IDEAS home Printed from https://ideas.repec.org/p/mib/wpaper/579.html

Symmetry-in-Asymmetry: On the Existence and Structure of Asymmetric Pure-Strategy Equilibria in Symmetric Tullock Two-Group Contests

Author

Listed:
  • Davide Bosco
  • Mario Gilli

Abstract

When information is complete, symmetric group-contest games `a la Tullock are known to display within-group-symmetric (WGS) pure-strategy equilibria. Far less is known about the possible existence and structure of their asymmetric counterparts—pure-strategy equilibria where, symmetry notwithstanding, members of the same group exert different effort levels. This note fills the gap by proving that within-group-asymmetric (WGA) pure-strategy equilibria indeed exist, but only for a specific subset of incentivisation schemes. All such equilibria display a peculiar ‘group-splitting’ structure: a subgroup of members free-ride exerting null effort, whereas the other members provide the same positive effort level. Because of this symmetry-in-asymmetry, the game displays the same aggregative structure identified in the context of WGS equilibria. Accordingly, an explicit characterisation is feasible also for their WGA counterparts.

Suggested Citation

  • Davide Bosco & Mario Gilli, 2026. "Symmetry-in-Asymmetry: On the Existence and Structure of Asymmetric Pure-Strategy Equilibria in Symmetric Tullock Two-Group Contests," Working Papers 579, University of Milano-Bicocca, Department of Economics.
    • Handle: RePEc:mib:wpaper:579
    as

    Download full text from publisher

    File URL: http://repec.dems.unimib.it/repec/pdf/mibwpaper579.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Nitzan, Shmuel, 1991. "Rent-Seeking with Non-identical Sharing Rules," Public Choice, Springer, vol. 71(1-2), pages 43-50, August.
    2. Lee, Sanghack, 1995. "Endogenous Sharing Rules in Collective-Group Rent-Seeking," Public Choice, Springer, vol. 85(1-2), pages 31-44, October.
    3. Kiminori Matsuyama, 2000. "Endogenous Inequality," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 67(4), pages 743-759.
    4. Russell Cooper & Andrew John, 1988. "Coordinating Coordination Failures in Keynesian Models," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 103(3), pages 441-463.
    5. Pau Balart & Sabine Flamand & Orestis Troumpounis, 2016. "Strategic choice of sharing rules in collective contests," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(2), pages 239-262, February.
    6. Kiminori Matsuyama, 2004. "Financial Market Globalization, Symmetry-Breaking, and Endogenous Inequality of Nations," Econometrica, Econometric Society, vol. 72(3), pages 853-884, May.
    7. Christian Ewerhart, 2024. "Solving the n-player Tullock contest," ECON - Working Papers 447, Department of Economics - University of Zurich, revised Jan 2026.
    8. Christian Ewerhart, 2026. "Solving the n $n$‐Player Tullock Contest," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 28(2), April.
    9. Kiminori Matsuyama, 2002. "Explaining Diversity: Symmetry-Breaking in Complementarity Games," American Economic Review, American Economic Association, vol. 92(2), pages 241-246, May.
    10. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
    11. Shmuel Nitzan, 2008. "Collective Rent Dissipation," Springer Books, in: Roger D. Congleton & Arye L. Hillman & Kai A. Konrad (ed.), 40 Years of Research on Rent Seeking 1, pages 309-321, Springer.
    12. Christian Ewerhart & Philip J. Reny, 2022. "Corrigendum to “On the Existence of Pure and Mixed Strategy Nash Equilibrium in Discontinuous Games”," Econometrica, Econometric Society, vol. 90(6), pages 1-2, November.
    13. Noh, Suk Jae, 1999. "A General Equilibrium Model of Two Group Conflict with Endogenous Intra-group Sharing Rules," Public Choice, Springer, vol. 98(3-4), pages 251-267, March.
    14. Amir, Rabah & Garcia, Filomena & Knauff, Malgorzata, 2010. "Symmetry-breaking in two-player games via strategic substitutes and diagonal nonconcavity: A synthesis," Journal of Economic Theory, Elsevier, vol. 145(5), pages 1968-1986, September.
    15. Davis, Douglas D & Reilly, Robert J, 1999. "Rent-Seeking with Non-identical Sharing Rules: An Equilibrium Rescued," Public Choice, Springer, vol. 100(1-2), pages 31-38, July.
    16. David Besanko & Ulrich Doraszelski, 2004. "Capacity Dynamics and Endogenous Asymmetries in Firm Size," RAND Journal of Economics, The RAND Corporation, vol. 35(1), pages 23-49, Spring.
    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. Dhritiman Gupta, 2020. "Prize sharing rules in collective contests: When does group size matter?," Discussion Papers 20-04, Indian Statistical Institute, Delhi.
    2. Amir, Rabah & Garcia, Filomena & Knauff, Malgorzata, 2010. "Symmetry-breaking in two-player games via strategic substitutes and diagonal nonconcavity: A synthesis," Journal of Economic Theory, Elsevier, vol. 145(5), pages 1968-1986, September.
    3. Friedhelm Hentschel, 2024. "Sharing rules in rent-seeking contests with third-party intervention," Economics of Governance, Springer, vol. 25(3), pages 401-414, September.
    4. Pau Balart & Sabine Flamand & Oliver Gürtler & Orestis Troumpounis, 2018. "Sequential choice of sharing rules in collective contests," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 20(5), pages 703-724, October.
    5. Changxia Ke, 2011. "Fight Alone or Together? The Need to Belong," Working Papers fight_alone_or_together, Max Planck Institute for Tax Law and Public Finance.
    6. Dhritiman Gupta, 2020. "Prize Sharing Rules in Collective Contests: Towards Strategic Foundations," Discussion Papers 20-01, Indian Statistical Institute, Delhi.
    7. Dhritiman Gupta, 2023. "Prize sharing rules in collective contests: when do social norms matter?," Review of Economic Design, Springer;Society for Economic Design, vol. 27(1), pages 221-244, February.
    8. Rabah Amir & Filomena Garcia & Malgorzata Knauff, 2006. "Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities," Working Papers Department of Economics 2006/29, ISEG - Lisbon School of Economics and Management, Department of Economics, Universidade de Lisboa.
    9. Baik, Kyung Hwan & Lee, Sanghack, 2007. "Collective rent seeking when sharing rules are private information," European Journal of Political Economy, Elsevier, vol. 23(3), pages 768-776, September.
    10. Pau Balart & Sabine Flamand & Orestis Troumpounis, 2016. "Strategic choice of sharing rules in collective contests," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(2), pages 239-262, February.
    11. Pau Balart & Sabine Flamand & Orestis Troumpounis, 2014. "Strategic choice of sharing rules in collective contests," Working Papers 64402108, Lancaster University Management School, Economics Department.
    12. Sun, Xutang & Wang, Youyou & Zheng, Jie, 2026. "Alliance formation in team contests with Blotto budgets," Journal of Economic Behavior & Organization, Elsevier, vol. 246(C).
    13. Vachadze, George, 2018. "Credit market imperfection, minimum investment requirement, and endogenous income inequality," Journal of Mathematical Economics, Elsevier, vol. 76(C), pages 62-79.
    14. Ke, Changxia & Konrad, Kai A. & Morath, Florian, 2013. "Brothers in arms – An experiment on the alliance puzzle," Games and Economic Behavior, Elsevier, vol. 77(1), pages 61-76.
    15. Crutzen, Benoît S Y & Flamand, Sabine & Sahuguet, Nicolas, 2020. "A model of a team contest, with an application to incentives under list proportional representation," Journal of Public Economics, Elsevier, vol. 182(C).
    16. Hwan Baik, Kyung & Lee, Sanghack, 1997. "Collective rent seeking with endogenous group sizes," European Journal of Political Economy, Elsevier, vol. 13(1), pages 121-130, February.
    17. Brian Roberson, 2026. "Existence of Equilibrium Mechanisms in Generalized Principal-Agent Problems with Interacting Teams," Papers 2602.20281, arXiv.org, revised May 2026.
    18. Cao, Zhigang & Yang, Xiaoguang, 2018. "Symmetric games revisited," Mathematical Social Sciences, Elsevier, vol. 95(C), pages 9-18.
    19. repec:elg:eechap:15325_6 is not listed on IDEAS
    20. Hausken, Kjell, 2000. "Migration and intergroup conflict," Economics Letters, Elsevier, vol. 69(3), pages 327-331, December.
    21. Barbieri, Stefano & Malueg, David A., 2016. "Private-information group contests: Best-shot competition," Games and Economic Behavior, Elsevier, vol. 98(C), pages 219-234.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D74 - Microeconomics - - Analysis of Collective Decision-Making - - - Conflict; Conflict Resolution; Alliances; Revolutions

    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:mib:wpaper:579. 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: Matteo Pelagatti (email available below). General contact details of provider: https://edirc.repec.org/data/dpmibit.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.