IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v269y2015icp863-884.html
   My bibliography  Save this article

Computing equilibria in discounted dynamic games

Author

Listed:
  • Burkov, Andriy
  • Chaib-draa, Brahim

Abstract

Game theory (GT) is an essential formal tool for interacting entities; however computing equilibria in GT is a hard problem. When the same game can be played repeatedly over time, the problem becomes even more complicated. The existence of multiple game states makes the problem of computing equilibria in such games extremely difficult. In this paper, we approach this problem by first proposing a method to compute a nonempty subset of approximate (up to any precision) subgame-perfect equilibria in repeated games. We then demonstrate how to extend this method to approximate all subgame-perfect equilibria in a repeated game, and also to solve more complex games, such as Markov chain games and stochastic games. We observe that in stochastic games, our algorithm requires additional strong assumptions to become tractable, while in repeated and Markov chain games it allows approximating all subgame-perfect equilibria reasonably fast and under considerably weaker assumptions than previous methods.

Suggested Citation

  • Burkov, Andriy & Chaib-draa, Brahim, 2015. "Computing equilibria in discounted dynamic games," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 863-884.
    • Handle: RePEc:eee:apmaco:v:269:y:2015:i:c:p:863-884
    DOI: 10.1016/j.amc.2015.07.068
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S009630031500987X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2015.07.068?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. Johannes Hörner & Wojciech Olszewski, 2006. "The Folk Theorem for Games with Private Almost-Perfect Monitoring," Econometrica, Econometric Society, vol. 74(6), pages 1499-1544, November.
    2. Cronshaw, Mark B, 1997. "Algorithms for Finding Repeated Game Equilibria," Computational Economics, Springer;Society for Computational Economics, vol. 10(2), pages 139-168, May.
    3. Drew Fudenberg & David Levine & Eric Maskin, 2008. "The Folk Theorem With Imperfect Public Information," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 12, pages 231-273, World Scientific Publishing Co. Pte. Ltd..
    4. Piccione, Michele, 1992. "Finite automata equilibria with discounting," Journal of Economic Theory, Elsevier, vol. 56(1), pages 180-193, February.
    5. Piccione Michele & Rubinstein Ariel, 1993. "Finite Automata Play a Repeated Extensive Game," Journal of Economic Theory, Elsevier, vol. 61(1), pages 160-168, October.
    6. Tao Wang & Keke Huang & Zhen Wang & Xiaoping Zheng, 2015. "Impact of Small Groups with Heterogeneous Preference on Behavioral Evolution in Population Evacuation," PLOS ONE, Public Library of Science, vol. 10(3), pages 1-13, March.
    7. Abreu, Dilip, 1988. "On the Theory of Infinitely Repeated Games with Discounting," Econometrica, Econometric Society, vol. 56(2), pages 383-396, March.
    8. Kalai, Ehud & Stanford, William, 1988. "Finite Rationality and Interpersonal Complexity in Repeated Games," Econometrica, Econometric Society, vol. 56(2), pages 397-410, March.
    9. Olivier Compte, 1998. "Communication in Repeated Games with Imperfect Private Monitoring," Econometrica, Econometric Society, vol. 66(3), pages 597-626, May.
    10. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    11. Aumann, Robert J, 1987. "Correlated Equilibrium as an Expression of Bayesian Rationality," Econometrica, Econometric Society, vol. 55(1), pages 1-18, January.
    12. Abreu, Dilip & Pearce, David & Stacchetti, Ennio, 1990. "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring," Econometrica, Econometric Society, vol. 58(5), pages 1041-1063, September.
    13. Lee, Jihong & Sabourian, Hamid, 2007. "Coase theorem, complexity and transaction costs," Journal of Economic Theory, Elsevier, vol. 135(1), pages 214-235, July.
    14. ENGLE-WARNICK, Jim & McCAUSLAND, William J. & MILLER, John H., 2004. "The Ghost in the Machine: Inferring Machine-Based Strategies from Observed Behavior," Cahiers de recherche 2004-11, Universite de Montreal, Departement de sciences economiques.
    15. Drew Fudenberg & David K. Levine, 2008. "An Approximate Folk Theorem with Imperfect Private Information," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 14, pages 309-330, World Scientific Publishing Co. Pte. Ltd..
    16. Kenneth L. Judd & Sevin Yeltekin & James Conklin, 2003. "Computing Supergame Equilibria," Econometrica, Econometric Society, vol. 71(4), pages 1239-1254, July.
    17. Busch, Lutz-Alexander & Wen, Quan, 1995. "Perfect Equilibria in Negotiation Model," Econometrica, Econometric Society, vol. 63(3), pages 545-565, May.
    18. Abreu, Dilip & Rubinstein, Ariel, 1988. "The Structure of Nash Equilibrium in Repeated Games with Finite Automata," Econometrica, Econometric Society, vol. 56(6), pages 1259-1281, November.
    19. Mailath, George J. & Samuelson, Larry, 2006. "Repeated Games and Reputations: Long-Run Relationships," OUP Catalogue, Oxford University Press, number 9780195300796.
    20. Rubinstein, Ariel, 1986. "Finite automata play the repeated prisoner's dilemma," Journal of Economic Theory, Elsevier, vol. 39(1), pages 83-96, June.
    21. Stahl, Dale II, 1991. "The graph of Prisoners' Dilemma supergame payoffs as a function of the discount factor," Games and Economic Behavior, Elsevier, vol. 3(3), pages 368-384, August.
    22. 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.
    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. Hamid Sabourian & Jihong Lee, 2004. "Complexity and Efficiency in Repeated Games with Negotiation," Econometric Society 2004 Far Eastern Meetings 401, Econometric Society.
    2. Luca Anderlini & Dino Gerardi & Roger Lagunoff, 2004. "The Folk Theorem in Dynastic Repeated Games," Cowles Foundation Discussion Papers 1490, Cowles Foundation for Research in Economics, Yale University.
    3. Mailath, George J. & Olszewski, Wojciech, 2011. "Folk theorems with bounded recall under (almost) perfect monitoring," Games and Economic Behavior, Elsevier, vol. 71(1), pages 174-192, January.
    4. Jihong Lee & Hamid Sabourian, 2005. "Efficiency in Negotiation: Complexity and Costly Bargaining," Birkbeck Working Papers in Economics and Finance 0505, Birkbeck, Department of Economics, Mathematics & Statistics.
    5. Kimmo Berg & Mitri Kitti, 2013. "Computing Equilibria in Discounted 2 × 2 Supergames," Computational Economics, Springer;Society for Computational Economics, vol. 41(1), pages 71-88, January.
    6. Lee, J. & Sabourian, H., 2004. "Complexity and Efficiency in Repeated Games and Negotiation," Cambridge Working Papers in Economics 0419, Faculty of Economics, University of Cambridge.
    7. Kimmo Berg & Gijs Schoenmakers, 2017. "Construction of Subgame-Perfect Mixed-Strategy Equilibria in Repeated Games," Games, MDPI, vol. 8(4), pages 1-14, November.
    8. Sugaya, Takuo & Wolitzky, Alexander, 2018. "Bounding payoffs in repeated games with private monitoring: n-player games," Journal of Economic Theory, Elsevier, vol. 175(C), pages 58-87.
    9. George J. Mailath & : Wojciech Olszewski, 2008. "Folk Theorems with Bounded Recall under (Almost) Perfect Monitoring, Second Version," PIER Working Paper Archive 08-027, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 28 Jul 2008.
    10. Roman, Mihai Daniel, 2008. "Entreprises behavior in cooperative and punishment‘s repeated negotiations," MPRA Paper 37527, University Library of Munich, Germany, revised 05 Jan 2009.
    11. Spiegler, Ran, 2004. "Simplicity of beliefs and delay tactics in a concession game," Games and Economic Behavior, Elsevier, vol. 47(1), pages 200-220, April.
    12. , J. & ,, 2006. "Coordination failure in repeated games with almost-public monitoring," Theoretical Economics, Econometric Society, vol. 1(3), pages 311-340, September.
    13. Binmore, Ken & Piccione, Michele & Samuelson, Larry, 1998. "Evolutionary Stability in Alternating-Offers Bargaining Games," Journal of Economic Theory, Elsevier, vol. 80(2), pages 257-291, June.
    14. Goldlücke, Susanne & Kranz, Sebastian, 2012. "Infinitely repeated games with public monitoring and monetary transfers," Journal of Economic Theory, Elsevier, vol. 147(3), pages 1191-1221.
    15. Lee, Jihong & Sabourian, Hamid, 2007. "Coase theorem, complexity and transaction costs," Journal of Economic Theory, Elsevier, vol. 135(1), pages 214-235, July.
    16. Laclau, M., 2014. "Communication in repeated network games with imperfect monitoring," Games and Economic Behavior, Elsevier, vol. 87(C), pages 136-160.
    17. Kimmo Berg & Markus Kärki, 2018. "Critical Discount Factor Values in Discounted Supergames," Games, MDPI, vol. 9(3), pages 1-17, July.
    18. Hamid Sabourian & Jihong Lee, 2004. "Complexity and Efficiency in the Negotiation Game," Econometric Society 2004 North American Winter Meetings 82, Econometric Society.
    19. Michihiro Kandori, 2011. "Weakly Belief‐Free Equilibria in Repeated Games With Private Monitoring," Econometrica, Econometric Society, vol. 79(3), pages 877-892, May.
    20. Laclau, Marie, 2012. "A folk theorem for repeated games played on a network," Games and Economic Behavior, Elsevier, vol. 76(2), pages 711-737.

    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:eee:apmaco:v:269:y:2015:i:c:p:863-884. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.