IDEAS home Printed from https://ideas.repec.org/p/cbt/econwp/10-59.html
   My bibliography  Save this paper

A New Existence and Uniqueness Theorem for Continuous Games

Author

Abstract

This paper derives a general sufficient condition for existence and uniqueness in continuous games using a variant of the contraction mapping theorem applied to mapping from a subset of the real line on to itself. We first prove this contraction mapping variant, and then show how the existence of a unique equilibrium in the general game can be shown by proving the existence of a unique equilibrium in an iterative sequence of games involving such R-to-R mappings. Finally, we show how a general condition for this to occur is that a matrix derived from the Jacobean matrix of best-response functions be have positive leading principal minors, and how this condition generalises some existing uniqueness theorems for particular games.

Suggested Citation

  • Seamus Hogan, 2010. "A New Existence and Uniqueness Theorem for Continuous Games," Working Papers in Economics 10/59, University of Canterbury, Department of Economics and Finance.
    • Handle: RePEc:cbt:econwp:10/59
    as

    Download full text from publisher

    File URL: https://repec.canterbury.ac.nz/cbt/econwp/1059.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, April.
    2. Szidarovszky, F & Yakowitz, S, 1977. "A New Proof of the Existence and Uniqueness of the Cournot Equilibrium," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(3), pages 787-789, October.
    3. 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.
    4. Charles D. Kolstad & Lars Mathiesen, 1987. "Necessary and Sufficient Conditions for Uniqueness of a Cournot Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 54(4), pages 681-690.
    5. Paul A. Samuelson, 1953. "Prices of Factors and Goods in General Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 21(1), pages 1-20.
    6. 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)

    Citations

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


    Cited by:

    1. Klis Anna A., 2019. "On the Openness of Unique Pure-Strategy Nash Equilibrium," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 19(1), pages 1-9, January.
    2. Jinji, Naoto, 2014. "Comparative statics for oligopoly: A generalized result," Economics Letters, Elsevier, vol. 124(1), pages 79-82.

    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. Ui, Takashi, 2016. "Bayesian Nash equilibrium and variational inequalities," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 139-146.
    4. 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.
    5. Ewerhart, Christian, 2014. "Cournot games with biconcave demand," Games and Economic Behavior, Elsevier, vol. 85(C), pages 37-47.
    6. 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.
    7. Koji Okuguchi & Takeshi Yamazaki, 2018. "Existence of Unique Equilibrium in Cournot Mixed Oligopoly," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-13, September.
    8. 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.
    9. 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).
    10. Belleflamme,Paul & Peitz,Martin, 2015. "Industrial Organization," Cambridge Books, Cambridge University Press, number 9781107687899, October.
    11. Pierre Fleckinger, 2007. "Collective Reputation and Market Structure: Regulating the Quality vs Quantity Trade-of," Working Papers hal-00243080, HAL.
    12. Peter Hubert Mathieu Mouche & Ferenc Szidarovszky, 2023. "Aggregative Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 1056-1092, March.
    13. R Cornes & R Hartley, 2005. "The Geometry of Aggregative Games," Economics Discussion Paper Series 0514, Economics, The University of Manchester.
    14. Yi, Sang-Seung, 1997. "On the existence of a unique correlated equilibrium in Cournot oligopoly," Economics Letters, Elsevier, vol. 54(3), pages 235-239, July.
    15. 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.
    16. Krishnendu Ghosh Dastidar & Sugata Marjit, 2022. "Market size, entry costs and free entry Cournot equilibrium," Journal of Economics, Springer, vol. 136(2), pages 97-114, July.
    17. Ngo Van Long & Antoine Soubeyran, 2001. "Emission Taxes and Standards for an Asymmetric Oligopoly," CIRANO Working Papers 2001s-07, CIRANO.
    18. Hefti, Andreas, 2016. "On the relationship between uniqueness and stability in sum-aggregative, symmetric and general differentiable games," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 83-96.
    19. Max Albert & Andreas Hildenbrand, 2012. "Industrial organization in the laboratory," MAGKS Papers on Economics 201205, Philipps-Universität Marburg, Faculty of Business Administration and Economics, Department of Economics (Volkswirtschaftliche Abteilung).
    20. repec:hok:dpaper:307 is not listed on IDEAS
    21. Cars H. Hommes & Marius I. Ochea & Jan Tuinstra, 2018. "Evolutionary Competition Between Adjustment Processes in Cournot Oligopoly: Instability and Complex Dynamics," Dynamic Games and Applications, Springer, vol. 8(4), pages 822-843, December.

    More about this item

    Keywords

    Existence; Uniqueness; Continuous Games; Contraction Mapping Theorem;
    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:10/59. 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.