Equilibria of Dynamic Games with Many Players: Existence, Approximation, and Market Structure

TL;DR

This paper establishes the existence and approximation of stationary equilibria in large-player dynamic games and links these equilibria to market fragmentation or concentration, providing insights into industry structure.

Contribution

It provides conditions ensuring stationary equilibrium existence and approximation to Markov perfect equilibrium, and connects these to market fragmentation in large industries.

Findings

Stationary equilibrium exists under certain conditions.

Stationary equilibrium approximates Markov perfect equilibrium in large games.

Market fragmentation occurs when conditions favor decreasing returns to larger states.

Abstract

In this paper we study stochastic dynamic games with many players; these are a fundamental model for a wide range of economic applications. The standard solution concept for such games is Markov perfect equilibrium (MPE), but it is well known that MPE computation becomes intractable as the number of players increases. We instead consider the notion of stationary equilibrium (SE), where players optimize assuming the empirical distribution of others' states remains constant at its long run average. We make two main contributions. First, we provide a rigorous justification for using SE. In particular, we provide a parsimonious collection of exogenous conditions over model primitives that guarantee existence of SE, and ensure that an appropriate approximation property to MPE holds, in a general model with possibly unbounded state spaces. Second, we draw a significant connection between the validity of SE, and market structure: under the same conditions that imply SE exist and approximates MPE well, the market becomes fragmented in the limit of many firms. To illustrate this connection, we study in detail a series of dynamic oligopoly examples. These examples show that our conditions enforce a form of "decreasing returns to larger states"; this yields fragmented industries in the limit. By contrast, violation of these conditions suggests "increasing returns to larger states" and potential market concentration. In that sense, our work uses a fully dynamic framework to also contribute to a longstanding issue in industrial organization: understanding the determinants of market structure in different industries.