IDEAS home Printed from https://ideas.repec.org/a/eee/gamebe/v158y2026icp348-366.html

Efficient equilibrium in submodular global games of entry

Author

Listed:
  • Amir, Rabah
  • Harrison, Rodrigo
  • Jara-Moroni, Pedro

Abstract

We propose a global games approach to the standard two-stage entry game. The entry decisions in the first stage are strategic substitutes. The second-stage game of product market competition reflects a fundamental common value “market attractiveness” parameter, about which firms get private noisy signals. The main result establishes the selection of a unique equilibrium in the entry game, as noise vanishes, in cut-off strategies implying efficient entry. This provides a theoretical foundation for the equilibrium selection commonly used in entry models in the empirical literature. In addition, using supermodularity techniques, we provide novel conditions of independent interest on the primitives (demand and cost functions) of market competition to justify our assumptions for Bertrand and Cournot competition. These include results on the effects of entry and demand shifts and highlight the critical relevance of the property of log-supermodularity of demand.

Suggested Citation

  • Amir, Rabah & Harrison, Rodrigo & Jara-Moroni, Pedro, 2026. "Efficient equilibrium in submodular global games of entry," Games and Economic Behavior, Elsevier, vol. 158(C), pages 348-366.
  • Handle: RePEc:eee:gamebe:v:158:y:2026:i:c:p:348-366
    DOI: 10.1016/j.geb.2026.03.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0899825626000680
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.geb.2026.03.014?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Xavier Vives & Orestis Vravosinos, 2025. "Free Entry in a Cournot Market with Overlapping Ownership," American Economic Journal: Microeconomics, American Economic Association, vol. 17(2), pages 292-320, May.
    2. Roy, Sunanda & Singh, Rajesh & Weninger, Quinn, 2024. "Entry under placement uncertainty," Games and Economic Behavior, Elsevier, vol. 145(C), pages 157-196.
    3. Amir, Rabah & De Castro, Luciano & Koutsougeras, Leonidas, 2014. "Free entry versus socially optimal entry," Journal of Economic Theory, Elsevier, vol. 154(C), pages 112-125.
    4. Frankel, David M. & Morris, Stephen & Pauzner, Ady, 2003. "Equilibrium selection in global games with strategic complementarities," Journal of Economic Theory, Elsevier, vol. 108(1), pages 1-44, January.
    5. Carlsson, Hans & van Damme, Eric, 1993. "Global Games and Equilibrium Selection," Econometrica, Econometric Society, vol. 61(5), pages 989-1018, September.
    6. Ulrich Doraszelski & Mark Satterthwaite, 2010. "Computable Markov‐perfect industry dynamics," RAND Journal of Economics, RAND Corporation, vol. 41(2), pages 215-243, June.
    7. Elberfeld, Walter & Wolfstetter, Elmar, 1999. "A dynamic model of Bertrand competition with entry," International Journal of Industrial Organization, Elsevier, vol. 17(4), pages 513-525, May.
    8. Boyarchenko, Svetlana & Machowska, Dominika & Topolyan, Iryna, 2025. "Supermodularity and incentive reversal in teams," Games and Economic Behavior, Elsevier, vol. 150(C), pages 93-105.
    9. Michele Polo, 2018. "Entry games and free entry equilibria," Chapters, in: Luis C. Corchón & Marco A. Marini (ed.), Handbook of Game Theory and Industrial Organization, Volume I, chapter 11, pages 312-342, Edward Elgar Publishing.
    10. Amir, Rabah & Lambson, Val E., 2003. "Entry, exit, and imperfect competition in the long run," Journal of Economic Theory, Elsevier, vol. 110(1), pages 191-203, May.
    11. Jonathan Weinstein & Muhamet Yildiz, 2007. "A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements," Econometrica, Econometric Society, vol. 75(2), pages 365-400, March.
    12. Hoffmann, Eric J. & Sabarwal, Tarun, 2019. "Global games with strategic complements and substitutes," Games and Economic Behavior, Elsevier, vol. 118(C), pages 72-93.
    13. Rabah Amir & Val E. Lambson, 2000. "On the Effects of Entry in Cournot Markets," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 67(2), pages 235-254.
    14. Berry, Steven T, 1992. "Estimation of a Model of Entry in the Airline Industry," Econometrica, Econometric Society, vol. 60(4), pages 889-917, July.
    15. Harrison, Rodrigo & Jara-Moroni, Pedro, 2015. "A dominance solvable global game with strategic substitutes," Journal of Mathematical Economics, Elsevier, vol. 57(C), pages 1-11.
    16. 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.
    17. Volker Nocke & Nicolas Schutz, 2018. "Multiproduct‐Firm Oligopoly: An Aggregative Games Approach," Econometrica, Econometric Society, vol. 86(2), pages 523-557, March.
    18. Steven C. Salop, 1979. "Monopolistic Competition with Outside Goods," Bell Journal of Economics, The RAND Corporation, vol. 10(1), pages 141-156, Spring.
    19. Duffy, John & Ochs, Jack, 2012. "Equilibrium selection in static and dynamic entry games," Games and Economic Behavior, Elsevier, vol. 76(1), pages 97-116.
    20. Michele Bisceglia & Jorge Padilla & Joe Perkins & Salvatore Piccolo, 2024. "On excessive entry in Bayes‐Cournot oligopoly," RAND Journal of Economics, RAND Corporation, vol. 55(4), pages 719-748, December.
    21. Amparo Urbano, 2018. "Learning in markets," Chapters, in: Luis C. Corchón & Marco A. Marini (ed.), Handbook of Game Theory and Industrial Organization, Volume I, chapter 17, pages 486-519, Edward Elgar Publishing.
    22. Nirvikar Singh & Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, The RAND Corporation, vol. 15(4), pages 546-554, Winter.
    23. Diego Moreno & John Wooders, 2011. "Auctions with heterogeneous entry costs," RAND Journal of Economics, RAND Corporation, vol. 42(2), pages 313-336, June.
    24. Amir, R., 1996. "Cournot oligopoly and theory of supermodular games," LIDAM Reprints CORE 1228, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    25. N. Gregory Mankiw & Michael D. Whinston, 1986. "Free Entry and Social Inefficiency," RAND Journal of Economics, The RAND Corporation, vol. 17(1), pages 48-58, Spring.
    26. Vives, Xavier, 1990. "Nash equilibrium with strategic complementarities," Journal of Mathematical Economics, Elsevier, vol. 19(3), pages 305-321.
    27. Amir, Rabah, 1996. "Cournot Oligopoly and the Theory of Supermodular Games," Games and Economic Behavior, Elsevier, vol. 15(2), pages 132-148, August.
    28. Roger Guesnerie & Pedro Jara-Moroni, 2011. "Expectational coordination in simple economic contexts," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 47(2), pages 205-246, June.
    29. Eric J. Hoffmann & Tarun Sabarwal, 2019. "Equilibrium existence in global games with general payoff structures," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(1), pages 105-115, May.
    30. Milgrom, Paul & Roberts, John, 1990. "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, Econometric Society, vol. 58(6), pages 1255-1277, November.
    31. José‐Antonio Espín‐Sánchez & Álvaro Parra & Yuzhou Wang, 2023. "Equilibrium uniqueness in entry games with private information," RAND Journal of Economics, RAND Corporation, vol. 54(3), pages 512-540, September.
    32. Rodrigo Harrison & Pedro Jara‐Moroni, 2021. "Global Games With Strategic Substitutes," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 62(1), pages 141-173, February.
    33. Stanley S. Reynolds & David Rietzke, 2018. "Price caps, oligopoly, and entry," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(3), pages 707-745, October.
    34. Levin, Dan & Smith, James L, 1994. "Equilibrium in Auctions with Entry," American Economic Review, American Economic Association, vol. 84(3), pages 585-599, June.
    35. Michele Bisceglia & Jorge Padilla & Joe Willie Pinetop Perkins & Salvatore Piccolo, 2024. "On Excessive Entry in Bayes-Cournot Oligopoly," Post-Print hal-05241436, HAL.
    36. Anne-Christine Barthel & Eric Hoffmann & Tarun Sabarwal, 2021. "A Unified Approach to p-Dominance and its Generalizations in Games with Strategic Complements and Substitutes," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 202109, University of Kansas, Department of Economics.
    37. Andrew Sweeting, 2009. "The strategic timing incentives of commercial radio stations: An empirical analysis using multiple equilibria," RAND Journal of Economics, RAND Corporation, vol. 40(4), pages 710-742, December.
    38. Hackner, Jonas, 2000. "A Note on Price and Quantity Competition in Differentiated Oligopolies," Journal of Economic Theory, Elsevier, vol. 93(2), pages 233-239, August.
    39. Quirmbach, Herman C, 1988. "Comparative Statics for Oligopoly: Demand Shift Effects," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 29(3), pages 451-459, August.
    40. Amir, Rabah & Erickson, Philip & Jin, Jim, 2017. "On the microeconomic foundations of linear demand for differentiated products," Journal of Economic Theory, Elsevier, vol. 169(C), pages 641-665.
    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. Roy, Sunanda & Singh, Rajesh & Weninger, Quinn, 2024. "Entry under placement uncertainty," Games and Economic Behavior, Elsevier, vol. 145(C), pages 157-196.
    2. Vives, Xavier & Vravosinos, Orestis, 2024. "Strategic complementarity in games," Journal of Mathematical Economics, Elsevier, vol. 113(C).
    3. Amir, Rabah & De Castro, Luciano & Koutsougeras, Leonidas, 2014. "Free entry versus socially optimal entry," Journal of Economic Theory, Elsevier, vol. 154(C), pages 112-125.
    4. Belleflamme,Paul & Peitz,Martin, 2015. "Industrial Organization," Cambridge Books, Cambridge University Press, number 9781107069978, June.
    5. Rabah Amir, 2019. "Supermodularity and Complementarity in Economic Theory," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(3), pages 487-496, April.
    6. Christian Ewerhart, 2020. "Ordinal potentials in smooth games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 70(4), pages 1069-1100, November.
    7. Rabah Amir & Adriana Gama, 2025. "On Cournot's theory of oligopoly with perfect complements," Serie documentos de trabajo del Centro de Estudios Económicos 2025-04, El Colegio de México, Centro de Estudios Económicos.
    8. Amir, Rabah & De Castro, Luciano, 2017. "Nash equilibrium in games with quasi-monotonic best-responses," Journal of Economic Theory, Elsevier, vol. 172(C), pages 220-246.
    9. Stanley S. Reynolds & David Rietzke, 2018. "Price caps, oligopoly, and entry," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(3), pages 707-745, October.
    10. Michele Polo, 2018. "Entry games and free entry equilibria," Chapters, in: Luis C. Corchón & Marco A. Marini (ed.), Handbook of Game Theory and Industrial Organization, Volume I, chapter 11, pages 312-342, Edward Elgar Publishing.
    11. Tesoriere, Antonio, 2017. "Stackelberg equilibrium with multiple firms and setup costs," Journal of Mathematical Economics, Elsevier, vol. 73(C), pages 86-102.
    12. Andrew J. Monaco & Tarun Sabarwal, 2016. "Games with strategic complements and substitutes," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 62(1), pages 65-91, June.
    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. Rabah Amir, 2018. "Special issue: supermodularity and monotone methods in economics," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(3), pages 547-556, October.
    15. Adriana Gama & Rim Lahmandi-Ayed & Ana Elisa Pereira, 2020. "Entry and mergers in oligopoly with firm-specific network effects," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 70(4), pages 1139-1164, November.
    16. Charlene Cosandier & Filomena Garcia & Malgorzata Knauff, 2018. "Price competition with differentiated goods and incomplete product awareness," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(3), pages 681-705, October.
    17. Alexandrov, Alexei & Bedre-Defolie, Özlem, 2017. "LeChatelier–Samuelson principle in games and pass-through of shocks," Journal of Economic Theory, Elsevier, vol. 168(C), pages 44-54.
    18. Rabah Amir & Igor Evstigneev & Adriana Gama, 2021. "Oligopoly with network effects: firm-specific versus single network," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(3), pages 1203-1230, April.
    19. Hongkun Ma & Cheng-Zhong Qin & Chenhang Zeng, 2024. "Incentive and welfare implications of cross-holdings in oligopoly," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 77(4), pages 975-997, June.
    20. Boyarchenko, Svetlana & Machowska, Dominika & Topolyan, Iryna, 2025. "Supermodularity and incentive reversal in teams," Games and Economic Behavior, Elsevier, vol. 150(C), pages 93-105.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

    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:eee:gamebe:v:158:y:2026:i:c:p:348-366. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/622836 .

    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.