IDEAS home Printed from https://ideas.repec.org/p/bie/wpaper/650.html
   My bibliography  Save this paper

Stationary Discounted and Ergodic Mean Field Games of Singular Control

Author

Listed:
  • Cao, Haoyang

    (Center for Mathematical Economics, Bielefeld University)

  • Dianetti, Jodi

    (Center for Mathematical Economics, Bielefeld University)

  • Ferrari, Giorgio

    (Center for Mathematical Economics, Bielefeld University)

Abstract

We study stationary mean field games with singular controls in which the representative player interacts with a long-time weighted average of the population through a discounted and an erodic performance criterion. This class of games finds natural applications in the context of optimal productivity expansion in dynamic oligopolies. We prove existence and uniqueness of the mean field equilibria for the discounted and the ergodic games by showing the validity of an Abelian limit. The latter allows also to approximate Nash equilibria of - so far unexplored - symmetric N-player ergodic singular control games through the mean field equilibrium of the discounted game. Numerical examples finally illustrate in a case study the dependency of the mean field equilibria with respect to the parameters of the games.

Suggested Citation

  • Cao, Haoyang & Dianetti, Jodi & Ferrari, Giorgio, 2021. "Stationary Discounted and Ergodic Mean Field Games of Singular Control," Center for Mathematical Economics Working Papers 650, Center for Mathematical Economics, Bielefeld University.
  • Handle: RePEc:bie:wpaper:650
    as

    Download full text from publisher

    File URL: https://pub.uni-bielefeld.de/download/2955165/2955166
    File Function: First Version, 2021
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. H. Dharma Kwon & Hongzhong Zhang, 2015. "Game of Singular Stochastic Control and Strategic Exit," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 869-887, October.
    2. Fu, Guanxing & Horst, Ulrich, 2017. "Mean Field Games with Singular Controls," Rationality and Competition Discussion Paper Series 22, CRC TRR 190 Rationality and Competition.
    3. Haoyang Cao & Xin Guo, 2019. "MFGs for partially reversible investment," Papers 1908.10916, arXiv.org, revised Aug 2020.
    4. Dixit, Avinash K & Stiglitz, Joseph E, 1977. "Monopolistic Competition and Optimum Product Diversity," American Economic Review, American Economic Association, vol. 67(3), pages 297-308, June.
    5. Luis H. R. Alvarez, 2001. "Reward functionals, salvage values, and optimal stopping," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 54(2), pages 315-337, December.
    6. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    7. Jack, Andrew & Johnson, Timothy C. & Zervos, Mihail, 2008. "A singular control model with application to the goodwill problem," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 2098-2124, November.
    8. Giorgio Ferrari & Frank Riedel & Jan-Henrik Steg, 2013. "Continuous-Time Public Good Contribution under Uncertainty: A Stochastic Control Approach," Papers 1307.2849, arXiv.org, revised Oct 2015.
    9. A. Bensoussan & K. C. J. Sung & S. C. P. Yam & S. P. Yung, 2016. "Linear-Quadratic Mean Field Games," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 496-529, May.
    10. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    11. Andrew Jack & Mihail Zervos, 2006. "A singular control problem with an expected and a pathwise ergodic performance criterion," International Journal of Stochastic Analysis, Hindawi, vol. 2006, pages 1-19, June.
    12. Jianjun Miao, 2005. "Optimal Capital Structure and Industry Dynamics," Journal of Finance, American Finance Association, vol. 60(6), pages 2621-2659, December.
    13. Bertola, Giuseppe, 1998. "Irreversible investment," Research in Economics, Elsevier, vol. 52(1), pages 3-37, March.
    14. Weintraub, Gabriel Y. & Benkard, C. Lanier & Van Roy, Benjamin, 2011. "Industry dynamics: Foundations for models with an infinite number of firms," Journal of Economic Theory, Elsevier, vol. 146(5), pages 1965-1994, September.
    15. 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.
    16. 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.
    17. Olivier Guéant & Pierre Louis Lions & Jean-Michel Lasry, 2011. "Mean Field Games and Applications," Post-Print hal-01393103, HAL.
    18. Pui Chan Lon & Mihail Zervos, 2011. "A Model for Optimally Advertising and Launching a Product," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 363-376, May.
    19. Xin Guo & Wenpin Tang & Renyuan Xu, 2018. "A class of stochastic games and moving free boundary problems," Papers 1809.03459, arXiv.org, revised Oct 2021.
    20. Erzo G. J. Luttmer, 2007. "Selection, Growth, and the Size Distribution of Firms," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 122(3), pages 1103-1144.
    21. Hopenhayn, Hugo A, 1992. "Entry, Exit, and Firm Dynamics in Long Run Equilibrium," Econometrica, Econometric Society, vol. 60(5), pages 1127-1150, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Guanxing Fu & Ulrich Horst & Xiaonyu Xia, 2022. "A Mean-Field Control Problem of Optimal Portfolio Liquidation with Semimartingale Strategies," Papers 2207.00446, arXiv.org, revised Sep 2023.
    2. Dianetti, Jodi & Ferrari, Giorgio & Fischer, Markus & Nendel, Max, 2022. "A Unifying Framework for Submodular Mean Field Games," Center for Mathematical Economics Working Papers 661, Center for Mathematical Economics, Bielefeld University.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Haoyang Cao & Jodi Dianetti & Giorgio Ferrari, 2021. "Stationary Discounted and Ergodic Mean Field Games of Singular Control," Papers 2105.07213, arXiv.org.
    2. Ren'e Aid & Matteo Basei & Giorgio Ferrari, 2023. "A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy," Papers 2305.00541, arXiv.org.
    3. Erzo G.J. Luttmer, 2010. "Models of Growth and Firm Heterogeneity," Annual Review of Economics, Annual Reviews, vol. 2(1), pages 547-576, September.
    4. Dianetti, Jodi & Ferrari, Giorgio & Fischer, Markus & Nendel, Max, 2022. "A Unifying Framework for Submodular Mean Field Games," Center for Mathematical Economics Working Papers 661, Center for Mathematical Economics, Bielefeld University.
    5. Régis Chenavaz & Corina Paraschiv & Gabriel Turinici, 2017. "Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach," Working Papers hal-01592958, HAL.
    6. Barney Hartman‐Glaser & Hanno Lustig & Mindy Z. Xiaolan, 2019. "Capital Share Dynamics When Firms Insure Workers," Journal of Finance, American Finance Association, vol. 74(4), pages 1707-1751, August.
    7. Gualdi, Stanislao & Mandel, Antoine, 2016. "On the emergence of scale-free production networks," Journal of Economic Dynamics and Control, Elsevier, vol. 73(C), pages 61-77.
    8. Aïd, René & Basei, Matteo & Ferrari, Giorgio, 2023. "A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy," Center for Mathematical Economics Working Papers 679, Center for Mathematical Economics, Bielefeld University.
    9. Dianetti, Jodi & Ferrari, Giorgio & Tzouanas, Ioannis, 2023. "Ergodic Mean-Field Games of Singular Control with Regime-Switching (extended version)," Center for Mathematical Economics Working Papers 681, Center for Mathematical Economics, Bielefeld University.
    10. Costas Arkolakis, 2016. "A Unified Theory of Firm Selection and Growth," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 131(1), pages 89-155.
    11. 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.
    12. Dianetti, Jodi & Ferrari, Giorgio, 2019. "Nonzero-Sum Submodular Monotone-Follower Games. Existence and Approximation of Nash Equilibria," Center for Mathematical Economics Working Papers 605, Center for Mathematical Economics, Bielefeld University.
    13. Régis Chenavaz & Corina Paraschiv & Gabriel Turinici, 2021. "Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach," Dynamic Games and Applications, Springer, vol. 11(3), pages 463-490, September.
    14. José-María Da-Rocha & Jaume Sempere, 2017. "ITQs, Firm Dynamics and Wealth Distribution: Does Full Tradability Increase Inequality?," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 68(2), pages 249-273, October.
    15. Monika Mrázová & J. Peter Neary & Mathieu Parenti, 2021. "Sales and Markup Dispersion: Theory and Empirics," Econometrica, Econometric Society, vol. 89(4), pages 1753-1788, July.
    16. Weintraub, Gabriel Y. & Benkard, C. Lanier & Van Roy, Benjamin, 2011. "Industry dynamics: Foundations for models with an infinite number of firms," Journal of Economic Theory, Elsevier, vol. 146(5), pages 1965-1994, September.
    17. 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.
    18. Cao, Haoyang & Guo, Xin, 2022. "MFGs for partially reversible investment," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 995-1014.
    19. Andrea L. Eisfeldt & Hanno Lustig & Lei Zhang, 2017. "Complex Asset Markets," NBER Working Papers 23476, National Bureau of Economic Research, Inc.
    20. Jian Yang, 2021. "Analysis of Markovian Competitive Situations Using Nonatomic Games," Dynamic Games and Applications, Springer, vol. 11(1), pages 184-216, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bie:wpaper:650. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Bettina Weingarten (email available below). General contact details of provider: https://edirc.repec.org/data/imbiede.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.