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Stochastic games: Recent results

In: Handbook of Game Theory with Economic Applications

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  • Vieille, Nicolas

Abstract

This chapter presents developments in the theory of stochastic games that have taken place in recent years. It complements the contribution by Mertens. Major emphasis is put on stochastic games with finite state and action sets. In the zero-sum case, a classical result of Mertens and Neyman states that given [epsilon] > 0, each player has a strategy that is [epsilon]-optimal for all discount factors close to zero. Extensions to non-zero-sum games are dealt with here. In particular, the proof of existence of uniform equilibrium payoffs for two-player games is discussed, as well as the results available for more-than-two-player games. Important open problems related to N-player games are introduced by means of a class of simple stochastic games, called quitting, or stopping, games. Finally, recent results on zero-sum games with imperfect monitoring and on zero-sum games with incomplete information are surveyed.

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  • 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.
    • Handle: RePEc:eee:gamchp:3-48
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    1. Robert J. Aumann, 1995. "Repeated Games with Incomplete Information," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262011476, April.
    2. R.J. Aumann & S. Hart (ed.), 2002. "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3.
    3. Frank Thuijsman & Thirukkannamangai E. S. Raghavan, 1997. "Perfect Information Stochastic Games and Related Classes," International Journal of Game Theory, Springer;Game Theory Society, vol. 26(3), pages 403-408.
    4. SORIN, Sylvain, 1984. "'Big match' with lack of information on one side (part 1)," LIDAM Reprints CORE 601, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Sorin, Sylvain, 1992. "Repeated games with complete information," 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 4, pages 71-107, Elsevier.
    6. Solan, Eilon & Vieille, Nicolas, 2002. "Correlated Equilibrium in Stochastic Games," Games and Economic Behavior, Elsevier, vol. 38(2), pages 362-399, February.
    7. Aumann, Robert J. & Heifetz, Aviad, 2002. "Incomplete information," 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 43, pages 1665-1686, Elsevier.
    8. 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.
    9. Eilon Solan & Nicolas Vieille, 1998. "Quitting Games," Discussion Papers 1227, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    10. repec:dau:papers:123456789/6231 is not listed on IDEAS
    11. Nicolas Vieille & Dinah Rosenberg, 2000. "The Maxmin of Recursive Games with Incomplete Information on one Side," Post-Print hal-00481429, HAL.
    12. Sergiu Hart, 1985. "Nonzero-Sum Two-Person Repeated Games with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 10(1), pages 117-153, February.
    13. Mertens,Jean-François & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107030206, September.
      • Mertens,Jean-François & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107662636, September.
    14. Eilon Solan, 1999. "Three-Player Absorbing Games," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 669-698, August.
    15. 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.
    16. Dinah Rosenberg & Nicolas Vieille, 2000. "The Maxmin of Recursive Games with Incomplete Information on one Side," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 23-35, February.
    17. Donald A. Walker (ed.), 2000. "Equilibrium," Books, Edward Elgar Publishing, volume 0, number 1585.
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    1. Laraki, Rida & Sorin, Sylvain, 2015. "Advances in Zero-Sum Dynamic Games," Handbook of Game Theory with Economic Applications,, Elsevier.
    2. Burhaneddin Sandıkçı & Lisa M. Maillart & Andrew J. Schaefer & Oguzhan Alagoz & Mark S. Roberts, 2008. "Estimating the Patient's Price of Privacy in Liver Transplantation," Operations Research, INFORMS, vol. 56(6), pages 1393-1410, December.
    3. Boyarchenko, Svetlana & Levendorskiĭ, Sergei, 2014. "Preemption games under Lévy uncertainty," Games and Economic Behavior, Elsevier, vol. 88(C), pages 354-380.
    4. Andrzej Nowak & Eilon Solan & Sylvain Sorin, 2013. "Preface: Special Issue on Stochastic Games," Dynamic Games and Applications, Springer, vol. 3(2), pages 125-127, June.
    5. Walker, Mark & Wooders, John & Amir, Rabah, 2011. "Equilibrium play in matches: Binary Markov games," Games and Economic Behavior, Elsevier, vol. 71(2), pages 487-502, March.

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