Skip to main content

Advertisement

Springer Nature Link
Log in

Correlated Equilibria in Stochastic Games with Borel Measurable Payoffs

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

An autonomous correlation device in a multistage game is a device that, before every stage, chooses for each player a private signal, possibly in a correlated way, and reveals to each player the signal chosen for him. The chosen signals depend only on previous signals, and not on the actions of the players. An extensive-form correlated \(\varepsilon \)-equilibrium in a multistage game is an \(\varepsilon \)-equilibrium in an extended game that includes an autonomous correlation device. In this paper we prove that every stochastic game with Borel measurable bounded payoffs has an extensive-form correlated \(\varepsilon \)-equilibrium, for every \(\varepsilon >0\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. A Blackwell game is a two-player zero-sum game that is played for infinitely many stages. At each stage the players simultaneously make their moves and are then informed of each other’s moves. Payoff is determined by a Borel measurable function on the set of possible resulting sequences of moves.

  2. One could assume that play goes on without any further influence on the payoffs, so as to make the horizon infinite.

  3. If \(|I|>2\), the players may use a correlated profile to punish a deviator. In this case, the device is also used to implement the punishment phase.

  4. Throughout the paper, we use the notation \(g^i_h\) also for the expected payoff of player \(i\) under a mixed or a correlated profile.

References

  1. Aumann RJ (1974) Subjectivity and correlation in randomized strategies. J Math Econ 1:67–96

    Article  MATH  MathSciNet  Google Scholar 

  2. Aumann RJ (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55:1–18

    Article  MATH  MathSciNet  Google Scholar 

  3. Blackwell D (1969) Infinite \(G_\delta \) games with imperfect information. Zastosowania Matematyki (Appl Math) 10:99–101

    MATH  MathSciNet  Google Scholar 

  4. Blackwell D, Ferguson TS (1968) The big match. Ann Math Stat 39:159–163

    Article  MATH  MathSciNet  Google Scholar 

  5. Flesch J, Kuipers J, Mashiah-Yaakovi A, Schoenmakers G, Shmaya E, Solan E, Vrieze K (2014) Non-existence of subgame-perfect \(ep\)-equilibrium in perfect-information games with infinite horizon. Int J Game Theory doi:10.1007/s00182-014-0412-3

  6. Forges F (1986) An approach to communication equilibria. Econometrica 54:1375–1385

    Article  MATH  MathSciNet  Google Scholar 

  7. Forges F (1988) Communication equilibria in repeated games with incomplete information. Math Oper Res 13:77–117

    Article  MathSciNet  Google Scholar 

  8. Hart S, Schmeidler D (1989) Existence of correlated equilibria. Math Oper Res 14:18–25

    Article  MATH  MathSciNet  Google Scholar 

  9. Maitra A, Sudderth W (1998) Finitely additive stochastic games with borel measurable payoffs. Int J Game Theory 27:257–267

    Article  MATH  MathSciNet  Google Scholar 

  10. Martin DA (1975) Borel determinacy. Ann Math 2:363–370

    Article  Google Scholar 

  11. Martin DA (1998) The determinacy of Blackwell games. J Symb Logic 63(4):1565–1581

    Article  MATH  Google Scholar 

  12. Mertens J-F (1987) Repeated games. In: Proceedings of the International Congress of Mathematicians, 0, (Berkeley, California, 986), pp 0528–0577, American Mathical Society, Providence

  13. Mertens J-F (1994) Correlated and communication equilibria. In: Mertens JF, Sorin S (eds) Game theoretic methods in general equilibrium analysis. Kluwer Academic, Dordrecht

    Chapter  Google Scholar 

  14. Myerson RB (1986) Multistage games with communication. Econometrica 54:323–358

    Article  MATH  MathSciNet  Google Scholar 

  15. Nowak AS, Raghavan TES (1992) Existence of stationary correlated equilibria with symmetric information for discounted stochastic games. Math Oper Res 17:519–526

    Article  MATH  MathSciNet  Google Scholar 

  16. Solan E (1999) Three-player absorbing games. Math Oper Res 24:669–698

    Article  MATH  MathSciNet  Google Scholar 

  17. Solan E (2001) Characterization of correlated equilibria in stochastic games. Int J Game Theory 30:259–277

  18. Solan E, Vieille N (2001) Quitting games. Math Oper Res 26:265–285

    Article  MATH  MathSciNet  Google Scholar 

  19. Solan E, Vieille N (2002) Correlated equilibrium in Stochastic games. Games Econ Behav 38:362–399

    Article  MATH  MathSciNet  Google Scholar 

  20. Vieille N (2000a) Equilibrium in 2-person Stochastic games. I. A reduction. Israel J Math 119:55–91

    Article  MATH  MathSciNet  Google Scholar 

  21. Vieille N (2000b) Equilibrium in 2-person Stochastic games. II. The case of recursive games. Israel J Math 119:93–126

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ ERC Grant agreement no [249159]. I would like to thank Sergiu Hart and Eilon Solan for insightful discussions and many useful comments.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Ayala Mashiah-Yaakovi

Authors
  1. Ayala Mashiah-Yaakovi
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Ayala Mashiah-Yaakovi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mashiah-Yaakovi, A. Correlated Equilibria in Stochastic Games with Borel Measurable Payoffs. Dyn Games Appl 5, 120–135 (2015). https://doi.org/10.1007/s13235-014-0122-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-014-0122-2

Keywords