as in Lasry and Lions, 2007, Achdou et al, 2014, Bertucci et al, 2018, Light and Weintraub, 2019, Achdou et al, 2020, mean-field equilibria (as in Weintraub et al, 2008, Adlakha et al, 2015, and Ifrach and Weintraub, 2016, or imagined-continuum equilibria (as in Kalai and Shmaya, 2018), among others. 3 In this paper, we argue that such simplifications need not play a crucial role if one wants to analyze the equilibrium dynamics in a class of games with strategic complementarities we consider.…”

Equilibrium Analysis in the Behavioral Neoclassical Growth Model

“…See p.590 inAcemoglu and Jensen (2015) for additional discussion of such non-monotone equilibrium comparative statics results.4 See for example Lemma 1 (andFigures 1-3) inMilgrom and Roberts (1994) or Definition 5 inAcemoglu and Jensen (2015) Milgrom and Roberts (1994). also use local assumptions, but just to derive local comparative statics results (seeFigure 7and the surrounding discussion); this is different from our results, which are global despite being based on local assumptions.5 The literature on mean field games is also related, especially since some papers in this literature, in particularLight and Weintraub (2018), investigate comparative statics. Nevertheless, like the majority of the papers in robust comparative statics, these works also focus on changes in environments that correspond to uniform and global changes (for example, Theorem 3 inLight and Weintraub (2018)).…”

“…also use local assumptions, but just to derive local comparative statics results (seeFigure 7and the surrounding discussion); this is different from our results, which are global despite being based on local assumptions.5 The literature on mean field games is also related, especially since some papers in this literature, in particularLight and Weintraub (2018), investigate comparative statics. Nevertheless, like the majority of the papers in robust comparative statics, these works also focus on changes in environments that correspond to uniform and global changes (for example, Theorem 3 inLight and Weintraub (2018)). See alsoAhn, Kaplan, Moll, Winberry and Wolf (2018) andAchdou, Han, Lasry, Lions and Moll (2018) for applications of related ideas to the analysis of Bewley-Aiyagari-style models.…”