IDEAS home Printed from https://ideas.repec.org/a/wsi/igtrxx/v20y2018i03ns0219198918400017.html
   My bibliography  Save this article

On the Commitment Value and Commitment Optimal Strategies in Bimatrix Games

Author

Listed:
  • Stefanos Leonardos

    (Department of Mathematics, National & Kapodistrian University of Athens, Panepistimioupolis GR — 157 84, Athens, Greece)

  • Costis Melolidakis

    (Department of Mathematics, National & Kapodistrian University of Athens, Panepistimioupolis GR — 157 84, Athens, Greece)

Abstract

Given a bimatrix game, the associated leadership or commitment games are defined as the games at which one player, the leader, commits to a (possibly mixed) strategy and the other player, the follower, chooses his strategy after being informed of the irrevocable commitment of the leader (but not of its realization in case it is mixed). Based on a result by Von Stengel and Zamir [2010], the notions of commitment value and commitment optimal strategies for each player are discussed as a possible solution concept. It is shown that in nondegenerate bimatrix games (a) pure commitment optimal strategies together with the follower’s best response constitute Nash equilibria, and (b) strategies that participate in a completely mixed Nash equilibrium are strictly worse than commitment optimal strategies, provided they are not matrix game optimal. For various classes of bimatrix games that generalize zero-sum games, the relationship between the maximin value of the leader’s payoff matrix, the Nash equilibrium payoff and the commitment optimal value are discussed. For the Traveler’s Dilemma, the commitment optimal strategy and commitment value for the leader are evaluated and seem more acceptable as a solution than the unique Nash equilibrium. Finally, the relationship between commitment optimal strategies and Nash equilibria in 2 × 2 bimatrix games is thoroughly examined and in addition, necessary and sufficient conditions for the follower to be worse off at the equilibrium of the leadership game than at any Nash equilibrium of the simultaneous move game are provided.

Suggested Citation

  • Stefanos Leonardos & Costis Melolidakis, 2018. "On the Commitment Value and Commitment Optimal Strategies in Bimatrix Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-28, September.
    • Handle: RePEc:wsi:igtrxx:v:20:y:2018:i:03:n:s0219198918400017
    DOI: 10.1142/S0219198918400017
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219198918400017
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219198918400017?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 search for a different version of it.

    References listed on IDEAS

    as
    1. Bernhard von Stengel, 2016. "Recursive Inspection Games," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 935-952, August.
    2. van Damme, Eric & Hurkens, Sjaak, 1996. "Commitment Robust Equilibria and Endogenous Timing," Games and Economic Behavior, Elsevier, vol. 15(2), pages 290-311, August.
    3. Aumann, Robert J., 1974. "Subjectivity and correlation in randomized strategies," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 67-96, March.
    4. MOULIN, Hervé & VIAL, Jean-Philippe, 1978. "Strategically zero-sum games: the class of games whose completely mixed equilibria connot be improved upon," LIDAM Reprints CORE 359, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Bernheim, B Douglas, 1984. "Rationalizable Strategic Behavior," Econometrica, Econometric Society, vol. 52(4), pages 1007-1028, July.
    6. von Stengel, Bernhard & Zamir, Shmuel, 2010. "Leadership games with convex strategy sets," Games and Economic Behavior, Elsevier, vol. 69(2), pages 446-457, July.
    7. Kaushik Basu & Leonardo Becchetti & Luca Stanca, 2011. "Experiments with the Traveler’s Dilemma: welfare, strategic choice and implicit collusion," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 37(4), pages 575-595, October.
    8. Viossat, Yannick, 2006. "The Geometry of Nash Equilibria and Correlated Equilibria and a Generalization of Zero-Sum Games," SSE/EFI Working Paper Series in Economics and Finance 641, Stockholm School of Economics.
    9. Kats, Amoz & Thisse, Jacques-Francois, 1992. "Unilaterally Competitive Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(3), pages 291-299.
    10. Tilman Becker & Michael Carter & Jörg Naeve, 2005. "Experts Playing the Traveler's Dilemma," Diskussionspapiere aus dem Institut für Volkswirtschaftslehre der Universität Hohenheim 252/2005, Department of Economics, University of Hohenheim, Germany.
    11. Hichem Ben-El-Mechaiekh & Robert W. Dimand, 2010. "Von Neumann, Ville, And The Minimax Theorem," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 12(02), pages 115-137.
    12. Reny, Philip J. & Robson, Arthur J., 2004. "Reinterpreting mixed strategy equilibria: a unification of the classical and Bayesian views," Games and Economic Behavior, Elsevier, vol. 48(2), pages 355-384, August.
    13. Monderer, Dov & Shapley, Lloyd S., 1996. "Fictitious Play Property for Games with Identical Interests," Journal of Economic Theory, Elsevier, vol. 68(1), pages 258-265, January.
    14. Basu, Kaushik, 1994. "The Traveler's Dilemma: Paradoxes of Rationality in Game Theory," American Economic Review, American Economic Association, vol. 84(2), pages 391-395, May.
    15. DE WOLF, Olivier, 1999. "Optimal strategies in n-person unilaterally competitive games," LIDAM Discussion Papers CORE 1999049, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    16. Rosenthal, Robert W., 1991. "A note on robustness of equilibria with respect to commitment opportunities," Games and Economic Behavior, Elsevier, vol. 3(2), pages 237-243, May.
    17. Vitaly Pruzhansky, 2011. "Some interesting properties of maximin strategies," International Journal of Game Theory, Springer;Game Theory Society, vol. 40(2), pages 351-365, May.
    18. Michael Maschler, 1966. "A price leadership method for solving the inspector's non‐constant‐sum game," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 13(1), pages 11-33, March.
    19. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-1050, July.
    20. Halpern, Joseph Y. & Pass, Rafael, 2012. "Iterated regret minimization: A new solution concept," Games and Economic Behavior, Elsevier, vol. 74(1), pages 184-207.
    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. Gisèle Umbhauer, 2019. "Traveler’s dilemma : how the value of the luggage influences behavior," Working Papers of BETA 2019-13, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    2. Christian Bach & Andrés Perea, 2014. "Utility proportional beliefs," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(4), pages 881-902, November.
    3. Ismail, M.S., 2014. "Maximin equilibrium," Research Memorandum 037, Maastricht University, Graduate School of Business and Economics (GSBE).
    4. Mehmet S. Ismail, 2019. "Super-Nash performance in games," Papers 1912.00211, arXiv.org, revised Sep 2023.
    5. Halpern, Joseph Y. & Pass, Rafael, 2012. "Iterated regret minimization: A new solution concept," Games and Economic Behavior, Elsevier, vol. 74(1), pages 184-207.
    6. Ismail, Mehmet, 2014. "Maximin equilibrium," MPRA Paper 97322, University Library of Munich, Germany.
    7. Balkenborg, Dieter G. & Hofbauer, Josef & Kuzmics, Christoph, 2013. "Refined best-response correspondence and dynamics," Theoretical Economics, Econometric Society, vol. 8(1), January.
    8. Ismail, Mehmet, 2014. "Maximin equilibrium," MPRA Paper 97401, University Library of Munich, Germany.
    9. Mounir, Angie & Perea, Andrés & Tsakas, Elias, 2018. "Common belief in approximate rationality," Mathematical Social Sciences, Elsevier, vol. 91(C), pages 6-16.
    10. Jacob K. Goeree & Charles A. Holt, 2001. "Ten Little Treasures of Game Theory and Ten Intuitive Contradictions," American Economic Review, American Economic Association, vol. 91(5), pages 1402-1422, December.
    11. Cédric Wanko, 2008. "Approche Conceptuelle et Algorithmique des Equilibres de Nash Robustes Incitatifs," Working Papers 08-03, LAMETA, Universtiy of Montpellier, revised Feb 2008.
    12. Fabrizio Germano & Peio Zuazo-Garin, 2017. "Bounded rationality and correlated equilibria," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(3), pages 595-629, August.
    13. Joseph Y. Halpern & Rafael Pass, 2018. "Game theory with translucent players," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(3), pages 949-976, September.
    14. Amanda Friedenberg, 2006. "Can Hidden Variables Explain Correlation? (joint with Adam Brandenburger)," Theory workshop papers 815595000000000005, UCLA Department of Economics.
    15. Hillas, John & Samet, Dov, 2022. "Non-Bayesian correlated equilibrium as an expression of non-Bayesian rationality," Games and Economic Behavior, Elsevier, vol. 135(C), pages 1-15.
    16. Kaushik Basu & Leonardo Becchetti & Luca Stanca, 2011. "Experiments with the Traveler’s Dilemma: welfare, strategic choice and implicit collusion," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 37(4), pages 575-595, October.
    17. Adam Brandenburger & Amanda Friedenberg, 2014. "Intrinsic Correlation in Games," World Scientific Book Chapters, in: The Language of Game Theory Putting Epistemics into the Mathematics of Games, chapter 4, pages 59-111, World Scientific Publishing Co. Pte. Ltd..
    18. Lensberg, Terje & Schenk-Hoppé, Klaus Reiner, 2021. "Cold play: Learning across bimatrix games," Journal of Economic Behavior & Organization, Elsevier, vol. 185(C), pages 419-441.
    19. Antonio Penta & Peio Zuazo-Garin, 2022. "Rationalizability, Observability, and Common Knowledge," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 89(2), pages 948-975.
    20. R. J. Aumann & J. H. Dreze, 2005. "When All is Said and Done, How Should You Play and What Should You Expect?," Discussion Paper Series dp387, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.

    More about this item

    Keywords

    Bimatrix game; Nash equilibrium; subgame perfect; commitment optimal; commitment value; weakly unilaterally competitive games; pure strategy equilibrium; commitment advantageous games;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

    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:wsi:igtrxx:v:20:y:2018:i:03:n:s0219198918400017. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/igtr/igtr.shtml .

    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.