IDEAS home Printed from https://ideas.repec.org/a/eee/gamebe/v66y2009i1p315-325.html
   My bibliography  Save this article

Approachability in repeated games: Computational aspects and a Stackelberg variant

Author

Listed:
  • Mannor, Shie
  • Tsitsiklis, John N.

Abstract

We consider a finite two-player zero-sum game with vector-valued rewards. We study the question of whether a given polyhedral set D is "approachable," that is, whether Player 1 (the "decision maker") can guarantee that the long-term average reward belongs to D, for any strategy of Player 2 (the "adversary"). We examine Blackwell's necessary and sufficient conditions for approachability, and show that the problem of checking these conditions is NP-hard, even in the special case where D is a singleton. We then consider a Stackelberg variant whereby, at each stage, the adversary gets to act after observing the decision maker's action. We provide necessary and sufficient conditions for approachability, and again establish that checking these conditions is NP-hard, even when D is a singleton. On the other hand, if the dimension of the reward vector is fixed, an approximate version of these conditions can be checked in polynomial time.

Suggested Citation

  • Mannor, Shie & Tsitsiklis, John N., 2009. "Approachability in repeated games: Computational aspects and a Stackelberg variant," Games and Economic Behavior, Elsevier, vol. 66(1), pages 315-325, May.
    • Handle: RePEc:eee:gamebe:v:66:y:2009:i:1:p:315-325
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0899-8256(08)00080-8
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Sergiu Hart & Andreu Mas-Colell, 2013. "A General Class Of Adaptive Strategies," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 3, pages 47-76, World Scientific Publishing Co. Pte. Ltd..
    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. Sandholm,W.H., 2003. "Excess payoff dynamics, potential dynamics, and stable games," Working papers 5, Wisconsin Madison - Social Systems.
    2. Ehud Lehrer & Eilon Solan, 2007. "Learning to play partially-specified equilibrium," Levine's Working Paper Archive 122247000000001436, David K. Levine.
    3. Sergiu Hart & Andreu Mas-Colell, 2013. "A Simple Adaptive Procedure Leading To Correlated Equilibrium," World Scientific Book Chapters, in: Simple Adaptive Strategies From Regret-Matching to Uncoupled Dynamics, chapter 2, pages 17-46, World Scientific Publishing Co. Pte. Ltd..
    4. Karl Schlag & Andriy Zapechelnyuk, 2009. "Decision Making in Uncertain and Changing Environments," Discussion Papers 19, Kyiv School of Economics.
    5. Jean-François Laslier & Bernard Walliser, 2015. "Stubborn learning," Theory and Decision, Springer, vol. 79(1), pages 51-93, July.
    6. Andriy Zapechelnyuk, 2007. "Better-Reply Strategies with Bounded Recall," Levine's Bibliography 321307000000000961, UCLA Department of Economics.
    7. Fudenberg, Drew & Takahashi, Satoru, 2011. "Heterogeneous beliefs and local information in stochastic fictitious play," Games and Economic Behavior, Elsevier, vol. 71(1), pages 100-120, January.
    8. Giovanni Di Bartolomeo & Debora Di Gioacchino, 2004. "Fiscal- Monetary Policy and Debt Management: a Two Stage Dynamic Analysis," Working Papers in Public Economics 74, University of Rome La Sapienza, Department of Economics and Law.
    9. Beggs, A.W., 2005. "On the convergence of reinforcement learning," Journal of Economic Theory, Elsevier, vol. 122(1), pages 1-36, May.
    10. Mannor, Shie & Shimkin, Nahum, 2008. "Regret minimization in repeated matrix games with variable stage duration," Games and Economic Behavior, Elsevier, vol. 63(1), pages 227-258, May.
    11. Young, H. Peyton, 2009. "Learning by trial and error," Games and Economic Behavior, Elsevier, vol. 65(2), pages 626-643, March.
    12. William Sandholm, 2014. "Probabilistic Interpretations of Integrability for Game Dynamics," Dynamic Games and Applications, Springer, vol. 4(1), pages 95-106, March.
    13. Schlag, Karl H. & Zapechelnyuk, Andriy, 2017. "Dynamic benchmark targeting," Journal of Economic Theory, Elsevier, vol. 169(C), pages 145-169.
    14. Giovanni Di Bartolomeo & Debora Di Gioacchino, 2008. "Fiscal-monetary policy coordination and debt management: a two-stage analysis," Empirica, Springer;Austrian Institute for Economic Research;Austrian Economic Association, vol. 35(4), pages 433-448, September.
    15. Eli Ben-Sasson & Adam Tauman Kalai & Ehud Kalai, 2006. "An Approach to Bounded Rationality," Discussion Papers 1439, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    16. Shie Mannor & Nahum Shimkin, 2003. "The Empirical Bayes Envelope and Regret Minimization in Competitive Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 327-345, May.
    17. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2006. "Stochastic Approximations and Differential Inclusions, Part II: Applications," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 673-695, November.
    18. Ehud Lehrer & Eilon Solan, 2016. "A General Internal Regret-Free Strategy," Dynamic Games and Applications, Springer, vol. 6(1), pages 112-138, March.
    19. Ho, Teck H. & Camerer, Colin F. & Chong, Juin-Kuan, 2007. "Self-tuning experience weighted attraction learning in games," Journal of Economic Theory, Elsevier, vol. 133(1), pages 177-198, March.
    20. Mertikopoulos, Panayotis & Sandholm, William H., 2018. "Riemannian game dynamics," Journal of Economic Theory, Elsevier, vol. 177(C), pages 315-364.

    More about this item

    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:eee:gamebe:v:66:y:2009:i:1:p:315-325. 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: http://www.elsevier.com/locate/inca/622836 .

    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.