IDEAS home Printed from https://ideas.repec.org/p/cbt/econwp/09-06.html
   My bibliography  Save this paper

A New Sufficient Condition for Uniqueness in Continuous Games

Author

Abstract

Consider the class of games in which each player chooses a strategy from a connected subset of the real line. Many oligopoly models fall into this class. In many of these applications, it would be useful to show that an equilibrium was unique, or at least to have a set of conditions under which uniqueness would hold. In this paper, we first prove a uniqueness theorem that is slightly less restrictive than the contraction mapping theorem for mappings from the subsets of the real line onto itself, and then show how uniqueness in the general game can be shown by proving uniqueness using an iterative sequence of R-to-R mappings. This iterative approach works by considering the equilibrium for an m-player game holding the strategies of all other players fixed, starting with a two-player game. If one can show that the m-player game has a unique equilibrium for all possible values for the remaining players strategies, then one can add one player at a time and consider the R-to-R mapping from that player’s strategy on to the unique equilibrium of the first m players and back onto the (m+1)th player’s strategy. We then show how a general condition for each one of this sequence of mappings to have a unique equilibrium is that the leading principal minors of a matrix derived from the Jacobean matrix of best-response functions be positive, and how this general condition encompasses and generalises some existing uniqueness theorems for particular games

Suggested Citation

  • Seamus Hogan, 2009. "A New Sufficient Condition for Uniqueness in Continuous Games," Working Papers in Economics 09/06, University of Canterbury, Department of Economics and Finance.
  • Handle: RePEc:cbt:econwp:09/06
    as

    Download full text from publisher

    File URL: https://repec.canterbury.ac.nz/cbt/econwp/0906.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Van Long, Ngo & Soubeyran, Antoine, 2000. "Existence and uniqueness of Cournot equilibrium: a contraction mapping approach," Economics Letters, Elsevier, vol. 67(3), pages 345-348, June.
    2. Gérard Gaudet & Stephen W. Salant, 1991. "Uniqueness of Cournot Equilibrium: New Results From Old Methods," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 58(2), pages 399-404.
    Full references (including those not matched with items on IDEAS)

    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. Cherchye, Laurens & Demuynck, Thomas & De Rock, Bram, 2013. "The empirical content of Cournot competition," Journal of Economic Theory, Elsevier, vol. 148(4), pages 1552-1581.
    2. Long, Ngo Van & Soubeyran, Antoine, 2001. "Cost Manipulation Games in Oligopoly, with Costs of Manipulating," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 42(2), pages 505-533, May.
    3. Ngo Van Long & Antoine Soubeyran, 2001. "Emission Taxes and Standards for an Asymmetric Oligopoly," CIRANO Working Papers 2001s-07, CIRANO.
    4. Ui, Takashi, 2016. "Bayesian Nash equilibrium and variational inequalities," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 139-146.
    5. Colombo, Luca & Labrecciosa, Paola & Long, Ngo Van, 2019. "A Dynamic Analysis of Climate Change Mitigation with Endogenous Number of Contributors: Loose vs Tight Cooperation," Discussion paper series HIAS-E-92, Hitotsubashi Institute for Advanced Study, Hitotsubashi University.
    6. Ngo Long & Antoine Soubeyran, 2005. "Selective penalization of polluters: an inf-convolution approach," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 25(2), pages 421-454, February.
    7. Bruno Badia & Yair Tauman & Biligbaatar Tumendemberel, 2014. "A note on Cournot equilibrium with positive price," Economics Bulletin, AccessEcon, vol. 34(2), pages 1229-1234.
    8. Awi Federgruen & Ming Hu, 2019. "Stability in a general oligopoly model," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(1), pages 90-102, February.
    9. Joan Canton & Antoine Soubeyran & Hubert Stahn, 2008. "Environmental Taxation and Vertical Cournot Oligopolies: How Eco-industries Matter," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 40(3), pages 369-382, July.
    10. Chaiken, Benjamin & Duggan, Joseph E. & Sioshansi, Ramteen, 2021. "Paid to produce absolutely nothing? A Nash-Cournot analysis of a proposed power purchase agreement," Energy Policy, Elsevier, vol. 156(C).
    11. Hogan Seamus D, 2011. "A New Existence and Uniqueness Theorem for Continuous Games," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 11(1), pages 1-28, August.
    12. Pierre Fleckinger, 2007. "Collective Reputation and Market Structure: Regulating the Quality vs Quantity Trade-of," Working Papers hal-00243080, HAL.
    13. Taisuke Otsu & Martin Pesendorfer & Yuya Sasaki & Yuya Takahashi, 2022. "Estimation Of (Static Or Dynamic) Games Under Equilibrium Multiplicity," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 63(3), pages 1165-1188, August.
    14. Georg Götz, 2002. "Existence, Uniqueness, and Symmetry of Free-Entry Cournot Equilibrium: The Importance of Market Size and Technoligy Choice," Vienna Economics Papers vie0214, University of Vienna, Department of Economics.
    15. Sen, Debapriya & Stamatopoulos, Giorgos, 2016. "Licensing under general demand and cost functions," European Journal of Operational Research, Elsevier, vol. 253(3), pages 673-680.
    16. Maarten Janssen & Eric Rasmusen, 2002. "Bertrand Competition Under Uncertainty," Journal of Industrial Economics, Wiley Blackwell, vol. 50(1), pages 11-21, March.
    17. Christos Koulovatianos & Leonard J. Mirman, 2003. "The Effects of Market Structure on Industry Growth," University of Cyprus Working Papers in Economics 7-2003, University of Cyprus Department of Economics.
    18. Cumbul, Eray & Virág, Gábor, 2018. "Multilateral limit pricing in price-setting games," Games and Economic Behavior, Elsevier, vol. 111(C), pages 250-273.
    19. SANIN, Maria Eugenia & ZANAJ, Skerdilajda, 2007. "Environmental innovation under Cournot competition," LIDAM Discussion Papers CORE 2007050, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    20. Fischer, Carolyn, 2011. "Market power and output-based refunding of environmental policy revenues," Resource and Energy Economics, Elsevier, vol. 33(1), pages 212-230, January.

    More about this item

    Keywords

    Uniqueness; Continuous Games; Oligopoly;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:cbt:econwp:09/06. 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: Albert Yee (email available below). General contact details of provider: https://edirc.repec.org/data/decannz.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.