IDEAS home Printed from https://ideas.repec.org/a/the/publsh/3423.html
   My bibliography  Save this article

Justifying optimal play via consistency

Author

Listed:
  • Brandl, Florian

    (Computer Science Department, Technische Universität München)

  • Brandt, Felix

    (Computer Science Department, Technische Universität München)

Abstract

Developing normative foundations for optimal play in two-player zero-sum games has turned out to be surprisingly difficult, despite the powerful strategic implications of the Minimax Theorem. We characterize maximin strategies by postulating coherent behavior in varying games. The first axiom, called consequentialism, states that how probability is distributed among completely indistinguishable actions is irrelevant. The second axiom, consistency, demands that strategies that are optimal in two different games should still be optimal when there is uncertainty which of the two games will actually be played. Finally, we impose a very mild rationality assumption, which merely requires that strictly dominated actions will not be played. Our characterization shows that a rational and consistent consequentialist who ascribes the same properties to his opponent has to play maximin strategies. This result can be extended to characterize Nash equilibrium in bimatrix games whenever the set of equilibria is interchangeable.

Suggested Citation

  • Brandl, Florian & Brandt, Felix, 2019. "Justifying optimal play via consistency," Theoretical Economics, Econometric Society, vol. 14(4), November.
    • Handle: RePEc:the:publsh:3423
    as

    Download full text from publisher

    File URL: http://econtheory.org/ojs/index.php/te/article/viewFile/20191185/25665/725
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Robert J. Aumann & Jacques H. Dreze, 2008. "Rational Expectations in Games," American Economic Review, American Economic Association, vol. 98(1), pages 72-86, March.
    2. Itzhak Gilboa & D. Schmeidler, 2003. "Expected Utility in the Context of a Game," Post-Print hal-00752136, HAL.
    3. 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).
    4. Bernheim, B Douglas, 1984. "Rationalizable Strategic Behavior," Econometrica, Econometric Society, vol. 52(4), pages 1007-1028, July.
    5. Norde, Henk & Potters, Jos & Reijnierse, Hans & Vermeulen, Dries, 1996. "Equilibrium Selection and Consistency," Games and Economic Behavior, Elsevier, vol. 12(2), pages 219-225, February.
    6. Tijs, S.H., 1981. "A characterization of the value of zero-sum two person games," Other publications TiSEM dc8d850f-f026-4f07-8049-1, Tilburg University, School of Economics and Management.
    7. Smith, John H, 1973. "Aggregation of Preferences with Variable Electorate," Econometrica, Econometric Society, vol. 41(6), pages 1027-1041, November.
    8. Norde, Henk & Voorneveld, Mark, 2004. "Characterizations of the value of matrix games," Mathematical Social Sciences, Elsevier, vol. 48(2), pages 193-206, September.
    9. R. J. Aumann & M. Maschler, 1972. "Some Thoughts on the Minimax Principle," Management Science, INFORMS, vol. 18(5-Part-2), pages 54-63, January.
    10. Hart, Sergiu & Modica, Salvatore & Schmeidler, David, 1990. "A Neo2Bayesian Foundation of the Maximin Value for Two-Person Zero- Sum Games," Foerder Institute for Economic Research Working Papers 275500, Tel-Aviv University > Foerder Institute for Economic Research.
    11. Gilboa, Itzhak & Schmeidler, David, 2003. "A derivation of expected utility maximization in the context of a game," Games and Economic Behavior, Elsevier, vol. 44(1), pages 172-182, July.
    12. S. H. Tijs, 1981. "A characterization of the value of zero‐sum two‐person games," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 28(1), pages 153-156, March.
    13. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-1050, July.
    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. Florian Brandl & Felix Brandt, 2023. "A Robust Characterization of Nash Equilibrium," Papers 2307.03079, arXiv.org.
    2. St'ephane Gonzalez & Nikolaos Pnevmatikos, 2023. "A Story of Consistency: Bridging the Gap between Bentham and Rawls Foundations," Papers 2303.07488, arXiv.org, revised Oct 2023.
    3. Fabien Gensbittel & Marcin Peski & Jérôme Renault, 2021. "Value-Based Distance Between Information Structures," Working Papers hal-01869139, HAL.

    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. Mosquera, M.A. & Borm, P. & Fiestras-Janeiro, M.G. & García-Jurado, I. & Voorneveld, M., 2008. "Characterizing cautious choice," Mathematical Social Sciences, Elsevier, vol. 55(2), pages 143-155, March.
    2. Mehmet S. Ismail, 2019. "Super-Nash performance in games," Papers 1912.00211, arXiv.org, revised Sep 2023.
    3. 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.
    4. 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.
    5. 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.
    6. Edouard Kujawski, 2015. "Accounting for Terrorist Behavior in Allocating Defensive Counterterrorism Resources," Systems Engineering, John Wiley & Sons, vol. 18(4), pages 365-376, July.
    7. Ismail, M.S., 2014. "Maximin equilibrium," Research Memorandum 037, Maastricht University, Graduate School of Business and Economics (GSBE).
    8. Guilhem Lecouteux, 2018. "Bayesian game theorists and non-Bayesian players," The European Journal of the History of Economic Thought, Taylor & Francis Journals, vol. 25(6), pages 1420-1454, November.
    9. Azrieli, Yaron, 2011. "Axioms for Euclidean preferences with a valence dimension," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 545-553.
    10. Morgan, John & Sefton, Martin, 2002. "An Experimental Investigation of Unprofitable Games," Games and Economic Behavior, Elsevier, vol. 40(1), pages 123-146, July.
    11. Ismail, Mehmet, 2014. "Maximin equilibrium," MPRA Paper 97322, University Library of Munich, Germany.
    12. Florian Brandl & Felix Brandt, 2023. "A Robust Characterization of Nash Equilibrium," Papers 2307.03079, arXiv.org.
    13. Grant, Simon & Meneghel, Idione & Tourky, Rabee, 2016. "Savage games," Theoretical Economics, Econometric Society, vol. 11(2), May.
    14. Ellingsen, Tore & Östling, Robert & Wengström, Erik, 2018. "How does communication affect beliefs in one-shot games with complete information?," Games and Economic Behavior, Elsevier, vol. 107(C), pages 153-181.
    15. Binmore, Ken, 2015. "Rationality," Handbook of Game Theory with Economic Applications,, Elsevier.
    16. Ismail, Mehmet, 2014. "Maximin equilibrium," MPRA Paper 97401, University Library of Munich, Germany.
    17. Du, Songzi, 2009. "Correlated Equilibrium via Hierarchies of Beliefs," MPRA Paper 16926, University Library of Munich, Germany.
    18. Vincent J. Vannetelbosch & P. Jean-Jacques Herings, 2000. "The equivalence of the Dekel-Fudenberg iterative procedure and weakly perfect rationalizability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 15(3), pages 677-687.
    19. Ambrus, Attila, 2006. "Coalitional Rationalizability," Scholarly Articles 3200266, Harvard University Department of Economics.
    20. Pei, Ting & Takahashi, Satoru, 2019. "Rationalizable strategies in random games," Games and Economic Behavior, Elsevier, vol. 118(C), pages 110-125.

    More about this item

    Keywords

    Zero-sum games; axiomatic characterization; maximin strategies;
    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:the:publsh:3423. 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: Martin J. Osborne (email available below). General contact details of provider: http://econtheory.org .

    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.