Markov perfect equilibria in industries with complementarities
TL;DR: In this paper, the existence and computation of Markov perfect equilibria in games with a "monotone" structure are studied. But the authors focus on games where agents are typically distributed across many states and will typically take different actions.
Abstract: This paper considers the existence and computation of Markov perfect equilibria in games with a “monotone” structure. Specifically, it provides a constructive proof of the existence of Markov perfect equilibria for a class of games in which a) there is a continuum of players, b) each player has the same per period payoff function and c) these per period payoff functions are supermodular in the player's current and past action and have increasing differences in the player's current action and the entire distribution of actions chosen by other players. The Markov perfect equilibria that are analyzed are symmetric, not in the sense that each player adopts the same action in any period, but rather in the sense that each player uses the same policy function. Since agents are typically distributed across many states they will typically take different actions. The formal environment considered has particular application to models of industries (or economies) in which firms face costs of price adjustment. It is in this context that the results are developed.
Citations
More filters
Posted Content•
TL;DR: In this article, the theory of monotone comparative statics and supermodular games is presented as the appropriate tool to model complementarities, which makes the analysis intuitive and simple, helps in deriving new results and in casting new light on old ones.
Abstract: The theory of monotone comparative statics and supermodular games is presented as the appropriate tool to model complementarities. The approach, which has not yet been fully incorporated into the standard toolbox of researchers, makes the analysis intuitive and simple, helps in deriving new results and in casting new light on old ones. The paper takes stock of recent contributions and develops applications to industrial organization (oligopoly, R&D, and dynamics), finance (currency and banking crisis) and macroeconomics (adjustment and menu costs). Particular attention is devoted to Markov games and to games of incomplete information (including global games).
265 citations
Cites background or methods from "Markov perfect equilibria in indust..."
...The lattice approach provides the key assumptions needed to answer these questions (Christopher Sleet 2001 and Byoung Jun and Vives 2004)....
[...]
...Another successful application of the techniques used to prove the existence of an MPE is provided by Sleet (2001)....
[...]
...The basic theory was developed by Donald M. Topkis (1978, 1979) and further developed and applied to economics by Xavier Vives (1985a, 1990a) and Paul Milgrom and John Roberts (1990a)....
[...]
...Finally, the problem of existence and characterization of Markov perfect equilibria is addressed (Laurent O. Curtat 1996, Sleet 2001)....
[...]
TL;DR: In this paper, the theory of monotone comparative statics and supermodular games is presented as the appropriate tool to model complementarities, which makes the analysis intuitive and simple, helps in deriving new results and in casting new light on old ones.
Abstract: The theory of monotone comparative statics and supermodular games is presented as the appropriate tool to model complementarities. The approach, which has not yet been fully incorporated into the standard toolbox of researchers, makes the analysis intuitive and simple, helps in deriving new results and in casting new light on old ones. The paper takes stock of recent contributions and develops applications to industrial organization (oligopoly, R&D, and dynamics), finance (currency and banking crisis) and macroeconomics (adjustment and menu costs). Particular attention is devoted to Markov games and to games of incomplete information (including global games).
250 citations
TL;DR: In this article, a limiting mean field model is proposed for symmetric games where a large number of players can be in any one of d states and its main properties are characterized.
Abstract: In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study the $N+1$-player problem, which the mean field model attempts to approximate. Our main result is the convergence as $N\to \infty$ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.
109 citations
TL;DR: In this paper, a limiting mean field model is proposed for symmetric games where a large number of players can be in any one of d states and its main properties are characterized.
Abstract: In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N→∞ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.
85 citations
TL;DR: This work finds necessary conditions for the existence of a mean field equilibrium in stochastic dynamic games that exhibit strategic complementarities between players, and shows that there exist a “largest” and a ”smallest” equilibrium among all those where the equilibrium strategy used by a player is nondecreasing.
Abstract: We study a class of stochastic dynamic games that exhibit strategic complementarities between players; formally, in the games we consider, the payoff of a player has increasing differences between her own state and the empirical distribution of the states of other players. Such games can be used to model a diverse set of applications, including network security models, recommender systems, and dynamic search in markets. Stochastic games are generally difficult to analyze, and these difficulties are only exacerbated when the number of players is large (as might be the case in the preceding examples).
We consider an approximation methodology called mean field equilibrium to study these games. In such an equilibrium, each player reacts to only the long-run average state of other players. We find necessary conditions for the existence of a mean field equilibrium in such games. Furthermore, as a simple consequence of this existence theorem, we obtain several natural monotonicity properties. We show that there exist a “largest” and a “smallest” equilibrium among all those where the equilibrium strategy used by a player is nondecreasing, and we also show that players converge to each of these equilibria via natural myopic learning dynamics; as we argue, these dynamics are more reasonable than the standard best-response dynamics. We also provide sensitivity results, where we quantify how the equilibria of such games move in response to changes in parameters of the game (for example, the introduction of incentives to players).
80 citations
Cites background or methods from "Markov perfect equilibria in indust..."
...Much of the attention in prior work on such games has focused on developing existence proofs for equilibrium; see, for example, Curtat (1996), Amir (2002, 2005), Vives (2009), Horst (2005), Nowak (2007), and Sleet (2001) for such results....
[...]
...We note that one alternative to L-BRD and U -BRD is presented by Sleet (2001)....
[...]
...Most closely related to our paper is the work of Sleet (2001), who considers mean field equilibria of a dynamic price-setting game with stochastic, exogenous firm-specific demand shocks per period, that exhibits strategic complementarities....
[...]
References
More filters
Book•
01 Jan 1998
TL;DR: In this article, the authors present techniques from the numerical analysis and applied mathematics literatures and show how to use them in economic analyses, including linear equations, iterative methods, optimization, nonlinear equations, approximation methods, numerical integration and differentiation, and Monte Carlo methods.
Abstract: To harness the full power of computer technology, economists need to use a broad range of mathematical techniques. In this book, Kenneth Judd presents techniques from the numerical analysis and applied mathematics literatures and shows how to use them in economic analyses. The book is divided into five parts. Part I provides a general introduction. Part II presents basics from numerical analysis on R^n, including linear equations, iterative methods, optimization, nonlinear equations, approximation methods, numerical integration and differentiation, and Monte Carlo methods. Part III covers methods for dynamic problems, including finite difference methods, projection methods, and numerical dynamic programming. Part IV covers perturbation and asymptotic solution methods. Finally, Part V covers applications to dynamic equilibrium analysis, including solution methods for perfect foresight models and rational expectation models. A web site contains supplementary material including programs and answers to exercises.
2,880 citations
TL;DR: In this article, a dynamic stochastic model for a competitive industry is developed in which entry, exit, and the growth of firms' output and employment is determined, and conditions under which there is entry and exit in the long run are developed.
Abstract: A dynamic stochastic model for a competitive industry is developed in which entry, exit, and the growth of firms' output and employment is determined. The paper extends long-run industry equilibrium theory to account for entry, exit, and heterogeneity in the size and growth rate of firms. Conditions under which there is entry and exit in the long run are developed. Cross sectional implications and distributions of profits and value of firms are derived. Comparative statics on the equilibrium size distribution and turnover rates are analyzed. Copyright 1992 by The Econometric Society.
2,778 citations
TL;DR: In this article, the authors present a qualitative and quantitative analysis of the standard growth model modified to include precautionary saving motives and liquidity constraints, and address the impact on the aggregate saving rate, the importance of asset trading to individuals, and the relative inequality of wealth and income distributions.
Abstract: We present a qualitative and quantitative analysis of the standard growth model modified to include precautionary saving motives and liquidity constraints. We address the impact on the aggregate saving rate, the importance of asset trading to individuals, and the relative inequality of wealth and income distributions.
2,738 citations
TL;DR: In this paper, the authors provide a model of firm and industry dynamics that allows for entry, exit, and firm-specific uncertainty generating variability in the fortunes of firms, focusing on the impact of uncertainty arising from investment in research and exploration.
Abstract: This paper provides a model of firm and industry dynamics that allows for entry, exit, and firm-specific uncertainty generating variability in the fortunes of firms. It focuses on the impact of uncertainty arising from investment in research and exploration. It analyzes the behavior of individual firms in an evolving market place and derives optimal policies, including exit. Then it adds an entry process and aggregates the optimal behavior of all firms, including potential entrants, into a rational expectations Markov-perfect industry equilibrium and proves ergodicity of the equilibrium process. Numerical examples illustrate the detailed characteristics of the stochastic process generating industry structures.
2,423 citations
TL;DR: In this article, a rich class of non-cooperative games, including models of oligopoly competition, macroeconomic coordination failures, arms races, bank runs, technology adoption and diffusion, R&D competition, pretrial bargaining, coordination in teams, and many others, are studied.
Abstract: We study a rich class of noncooperative games that includes models of oligopoly competition, macroeconomic coordination failures, arms races, bank runs, technology adoption and diffusion, R&D competition, pretrial bargaining, coordination in teams, and many others. For all these games, the sets of pure strategy Nash equilibria, correlated equilibria, and rationalizable strategies have identical bounds. Also, for a class of models of dynamic adaptive choice behavior that encompasses both best-response dynamics and Bayesian learning, the players' choices lie eventually within the same bounds. These bounds are shown to vary monotonically with certain exogenous parameters. WE STUDY THE CLASS of (noncooperative) supermodular games introduced by Topkis (1979) and further analyzed by Vives (1985, 1989), who also pointed out the importance of these games in industrial economics. Supermodular games are games in which each player's strategy set is partially ordered, the marginal returns to increasing one's strategy rise with increases in the competitors' strategies (so that the game exhibits "strategic complementarity"2) and, if a player's strategies are multidimensional, marginal returns to any one com- ponent of the player's strategy rise with increases in the other components. This class turns out to encompass many of the most important economic applications of noncooperative game theory. In macroeconomics, Diamond's (1982) search model and Bryant's (1983, 1984) rational expectations models can be represented as supermodular games. In each of these models, more activity by some members of the economy raises the returns to increased levels of activity by others. In oligopoly theory, some models of Bertrand oligopoly with differentiated products qualify as supermodu- lar games. In these games, when a firm's competitors raise their prices, the marginal profitability of the firm's own price increase rises. A similar structure is present in games of new technology adoption such as those of Dybvig and Spatt (1983), Farrell and Saloner (1986), and Katz and Shapiro (1986). When more users hook into a communication system or more manufacturers adopt an interface standard, the marginal return to others of doing the same often rises. Similarly, in some specifications of the bank runs model introduced by Diamond and Dybvig (1983), when more depositors withdraw their funds from a bank, it is more worthwhile for other depositors to do the same. In the warrant exercise
1,795 citations