An approximate dynamic programming approach to solving dynamic oligopoly models

In this article, we introduce a new method to approximate Markov perfect equilibrium in largescale Ericson and Pakes (1995)-style dynamic oligopoly models that are not amenable to exact solution due to the curse of dimensionality. The method is based on an algorithm that iterates an approximate best response operator using an approximate dynamic programming approach. The method, based on mathematical programming, approximates the value function with a linear combination of basis functions. We provide results that lend theoretical support to our approach. We introduce a rich yet tractable set of basis functions, and test our method on important classes of models. Our results suggest that the approach we propose significantly expands the set of dynamic oligopoly models that can be analyzed computationally.

[1]  Stanley E. Zin,et al.  SPLINE APPROXIMATIONS TO VALUE FUNCTIONS: Linear Programming Approach , 1997 .

[2]  Steven T. Berry,et al.  Simple Estimators for the Parameters of Discrete Dynamic Games (with Entry/Exit Examples) , 2004 .

[3]  Benjamin Van Roy,et al.  Computational Methods for Oblivious Equilibrium , 2010, Oper. Res..

[4]  A. Pakes,et al.  Computing Markov Perfect Nash Equilibria: Numerical Implications of a Dynamic Differentiated Product Model , 1992 .

[5]  K. Judd.  Numerical methods in economics , 1998 .

[6]  Andrew Caplin,et al.  Aggregation and Imperfect Competition: On the Existence of Equilibrium , 1991 .

[7]  Martin Pesendorfer,et al.  Identification and Estimation of Dynamic Games , 2003 .

[8]  C. L. Benkard,et al.  Estimating Dynamic Models of Imperfect Competition , 2004 .

[9]  E. Maskin,et al.  A theory of dynamic oligopoly. , 1989 .

[10]  Per Krusell,et al.  Income and Wealth Heterogeneity in the Macroeconomy , 1998, Journal of Political Economy.

[11]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[12]  W. Whitt.  Representation and Approximation of Noncooperative Sequential Games , 1980 .

[13]  XNQ YV ZVTJ,et al.  Capacity dynamics and endogenous asymmetries in firm size , 2003 .

[14]  Benjamin Van Roy,et al.  On Constraint Sampling in the Linear Programming Approach to Approximate Dynamic Programming , 2004, Math. Oper. Res..

[15]  Giuseppe Carlo Calafiore,et al.  Uncertain convex programs: randomized solutions and confidence levels , 2005, Math. Program..

[16]  Victor Aguirregabiria,et al.  Sequential Estimation of Dynamic Discrete Games , 2007 .

[17]  A. Pakes,et al.  Stochastic Algorithms, Symmetric Markov Perfect Equilibrium, and the , 2001 .

[18]  Benjamin Van Roy,et al.  The Linear Programming Approach to Approximate Dynamic Programming , 2003, Oper. Res..

[19]  Kenneth L. Judd,et al.  Avoiding the curse of dimensionality in dynamic stochastic games , 2012 .

[20]  Benjamin Van Roy,et al.  Feature-based methods for large scale dynamic programming , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[21]  William J. Cook,et al.  The Traveling Salesman Problem: A Computational Study , 2007 .

[22]  C. L. Benkard,et al.  Markov Perfect Industry Dynamics with Many Firms , 2005 .

[23]  Gabriel Y. Weintraub,et al.  Oblivious Equilibrium for Concentrated Industries , 2013 .

[24]  M. Satterthwaite,et al.  Computable Markov-Perfect Industry Dynamics: Existence, Purification, and Multiplicity , 2007 .

[25]  John N. Tsitsiklis,et al.  Neuro-Dynamic Programming , 1996, Encyclopedia of Machine Learning.

[26]  M. Dufwenberg.  Game theory. , 2011, Wiley interdisciplinary reviews. Cognitive science.

[27]  A. Pakes,et al.  Markov-Perfect Industry Dynamics: A Framework for Empirical Work , 1995 .

[28]  E. Maskin,et al.  Overview and quantity competition with large fixed costs , 1988 .

[29]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[30]  M. Satterthwaite,et al.  Computable Markov-Perfect Industry Dynamics: Existence, Purification, and Multiplicity , 2007 .

[31]  U. Doraszelski.  An R&D Race with Knowledge Accumulation , 2003 .

[32]  John Rust,et al.  Estimation of Dynamic Structural Models: Problems and Prospects , 1991 .

[33]  C. Fershtman,et al.  Finite State Dynamic Games with Asymmetric Information: A Framework for Applied Work , 2009 .

[34]  Ariel Pakes,et al.  A Framework for Applied Dynamic Analysis in I.O , 2000 .

[35]  Benjamin Van Roy,et al.  MARKOV PERFECT INDUSTRY DYNAMICS WITH MANY FIRMS , 2008 .

[36]  Martin Pesendorfer,et al.  Asymptotic Least Squares Estimators for Dynamic Games , 2008 .

[37]  Bertel Schjerning,et al.  A Dynamic Model of Leap-Frogging Investments and Bertrand Price Competition , 2012 .