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Mean Field Equilibrium: Uniqueness, Existence, and Comparative Statics

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  • Light, Bar

    (Graduate School of Business, Stanford University)

  • Weintraub, Gabriel

    (Graduate School of Business, Stanford University)

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 agents. We make three main contributions. First, our main result in the paper provides conditions that ensure the uniqueness of an MFE. 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. We believe our uniqueness result is the first of its nature in the class of models we study.

Suggested Citation

  • Light, Bar & Weintraub, Gabriel, 2018. "Mean Field Equilibrium: Uniqueness, Existence, and Comparative Statics," Research Papers 3731, Stanford University, Graduate School of Business.
    • Handle: RePEc:ecl:stabus:3731
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    1. David Besanko & Ulrich Doraszelski, 2004. "Capacity Dynamics and Endogenous Asymmetries in Firm Size," RAND Journal of Economics, The RAND Corporation, vol. 35(1), pages 23-49, Spring.
    2. Doraszelski, Ulrich & Pakes, Ariel, 2007. "A Framework for Applied Dynamic Analysis in IO," Handbook of Industrial Organization, in: Mark Armstrong & Robert Porter (ed.), Handbook of Industrial Organization, edition 1, volume 3, chapter 30, pages 1887-1966, Elsevier.
    3. Jovanovic, Boyan & Rosenthal, Robert W., 1988. "Anonymous sequential games," Journal of Mathematical Economics, Elsevier, vol. 17(1), pages 77-87, February.
    4. Richard Ericson & Ariel Pakes, 1995. "Markov-Perfect Industry Dynamics: A Framework for Empirical Work," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 62(1), pages 53-82.
    5. Santiago R. Balseiro & Omar Besbes & Gabriel Y. Weintraub, 2015. "Repeated Auctions with Budgets in Ad Exchanges: Approximations and Design," Management Science, INFORMS, vol. 61(4), pages 864-884, April.
    6. William S. Lovejoy, 1987. "Ordered Solutions for Dynamic Programs," Mathematics of Operations Research, INFORMS, vol. 12(2), pages 269-276, May.
    7. Ariel Pakes & Paul McGuire, 1994. "Computing Markov-Perfect Nash Equilibria: Numerical Implications of a Dynamic Differentiated Product Model," RAND Journal of Economics, The RAND Corporation, vol. 25(4), pages 555-589, Winter.
    8. Jaume Ventura, 1997. "Growth and Interdependence," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 112(1), pages 57-84.
    9. S. Rao Aiyagari, 1994. "Uninsured Idiosyncratic Risk and Aggregate Saving," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 109(3), pages 659-684.
    10. Maskin, Eric & Tirole, Jean, 2001. "Markov Perfect Equilibrium: I. Observable Actions," Journal of Economic Theory, Elsevier, vol. 100(2), pages 191-219, October.
    11. Kanoria, Yash & Saban, Daniela, 2017. "Facilitating the Search for Partners on Matching Platforms: Restricting Agents' Actions," Research Papers 3572, Stanford University, Graduate School of Business.
    12. Darrell Duffie & Semyon Malamud & Gustavo Manso, 2009. "Information Percolation With Equilibrium Search Dynamics," Econometrica, Econometric Society, vol. 77(5), pages 1513-1574, September.
    13. Caplin, Andrew & Nalebuff, Barry, 1991. "Aggregation and Imperfect Competition: On the Existence of Equilibrium," Econometrica, Econometric Society, vol. 59(1), pages 25-59, January.
    14. Jonathan Heathcote & Kjetil Storesletten & Giovanni L. Violante, 2009. "Quantitative Macroeconomics with Heterogeneous Households," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 319-354, May.
    15. Krishnamurthy Iyer & Ramesh Johari & Mukund Sundararajan, 2014. "Mean Field Equilibria of Dynamic Auctions with Learning," Management Science, INFORMS, vol. 60(12), pages 2949-2970, December.
    16. Daron Acemoglu & Martin Kaae Jensen, 2015. "Robust Comparative Statics in Large Dynamic Economies," Journal of Political Economy, University of Chicago Press, vol. 123(3), pages 587-640.
    17. Adlakha, Sachin & Johari, Ramesh & Weintraub, Gabriel Y., 2015. "Equilibria of dynamic games with many players: Existence, approximation, and market structure," Journal of Economic Theory, Elsevier, vol. 156(C), pages 269-316.
    18. Sachin Adlakha & Ramesh Johari, 2013. "Mean Field Equilibrium in Dynamic Games with Strategic Complementarities," Operations Research, INFORMS, vol. 61(4), pages 971-989, August.
    19. Huggett, Mark, 1993. "The risk-free rate in heterogeneous-agent incomplete-insurance economies," Journal of Economic Dynamics and Control, Elsevier, vol. 17(5-6), pages 953-969.
    20. , & ,, 2014. "Stochastic stability in monotone economies," Theoretical Economics, Econometric Society, vol. 9(2), May.
    21. Benveniste, L M & Scheinkman, J A, 1979. "On the Differentiability of the Value Function in Dynamic Models of Economics," Econometrica, Econometric Society, vol. 47(3), pages 727-732, May.
    22. Gabriel Y. Weintraub & C. Lanier Benkard & Benjamin Van Roy, 2008. "Markov Perfect Industry Dynamics With Many Firms," Econometrica, Econometric Society, vol. 76(6), pages 1375-1411, November.
    23. He, Wei & Sun, Yeneng, 2017. "Stationary Markov perfect equilibria in discounted stochastic games," Journal of Economic Theory, Elsevier, vol. 169(C), pages 35-61.
    24. Terry L. Friesz & David Bernstein & Tony E. Smith & Roger L. Tobin & B. W. Wie, 1993. "A Variational Inequality Formulation of the Dynamic Network User Equilibrium Problem," Operations Research, INFORMS, vol. 41(1), pages 179-191, February.
    25. Bar Ifrach & Gabriel Y. Weintraub, 2017. "A Framework for Dynamic Oligopoly in Concentrated Industries," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 84(3), pages 1106-1150.
    26. Besanko, David & Perry, Martin K & Spady, Richard H, 1990. "The Logit Model of Monopolistic Competition: Brand Diversity," Journal of Industrial Economics, Wiley Blackwell, vol. 38(4), pages 397-415, June.
    27. Hopenhayn, Hugo A & Prescott, Edward C, 1992. "Stochastic Monotonicity and Stationary Distributions for Dynamic Economies," Econometrica, Econometric Society, vol. 60(6), pages 1387-1406, November.
    28. Milgrom, Paul & Roberts, John, 1990. "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, Econometric Society, vol. 58(6), pages 1255-1277, November.
    29. Hopenhayn, Hugo A, 1992. "Entry, Exit, and Firm Dynamics in Long Run Equilibrium," Econometrica, Econometric Society, vol. 60(5), pages 1127-1150, September.
    30. Acikgoz, Omer, 2015. "On the Existence and Uniqueness of Stationary Equilibrium in Bewley Economies with Production," MPRA Paper 71066, University Library of Munich, Germany, revised 30 Apr 2016.
    31. Wei He & Yeneng Sun, 2013. "Stationary Markov Perfect Equilibria in Discounted Stochastic Games," Papers 1311.1562, arXiv.org, revised Jan 2017.
    32. Donald M. Topkis, 1978. "Minimizing a Submodular Function on a Lattice," Operations Research, INFORMS, vol. 26(2), pages 305-321, April.
    33. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, October.
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    Cited by:

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    3. Daron Acemoglu & Martin Kaae Jensen, 2018. "Equilibrium Analysis in the Behavioral Neoclassical Growth Model," NBER Working Papers 25363, National Bureau of Economic Research, Inc.
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    5. Bar Light, 2020. "Uniqueness of equilibrium in a Bewley–Aiyagari model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 69(2), pages 435-450, March.

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