IDEAS home Printed from https://ideas.repec.org/p/sef/csefwp/661.html
   My bibliography  Save this paper

Asymptotic Behavior of Subgame Perfect Nash Equilibria in Stackelberg Games

Author

Listed:

Abstract

The study on how equilibria behave when perturbations occur in the data of a game is a fundamental theme, since actions and payoffs of the players may be affected by uncertainty or trembles. In this paper we investigate the asymptotic behavior of the subgame perfect Nash equilibrium (SPNE) in one-leader one-follower Stackelberg games under perturbations both of the action sets and of the payoff functions. To pursue this aim, we consider a general sequence of perturbed Stackelberg games and a set of assumptions that fit the usual types of perturbations. We study if the limit of SPNEs of the perturbed games is an SPNE of the original game and if the limit of SPNE outcomes of perturbed games is an SPNE outcome of the original game. We fully positively answer when the follower’s best reply correspondence is single valued. When the follower’s best reply correspondence is not single valued, the answer is positive only for the SPNEs outcomes; whereas the answer for SPNEs may be negative, even if the perturbation does not affect the sets and affects only one payoff function. However, we show that under suitable non-restrictive assumptions it is possible to obtain an SPNE starting from the limit of SPNEs of perturbed games, possibly modifying the limit at just one point.

Suggested Citation

  • Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2022. "Asymptotic Behavior of Subgame Perfect Nash Equilibria in Stackelberg Games," CSEF Working Papers 661, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
    • Handle: RePEc:sef:csefwp:661
    as

    Download full text from publisher

    File URL: https://www.csef.it/WP/wp661.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Guilherme Carmona, 2005. "On Games Of Perfect Information: Equilibria, Ε–Equilibria And Approximation By Simple Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 7(04), pages 491-499.
    2. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384, December.
    3. Francesco Caruso & M. Beatrice Lignola & Jacqueline Morgan, 2020. "Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 77-138, Springer.
    4. M. Beatrice Lignola & Jacqueline Morgan, 2017. "Inner Regularizations and Viscosity Solutions for Pessimistic Bilevel Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 183-202, April.
    5. Harris, Christopher, 1985. "A characterisation of the perfect equilibria of infinite horizon games," Journal of Economic Theory, Elsevier, vol. 37(1), pages 99-125, October.
    6. Drew Fudenberg & David Levine, 2008. "Subgame–Perfect Equilibria of Finite– and Infinite–Horizon Games," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 1, pages 3-20, World Scientific Publishing Co. Pte. Ltd..
    7. Lina Mallozzi & Panos M. Pardalos, 2022. "Entropic Regularization in Hierarchical Games," SN Operations Research Forum, Springer, vol. 3(1), pages 1-14, March.
    8. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2019. "Subgame Perfect Nash Equilibrium: A Learning Approach via Costs to Move," Dynamic Games and Applications, Springer, vol. 9(2), pages 416-432, June.
    9. Hellwig, Martin & Leininger, Wolfgang & Reny, Philip J. & Robson, Arthur J., 1990. "Subgame perfect equilibrium in continuous games of perfect information: An elementary approach to existence and approximation by discrete games," Journal of Economic Theory, Elsevier, vol. 52(2), pages 406-422, December.
    10. S. Flåm & E. Cavazzuti, 2005. "Entropic Penalties in Finite Games," Annals of Operations Research, Springer, vol. 137(1), pages 331-348, July.
    11. M. Beatrice Lignola & Jacqueline Morgan, 2019. "Further on Inner Regularizations in Bilevel Optimization," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 1087-1097, March.
    12. M. B. Lignola & J. Morgan, 1997. "Stability of Regularized Bilevel Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 93(3), pages 575-596, June.
    13. van Damme, E.E.C., 1984. "A relation between perfect equilibria in extensive form games and proper equilibria in normal form games," Other publications TiSEM 3734d89e-fd5c-4c80-a230-5, Tilburg University, School of Economics and Management.
    14. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    15. Harris, Christopher J, 1985. "Existence and Characterization of Perfect Equilibrium in Games of Perfect Information," Econometrica, Econometric Society, vol. 53(3), pages 613-628, May.
    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. Duggan, John, 2017. "Existence of stationary bargaining equilibria," Games and Economic Behavior, Elsevier, vol. 102(C), pages 111-126.
    2. János Flesch & Arkadi Predtetchinski, 2017. "A Characterization of Subgame-Perfect Equilibrium Plays in Borel Games of Perfect Information," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1162-1179, November.
    3. Sjur Didrik Flåm, 2021. "Games and cost of change," Annals of Operations Research, Springer, vol. 301(1), pages 107-119, June.
    4. Echenique, Federico, 2004. "Extensive-form games and strategic complementarities," Games and Economic Behavior, Elsevier, vol. 46(2), pages 348-364, February.
    5. Wei He & Yeneng Sun, 2015. "Dynamic Games with Almost Perfect Information," Papers 1503.08900, arXiv.org.
    6. Mariotti, Thomas, 2000. "Subgame-perfect equilibrium outcomes in continuous games of almost perfect information1," Journal of Mathematical Economics, Elsevier, vol. 34(1), pages 99-128, August.
    7. Guilherme Carmona, 2005. "On Games Of Perfect Information: Equilibria, Ε–Equilibria And Approximation By Simple Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 7(04), pages 491-499.
    8. Mertens, J.-F., 1995. "Two examples of strategic equilibrium," Games and Economic Behavior, Elsevier, vol. 8(2), pages 378-388.
    9. Francesco Caruso & M. Beatrice Lignola & Jacqueline Morgan, 2020. "Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 77-138, Springer.
    10. Hannu Salonen & Hannu Vartiainen, 2011. "On the Existence of Markov Perfect Equilibria in Perfect Information Games," Discussion Papers 68, Aboa Centre for Economics.
    11. Carlos Alós-Ferrer & Klaus Ritzberger, 2017. "Characterizing existence of equilibrium for large extensive form games: a necessity result," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(2), pages 407-430, February.
    12. He, Wei & Sun, Yeneng, 2015. "Dynamic Games with Almost Perfect Information," MPRA Paper 63345, University Library of Munich, Germany.
    13. He, Wei & Sun, Yeneng, 2020. "Dynamic games with (almost) perfect information," Theoretical Economics, Econometric Society, vol. 15(2), May.
    14. Alós-Ferrer, Carlos & Ritzberger, Klaus, 2016. "Equilibrium existence for large perfect information games," Journal of Mathematical Economics, Elsevier, vol. 62(C), pages 5-18.
    15. Govindan, Srihari & Wilson, Robert B., 2005. "Refinements of Nash Equilibrium," Research Papers 1897, Stanford University, Graduate School of Business.
    16. Guilherme Carmona, 2004. "On Games of Perfect Information: Equilibria, epsilon-Equilibria and Approximation by Simple Games," Game Theory and Information 0402002, University Library of Munich, Germany.
    17. Dieter Balkenborg & Rosemarie Nagel, 2016. "An Experiment on Forward vs. Backward Induction: How Fairness and Level k Reasoning Matter," German Economic Review, Verein für Socialpolitik, vol. 17(3), pages 378-408, August.
    18. Srihari Govindan & Robert Wilson, 2009. "On Forward Induction," Econometrica, Econometric Society, vol. 77(1), pages 1-28, January.
    19. Keser Claudia & Gaudeul Alexia, 2016. "Foreword: Special Issue in Honor of Reinhard Selten’s 85th Birthday," German Economic Review, De Gruyter, vol. 17(3), pages 277-283, August.
    20. van Damme, E.E.C., 1995. "Game theory : The next stage," Other publications TiSEM 7779b0f9-bef5-45c7-ae6b-7, Tilburg University, School of Economics and Management.

    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:sef:csefwp:661. 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: Dr. Maria Carannante (email available below). General contact details of provider: https://edirc.repec.org/data/cssalit.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.