IDEAS home Printed from https://ideas.repec.org/h/eee/gamchp/v4y2015icp27-93.html
   My bibliography  Save this book chapter

Advances in Zero-Sum Dynamic Games

Author

Listed:
  • Laraki, Rida
  • Sorin, Sylvain

Abstract

The survey presents recent results in the theory of two-person zero-sum repeated games and their connections with differential and continuous-time games. The emphasis is made on the following(1)A general model allows to deal simultaneously with stochastic and informational aspects.(2)All evaluations of the stage payoffs can be covered in the same framework (and not only the usual Cesà ro and Abel means).(3)The model in discrete time can be seen and analyzed as a discretization of a continuous time game. Moreover, tools and ideas from repeated games are very fruitful for continuous time games and vice versa.(4)Numerous important conjectures have been answered (some in the negative).(5)New tools and original models have been proposed. As a consequence, the field (discrete versus continuous time, stochastic versus incomplete information models) has a much more unified structure, and research is extremely active.

Suggested Citation

  • Laraki, Rida & Sorin, Sylvain, 2015. "Advances in Zero-Sum Dynamic Games," Handbook of Game Theory with Economic Applications, Elsevier.
  • Handle: RePEc:eee:gamchp:v:4:y:2015:i:c:p:27-93
    DOI: 10.1016/B978-0-444-53766-9.00002-1
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/B9780444537669000021
    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. DE MEYER, Bernard & ROSENBERG, Dinah, 1997. "“Cav u” and the dual game," CORE Discussion Papers 1997048, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Lehrer Ehud & Monderer Dov, 1994. "Discounting versus Averaging in Dynamic Programming," Games and Economic Behavior, Elsevier, vol. 6(1), pages 97-113, January.
    3. MERTENS , Jean-François & SORIN , Sylvain & ZAMIR , Shmuel, 1994. "Repeated Games. Part B : The Central Results," CORE Discussion Papers 1994021, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. De Meyer, Bernard, 2010. "Price dynamics on a stock market with asymmetric information," Games and Economic Behavior, Elsevier, vol. 69(1), pages 42-71, May.
    5. A. Maitra & W. Sudderth, 1998. "Finitely additive stochastic games with Borel measurable payoffs," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(2), pages 257-267.
    6. Robert J. Aumann, 1995. "Repeated Games with Incomplete Information," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262011476, May.
    7. Shmaya, Eran & Solan, Eilon, 2004. "Zero-sum dynamic games and a stochastic variation of Ramsey's theorem," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 319-329, August.
    8. Milman, Emanuel, 2006. "Approachable sets of vector payoffs in stochastic games," Games and Economic Behavior, Elsevier, vol. 56(1), pages 135-147, July.
    9. Vieille, Nicolas, 2002. "Stochastic games: Recent results," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 48, pages 1833-1850 Elsevier.
    10. Abraham Neyman, 2013. "Stochastic Games with Short-Stage Duration," Dynamic Games and Applications, Springer, vol. 3(2), pages 236-278, June.
    11. Dinah Rosenberg & Eilon Solan & Nicolas Vieille, 2003. "The MaxMin value of stochastic games with imperfect monitoring," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(1), pages 133-150, December.
    12. repec:dau:papers:123456789/5561 is not listed on IDEAS
    13. Levy, Yehuda, 2012. "Stochastic games with information lag," Games and Economic Behavior, Elsevier, vol. 74(1), pages 243-256.
    14. Dinah Rosenberg, 1998. "Duality and markovian strategies," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(4), pages 577-597.
    15. Abraham Neyman, 2012. "The value of two-person zero-sum repeated games with incomplete information and uncertain duration," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(1), pages 195-207, February.
    16. repec:eee:gamebe:v:104:y:2017:i:c:p:92-130 is not listed on IDEAS
    17. Friedman, Avner, 1994. "Differential games," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 2, chapter 22, pages 781-799 Elsevier.
    18. Monderer, Dov & Sorin, Sylvain, 1993. "Asymptotic Properties in Dynamic Programming," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(1), pages 1-11.
    19. Jean-FranÚois Mertens, 1998. "The speed of convergence in repeated games with incomplete information on one side," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(3), pages 343-357.
    20. Alexander Krausz & Ulrich Rieder, 1997. "Markov games with incomplete information," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(2), pages 263-279, June.
    21. repec:spr:compst:v:46:y:1997:i:2:p:263-279 is not listed on IDEAS
    22. Rainer Buckdahn & Juan Li & Marc Quincampoix, 2013. "Value function of differential games without Isaacs conditions. An approach with nonanticipative mixed strategies," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(4), pages 989-1020, November.
    23. Mertens, Jean-Francois, 2002. "Stochastic games," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 47, pages 1809-1832 Elsevier.
    24. De Meyer , B., 1995. "From Repeated Games to Brownian Games," CORE Discussion Papers 1995039, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    25. MERTENS, Jean-François & ZAMIR, Shmuel, 1995. "Incomplete Information Games and the Normal Distribution," CORE Discussion Papers 1995020, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    26. MERTENS , Jean-François & SORIN , Sylvain & ZAMIR , Shmuel, 1994. "Repeated Games. Part A : Background Material," CORE Discussion Papers 1994020, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    27. repec:dau:papers:123456789/6231 is not listed on IDEAS
    28. Dinah Rosenberg & Eilon Solan & Nicolas Vieille, 2002. "Stochastic Games with Imperfect Monitoring," Discussion Papers 1341, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    29. Ehud Lehrer & Eilon Solan, 2007. "Learning to play partially-specified equilibrium," Levine's Working Paper Archive 122247000000001436, David K. Levine.
    30. repec:dau:papers:123456789/6935 is not listed on IDEAS
    31. repec:dau:papers:123456789/6927 is not listed on IDEAS
    32. Coulomb, Jean-Michel, 1992. "Repeated Games with Absorbing States and No Signals," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(2), pages 161-174.
    33. Bernard De Meyer & Alexandre Marino, 2005. "Duality and optimal strategies in the finitely repeated zero-sum games with incomplete information on both sides," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00193996, HAL.
    34. DE MEYER , Bernard, 1993. "Repeated Games and the Central Limit Theorem," CORE Discussion Papers 1993003, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    35. Abraham Neyman & Sylvain Sorin, 1998. "Equilibria in repeated games of incomplete information: The general symmetric case," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(2), pages 201-210.
    36. Sorin, S, 1989. "On Repeated Games without a Recursive Structure: Existence of Lim Vn," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(1), pages 45-55.
    37. Rida Laraki, 2010. "Explicit formulas for repeated games with absorbing states," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 53-69, March.
    38. 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.
    39. Hadiza Moussa Saley & Bernard De Meyer, 2003. "On the strategic origin of Brownian motion in finance," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(2), pages 285-319.
    40. Pierre Cardaliaguet & Catherine Rainer, 2012. "Games with Incomplete Information in Continuous Time and for Continuous Types," Dynamic Games and Applications, Springer, vol. 2(2), pages 206-227, June.
    41. 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.
    42. A. Maitra & W. Sudderth, 2003. "Borel stay-in-a-set games," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(1), pages 97-108, December.
    43. Neyman, Abraham, 2017. "Continuous-time stochastic games," Games and Economic Behavior, Elsevier, vol. 104(C), pages 92-130.
    44. repec:dau:papers:123456789/6775 is not listed on IDEAS
    45. MERTENS, Jean-François & SORIN , Sylvain & ZAMIR , Shmuel, 1994. "Repeated Games. Part C : Further Developments," CORE Discussion Papers 1994022, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    46. Abraham Neyman, 2008. "Existence of optimal strategies in Markov games with incomplete information," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(4), pages 581-596, December.
    47. Guillaume Vigeral, 2013. "A Zero-Sum Stochastic Game with Compact Action Sets and no Asymptotic Value," Dynamic Games and Applications, Springer, vol. 3(2), pages 172-186, June.
    48. Zamir, Shmuel, 1992. "Repeated games of incomplete information: Zero-sum," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 5, pages 109-154 Elsevier.
    49. repec:dau:papers:123456789/10880 is not listed on IDEAS
    50. 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.
    51. Cardaliaguet, Pierre & Rainer, Catherine & Rosenberg, Dinah & Vieille , Nicolas, 2013. "Markov Games with Frequent Actions and Incomplete Information," HEC Research Papers Series 1007, HEC Paris.
    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. repec:eee:gamebe:v:108:y:2018:i:c:p:486-503 is not listed on IDEAS

    More about this item

    Keywords

    repeated; stochastic and differential games; discrete and continuous time; Shapley operator; incomplete information; imperfect monitoring; asymptotic and uniform value; dual game; weak and strong approachability.; C73.C61; C62; 91A5; 91A23; 91A25;

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium

    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:gamchp:v:4:y:2015:i:c:p:27-93. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu). General contact details of provider: http://www.elsevier.com/wps/find/bookseriesdescription.cws_home/BS_HE/description .

    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 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.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.