IDEAS home Printed from https://ideas.repec.org/p/ivi/wpasad/1995-18.html
   My bibliography  Save this paper

A Note On Stability Of Best Reply And Gradient Systems With Applications To Imperfectly Competitive Models

Author

Listed:
  • Luis Corchón

    (Universidad de Alicante)

  • Andreu Mas-Colell

    (Harvard University)

Abstract

In this paper we present a general result on the convergence to an equilibrium of a class of dynamic adjustment procedures -which includes Gradient Systems and Best Reply Dynamics as special cases- when there are two players and strategy sets are one dimensional. We also show that there are no restrictions on the form of Gradient or Best Reply dynamics, even under strong restrictions on the functional form of both demand and costs. This implies that we can construct examples with three players where the above dynamical procedures yield chaotic behavior.

Suggested Citation

  • Luis Corchón & Andreu Mas-Colell, 1995. "A Note On Stability Of Best Reply And Gradient Systems With Applications To Imperfectly Competitive Models," Working Papers. Serie AD 1995-18, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    • Handle: RePEc:ivi:wpasad:1995-18
    as

    Download full text from publisher

    File URL: http://www.ivie.es/downloads/docs/wpasad/wpasad-1995-18.pdf
    File Function: Fisrt version / Primera version, 1995
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Marhuenda, Francisco & Ortuno-Ortin, Ignacio, 1995. "Popular support for progressive taxation," Economics Letters, Elsevier, vol. 48(3-4), pages 319-324, June.
    2. Miguel Gines & Francisco Marhuenda, 1996. "Cost monotonic mechanisms," Investigaciones Economicas, Fundación SEPI, vol. 20(1), pages 89-103, January.
    3. Alcalde, Jose, 1996. "Implementation of Stable Solutions to Marriage Problems," Journal of Economic Theory, Elsevier, vol. 69(1), pages 240-254, April.
    4. Chakravorty, Bhaskar & Corchon, Luis C. & Wilkie, Simon, 2006. "Credible implementation," Games and Economic Behavior, Elsevier, vol. 57(1), pages 18-36, October.
    5. Marhuenda, F, 1995. "Distribution of Income and Aggregation of Demand," Econometrica, Econometric Society, vol. 63(3), pages 647-666, May.
    6. Josefa Tomás & Ramón Sirvent, 1995. "Expanded Version Of Regret Theory: Experimental Test," Working Papers. Serie AD 1995-16, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    7. Gaunersdorfer Andrea & Hofbauer Josef, 1995. "Fictitious Play, Shapley Polygons, and the Replicator Equation," Games and Economic Behavior, Elsevier, vol. 11(2), pages 279-303, November.
    8. Amparo Urbano Salvador & María Dolores Alepuz Domenech, 1994. "Market learning and price-dispersion," Working Papers. Serie AD 1994-14, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    9. K. Okuguchi, 1964. "The Stability of The Cournot Oligopoly Solution: A Further Generalization," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 31(2), pages 143-146.
    10. Dana, Rose-Anne & Montrucchio, Luigi, 1987. "On rational dynamic strategies in infinite horizon models where agents discount the future," Journal of Economic Behavior & Organization, Elsevier, vol. 8(3), pages 497-511, September.
    11. Takashi Negishi, 1961. "Monopolistic Competition and General Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 28(3), pages 196-201.
    12. Carmen Bevi?Author-Email: Carmen.Bevia@uab.es & Luis C. Corch?n & Simon Wilkie, "undated". "Implementation of the Walrasian Correspondence by Market Games," UFAE and IAE Working Papers 493.01, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
    13. Kennan, Donald & Rader, Trout, 1985. "Market Dynamics and the Law of Demand," Econometrica, Econometric Society, vol. 53(2), pages 465-471, March.
    14. Boldrin, Michele & Montrucchio, Luigi, 1986. "On the indeterminacy of capital accumulation paths," Journal of Economic Theory, Elsevier, vol. 40(1), pages 26-39, October.
    15. José Alcalde, 1994. "Exchange-proofness or divorce-proofness? Stability in one-sided matching markets," Review of Economic Design, Springer;Society for Economic Design, vol. 1(1), pages 275-287, December.
    16. Vives, Xavier, 1990. "Nash equilibrium with strategic complementarities," Journal of Mathematical Economics, Elsevier, vol. 19(3), pages 305-321.
    17. Moulin, Herve, 1994. "Social choice," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 31, pages 1091-1125, Elsevier.
    18. Dana, Rose-Anne & Montrucchio, Luigi, 1986. "Dynamic complexity in duopoly games," Journal of Economic Theory, Elsevier, vol. 40(1), pages 40-56, October.
    19. Debreu, Gerard, 1974. "Excess demand functions," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 15-21, March.
    20. Milgrom, Paul & Roberts, John, 1990. "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, Econometric Society, vol. 58(6), pages 1255-1277, November.
    21. F. H. Hahn, 1962. "The Stability of the Cournot Oligopoly Solution," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 29(4), pages 329-331.
    22. Sonnenschein, Hugo, 1972. "Market Excess Demand Functions," Econometrica, Econometric Society, vol. 40(3), pages 549-563, May.
    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. Hommes, C.H. & Ochea, M. & Tuinstra, J., 2011. "On the stability of the Cournot equilibrium: An evolutionary approach," CeNDEF Working Papers 11-10, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.

    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. Corchon, Luis C. & Mas-Colell, Andreu, 1996. "On the stability of best reply and gradient systems with applications to imperfectly competitive models," Economics Letters, Elsevier, vol. 51(1), pages 59-65, April.
    2. José Alcalde, 1994. "Exchange-proofness or divorce-proofness? Stability in one-sided matching markets," Review of Economic Design, Springer;Society for Economic Design, vol. 1(1), pages 275-287, December.
    3. Troy Tassier, 2013. "Handbook of Research on Complexity, by J. Barkley Rosser, Jr. and Edward Elgar," Eastern Economic Journal, Palgrave Macmillan;Eastern Economic Association, vol. 39(1), pages 132-133.
    4. Antonio Doria, Francisco, 2011. "J.B. Rosser Jr. , Handbook of Research on Complexity, Edward Elgar, Cheltenham, UK--Northampton, MA, USA (2009) 436 + viii pp., index, ISBN 978 1 84542 089 5 (cased)," Journal of Economic Behavior & Organization, Elsevier, vol. 78(1-2), pages 196-204, April.
    5. Carmen Bevi?Author-Email: Carmen.Bevia@uab.es & Luis C. Corch?n & Simon Wilkie, "undated". "Implementation of the Walrasian Correspondence by Market Games," UFAE and IAE Working Papers 493.01, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
    6. R Cornes & R Hartley, 2005. "The Geometry of Aggregative Games," Economics Discussion Paper Series 0514, Economics, The University of Manchester.
    7. María Dolores Guilló, 1995. "Terms-Of-Trade And The Current Account: A Two-Country/Two-Sector Growth Model," Working Papers. Serie AD 1995-03, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
    8. Antonio Romero-Medina & Matteo Triossi, 2023. "Take-it-or-leave-it contracts in many-to-many matching markets," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 75(2), pages 591-623, February.
    9. Börgers, Tilman & Janssen, Maarten C.W., 1995. "On the dominance solvability of large cournot games," Games and Economic Behavior, Elsevier, vol. 8(2), pages 297-321.
    10. Amir, Rabah & Lazzati, Natalia, 2011. "Network effects, market structure and industry performance," Journal of Economic Theory, Elsevier, vol. 146(6), pages 2389-2419.
    11. Emanuela Randon, "undated". "Multiple Equilibria with Externalities," Discussion Papers 04/09, Department of Economics, University of York.
    12. Shirai, Koji, 2010. "An existence theorem for Cournot-Walras equilibria in a monopolistically competitive economy," Journal of Mathematical Economics, Elsevier, vol. 46(6), pages 1093-1102, November.
    13. Jean J. Gabszewicz & Jacques-François Thisse, 2000. "Microeconomic theories of imperfect competition," Cahiers d'Économie Politique, Programme National Persée, vol. 37(1), pages 47-99.
    14. Dindos, Martin & Mezzetti, Claudio, 2006. "Better-reply dynamics and global convergence to Nash equilibrium in aggregative games," Games and Economic Behavior, Elsevier, vol. 54(2), pages 261-292, February.
    15. Bresnahan, Timothy F. & Trajtenberg, M., 1995. "General purpose technologies 'Engines of growth'?," Journal of Econometrics, Elsevier, vol. 65(1), pages 83-108, January.
    16. Acemoglu, Daron & Jensen, Martin Kaae, 2013. "Aggregate comparative statics," Games and Economic Behavior, Elsevier, vol. 81(C), pages 27-49.
    17. C-Rene DOMINIQUE & Luis Eduardo RIVERA-SOLIS, 2014. "On Market Economies: How Controllable Constructs Become Complex," Expert Journal of Economics, Sprint Investify, vol. 2(3), pages 100-108.
    18. Corchon, Luis C. & Fradera, Isabel, 2002. "Comparative statics in Cournot free entry equilibrium," Mathematical Social Sciences, Elsevier, vol. 44(2), pages 155-168, November.
    19. Berger, Ulrich, 2007. "Two more classes of games with the continuous-time fictitious play property," Games and Economic Behavior, Elsevier, vol. 60(2), pages 247-261, August.
    20. Hoernig, Steffen H., 2003. "Existence of equilibrium and comparative statics in differentiated goods Cournot oligopolies," International Journal of Industrial Organization, Elsevier, vol. 21(7), pages 989-1019, September.

    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:ivi:wpasad:1995-18. 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: Departamento de Edición (email available below). General contact details of provider: https://edirc.repec.org/data/ievages.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.