IDEAS home Printed from https://ideas.repec.org/p/hal/journl/halshs-03672222.html
   My bibliography  Save this paper

Splitting games over finite sets

Author

Listed:
  • Frédéric Koessler

    (PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris sciences et lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, PJSE - Paris Jourdan Sciences Economiques - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris sciences et lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Marie Laclau

    (HEC Paris - Ecole des Hautes Etudes Commerciales, GREGHEC - Groupement de Recherche et d'Etudes en Gestion - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique)

  • Jérôme Renault

    (TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Tristan Tomala

    (HEC Paris - Ecole des Hautes Etudes Commerciales, GREGHEC - Groupement de Recherche et d'Etudes en Gestion - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique)

Abstract

This paper studies zero-sum splitting games with finite sets of states. Players dynamically choose a pair of martingales {pt,qt}t, in order to control a terminal payoff u(p∞,q∞). A first part introduces the notion of "Mertens–Zamir transform" of a real-valued matrix and use it to approximate the solution of the Mertens–Zamir system for continuous functions on the square [0,1]2. A second part considers the general case of finite splitting games with arbitrary correspondences containing the Dirac mass on the current state: building on Laraki and Renault (Math Oper Res 45:1237–1257, 2020), we show that the value exists by constructing non Markovian ε-optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions.

Suggested Citation

  • Frédéric Koessler & Marie Laclau & Jérôme Renault & Tristan Tomala, 2022. "Splitting games over finite sets," Post-Print halshs-03672222, HAL.
    • Handle: RePEc:hal:journl:halshs-03672222
    DOI: 10.1007/s10107-022-01806-7
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-03672222
    as

    Download full text from publisher

    File URL: https://shs.hal.science/halshs-03672222/document
    Download Restriction: no
    File URL: https://libkey.io/10.1007/s10107-022-01806-7?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
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Forges, F., 1984. "Note on Nash equilibria in infinitely repeated games with incomplete information," LIDAM Reprints CORE 573, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Heuer, M, 1992. "Asymptotically Optimal Strategies in Repeated Games with Incomplete Information," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(4), pages 377-392.
    3. MERTENS, Jean-François & ZAMIR, Shmuel, 1977. "A duality theorem on a pair of simultaneous functional equations," LIDAM Reprints CORE 321, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. MERTENS, Jean-François & ZAMIR, Shmuel, 1971. "The value of two-person zero-sum repeated games with lack of information on both sides," LIDAM Reprints CORE 154, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Rida Laraki & Jérôme Renault, 2020. "Acyclic Gambling Games," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1237-1257, November.
    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. Koessler, Frederic & Laclau, Marie & Renault, Jérôme & Tomala, Tristan, 2022. "Long information design," Theoretical Economics, Econometric Society, vol. 17(2), May.
    2. Frédéric Koessler & Marie Laclau & Jerôme Renault & Tristan Tomala, 2022. "Long information design," Post-Print hal-03700394, HAL.
    3. Frédéric Koessler & Marie Laclau & Jerôme Renault & Tristan Tomala, 2022. "Long information design," PSE-Ecole d'économie de Paris (Postprint) hal-03700394, HAL.
    4. Koessler, Frederic & Laclau, Marie & Renault, Jérôme & Tomala, Tristan, 2022. "Long information design," Theoretical Economics, Econometric Society, vol. 17(2), May.
    5. Miquel Oliu-Barton, 2015. "Differential Games with Asymmetric and Correlated Information," Dynamic Games and Applications, Springer, vol. 5(3), pages 378-396, September.
    6. Laraki, Rida & Sorin, Sylvain, 2015. "Advances in Zero-Sum Dynamic Games," Handbook of Game Theory with Economic Applications,, Elsevier.
    7. Miquel Oliu-Barton, 2018. "The Splitting Game: Value and Optimal Strategies," Dynamic Games and Applications, Springer, vol. 8(1), pages 157-179, March.
    8. Fabien Gensbittel & Miquel Oliu-Barton, 2020. "Optimal Strategies in Zero-Sum Repeated Games with Incomplete Information: The Dependent Case," Dynamic Games and Applications, Springer, vol. 10(4), pages 819-835, December.
    9. Fabien Gensbittel & Jérôme Renault, 2015. "The Value of Markov Chain Games with Incomplete Information on Both Sides," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 820-841, October.
    10. Abraham Neyman & Sylvain Sorin, 2010. "Repeated games with public uncertain duration process," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 29-52, March.
    11. Salomon, Antoine & Forges, Françoise, 2015. "Bayesian repeated games and reputation," Journal of Economic Theory, Elsevier, vol. 159(PA), pages 70-104.
    12. Fudenberg, Drew & Yamamoto, Yuichi, 2011. "Learning from private information in noisy repeated games," Journal of Economic Theory, Elsevier, vol. 146(5), pages 1733-1769, September.
    13. Pierre Cardaliaguet & Rida Laraki & Sylvain Sorin, 2012. "A Continuous Time Approach for the Asymptotic Value in Two-Person Zero-Sum Repeated Games," Post-Print hal-00609476, HAL.
    14. Jérôme Renault, 2006. "The Value of Markov Chain Games with Lack of Information on One Side," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 490-512, August.
    15. Rainer Buckdahn & Marc Quincampoix & Catherine Rainer & Yuhong Xu, 2016. "Differential games with asymmetric information and without Isaacs’ condition," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(4), pages 795-816, November.
    16. Jérôme Bolte & Stéphane Gaubert & Guillaume Vigeral, 2015. "Definable Zero-Sum Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 171-191, February.
    17. repec:dau:papers:123456789/8023 is not listed on IDEAS
    18. Sylvain Sorin, 2011. "Zero-Sum Repeated Games: Recent Advances and New Links with Differential Games," Dynamic Games and Applications, Springer, vol. 1(1), pages 172-207, March.
    19. Frédéric Koessler & Marie Laclau & Tristan Tomala, 2022. "Interactive Information Design," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 153-175, February.
    20. Laraki, Rida & Renault, Jérôme, 2017. "Acyclic Gambling Games," TSE Working Papers 17-768, Toulouse School of Economics (TSE).
    21. VIEILLE, Nicolas & ROSENBERG, Dinah & SOLAN, Eilon, 2002. "Stochastic games with a single controller and incomplete information," HEC Research Papers Series 754, HEC Paris.

    More about this item

    Keywords

    Splitting games; Mertens-Zamir system; Repeated games with incomplete information; Bayesian persuasion; Information design;
    All these keywords.

    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:hal:journl:halshs-03672222. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.