Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Correlated Equilibria and Communication in Games

  • Françoise Forges
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_103-3

Definition of the Subject

The correlated equilibrium is a game theoretic solution concept. It was proposed by Aumann (1974, 1987) in order to capture the strategic correlation opportunities that the players face when they take into account the extraneous environment in which they interact. The notion is illustrated in section “Introduction.” A formal definition is given in section “Correlated Equilibrium: Definition and Basic Properties.” The correlated equilibrium also appears as the appropriate solution concept if preplay communication is allowed between the players. As shown in section “Correlated Equilibrium and Communication,” this property can be given several precise statements according to the constraints imposed on the players’ communication, which can go from plain conversation to exchange of messages through noisy channels. Originally designed for static games with complete information, the correlated equilibrium applies to any strategic form game. It is geometrically and...


Nash Equilibrium Solution Concept Cheap Talk Correlate Equilibrium Sequential Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.


Primary Literature

  1. Aumann RJ (1974) Subjectivity and correlation in randomized strategies. J Math Econ 1:67–96zbMATHMathSciNetCrossRefGoogle Scholar
  2. Aumann RJ (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55:1–18zbMATHMathSciNetCrossRefGoogle Scholar
  3. Aumann RJ, Maschler M, Stearns R (1968) Repeated games with incomplete information: an approach to the nonzero sum case. Reports to the US Arms Control and Disarmament Agency, ST-143, Chapter IV, 117–216 (reprinted In: Aumann RJ, Maschler M (1995) Repeated Games of Incomplete Information. M.I.T. Press, Cambridge)Google Scholar
  4. Bárány I (1992) Fair distribution protocols or how players replace fortune. Math Oper Res 17:327–340zbMATHMathSciNetCrossRefGoogle Scholar
  5. Ben-Porath E (1998) Correlation without mediation: expanding the set of equilibrium outcomes by cheap pre-play procedures. J Econ Theory 80:108–122zbMATHMathSciNetCrossRefGoogle Scholar
  6. Ben-Porath E (2003) Cheap talk in games with incomplete information. J Econ Theory 108:45–71zbMATHMathSciNetCrossRefGoogle Scholar
  7. Ben-Porath E (2006) A correction to “Cheap talk in games with incomplete information”. Mimeo, Hebrew University of Jerusalem, JerusalemGoogle Scholar
  8. Blackwell D (1951) Comparison of experiments. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, pp 93–102Google Scholar
  9. Blackwell D (1953) Equivalent comparison of experiments. Ann Math Stat 24:265–272zbMATHMathSciNetCrossRefGoogle Scholar
  10. Brandenburger A, Dekel E (1987) Rationalizability and correlated equilibria. Econometrica 55:1391–1402zbMATHMathSciNetCrossRefGoogle Scholar
  11. Dhillon A, Mertens JF (1996) Perfect correlated equilibria. J Econ Theory 68:279–302zbMATHMathSciNetCrossRefGoogle Scholar
  12. Dodis Y, Halevi S, Rabin T (2000) A cryptographic solution to a game theoretic problem. In: CRYPTO 2000: 20th international cryptology conference. Springer, Berlin, pp 112–130Google Scholar
  13. Evangelista F, Raghavan TES (1996) A note on correlated equilibrium. Int J Game Theory 25:35–41zbMATHMathSciNetCrossRefGoogle Scholar
  14. Forges F (1985) Correlated equilibria in a class of repeated games with incomplete information. Int J Game Theory 14:129–150zbMATHMathSciNetCrossRefGoogle Scholar
  15. Forges F (1986) An approach to communication equilibrium. Econometrica 54:1375–1385zbMATHMathSciNetCrossRefGoogle Scholar
  16. Forges F (1988) Communication equilibria in repeated games with incomplete information. Math Oper Res 13:191–231zbMATHMathSciNetCrossRefGoogle Scholar
  17. Forges F (1990) Universal mechanisms. Econometrica 58:1341–1364zbMATHMathSciNetCrossRefGoogle Scholar
  18. Forges F (1993) Five legitimate definitions of correlated equilibrium in games with incomplete information. Theor Decis 35:277–310zbMATHMathSciNetCrossRefGoogle Scholar
  19. Forges F (2006) Correlated equilibrium in games with incomplete information revisited. Theor Decis 61:329–344zbMATHMathSciNetCrossRefGoogle Scholar
  20. Foster D, Vohra R (1997) Calibrated learning and correlated equilibrium. Games Econ Behav 21:40–55zbMATHMathSciNetCrossRefGoogle Scholar
  21. Gerardi D (2000) Interim pre-play communication. Mimeo, Yale University, New HavenGoogle Scholar
  22. Gerardi D (2004) Unmediated communication in games with complete and incomplete information. J Econ Theory 114:104–131zbMATHMathSciNetCrossRefGoogle Scholar
  23. Gerardi D, Myerson R (2007) Sequential equilibria in Bayesian games with communication. Games Econ Behav 60:104–134zbMATHMathSciNetCrossRefGoogle Scholar
  24. Gilboa I, Zemel E (1989) Nash and correlated equilibria: some complexity considerations. Games Econ Behav 1:80–93zbMATHMathSciNetCrossRefGoogle Scholar
  25. Gomez-Canovas S, Hansen P, Jaumard B (1999) Nash Equilibria from the correlated equilibria viewpoint. Int Game Theory Rev 1:33–44zbMATHCrossRefGoogle Scholar
  26. Gossner O (1998) Secure protocols or how communication generates correlation. J Econ Theory 83:69–89zbMATHMathSciNetCrossRefGoogle Scholar
  27. Gossner O (2000) Comparison of information structures. Games Econ Behav 30:44–63zbMATHMathSciNetCrossRefGoogle Scholar
  28. Gossner O, Tomala T (2007) Secret correlation in repeated games with signals. Math Oper Res 32:413–424zbMATHMathSciNetCrossRefGoogle Scholar
  29. Halpern JY (2007) Computer science and game theory. In: Durlauf SN, Blume LE (eds) The New Palgrave dictionary of economics, 2nd edn. Palgrave Macmillan. The New Palgrave dictionary of economics online. http://www.dictionaryofeconomics.com/article?id=pde2008_C000566. Accessed 24 May 2008
  30. Hart S, Schmeidler D (1989) Existence of correlated equilibria. Math Oper Res 14:18–25zbMATHMathSciNetCrossRefGoogle Scholar
  31. Hart S, Mas-Colell A (2000) A simple adaptive procedure leading to correlated equilibrium. Econometrica 68:1127–1150zbMATHMathSciNetCrossRefGoogle Scholar
  32. Hart S (2005) Adaptative heuristics. Econometrica 73:1401–1430zbMATHMathSciNetCrossRefGoogle Scholar
  33. Krishna RV (2007) Communication in games of incomplete information: two players. J Econ Theory 132:584–592zbMATHMathSciNetCrossRefGoogle Scholar
  34. Lehrer E (1991) Internal correlation in repeated games. Int J Game Theory 19:431–456zbMATHMathSciNetCrossRefGoogle Scholar
  35. Lehrer E (1992) Correlated equilibria in two-player repeated games with non-observable actions. Math Oper Res 17:175–199zbMATHMathSciNetCrossRefADSGoogle Scholar
  36. Lehrer E (1996) Mediated talk. Int J Game Theory 25:177–188zbMATHMathSciNetCrossRefGoogle Scholar
  37. Lehrer E, Sorin S (1997) One-shot public mediated talk. Games Econ Behav 20:131–148zbMATHMathSciNetCrossRefGoogle Scholar
  38. Lehrer E, Rosenberg D, Shmaya E (2006) Signaling and mediation in Bayesian games. Mimeo, Tel Aviv University, Tel AvivGoogle Scholar
  39. Milgrom P, Roberts J (1996) Coalition-proofness and correlation with arbitrary communication possibilities. Games Econ Behav 17:113–128MathSciNetCrossRefGoogle Scholar
  40. Moreno D, Wooders J (1996) Coalition-proof equilibrium. Games Econ Behav 17:80–112MathSciNetCrossRefGoogle Scholar
  41. Myerson R (1982) Optimal coordination mechanisms in generalized principal-agent problems. J Math Econ 10:67–81zbMATHMathSciNetCrossRefGoogle Scholar
  42. Myerson R (1986a) Multistage games with communication. Econometrica 54:323–358zbMATHMathSciNetCrossRefGoogle Scholar
  43. Myerson R (1986b) Acceptable and predominant correlated equilibria. Int J Game Theory 15:133–154zbMATHMathSciNetCrossRefGoogle Scholar
  44. Myerson R (1997) Dual reduction and elementary games. Games Econ Behav 21:183–202zbMATHMathSciNetCrossRefGoogle Scholar
  45. Nash J (1951) Non-cooperative games. Ann Math 54:286–295zbMATHMathSciNetCrossRefGoogle Scholar
  46. Nau RF (1992) Joint coherence in games with incomplete information. Manag Sci 38:374–387zbMATHCrossRefGoogle Scholar
  47. Nau RF, McCardle KF (1990) Coherent behavior in noncooperative games. J Econ Theory 50(2):424–444zbMATHMathSciNetCrossRefGoogle Scholar
  48. Nau RF, McCardle KF (1991) Arbitrage, rationality and equilibrium. Theor Decis 31:199–240zbMATHCrossRefGoogle Scholar
  49. Nau RF, Gomez-Canovas S, Hansen P (2004) On the geometry of Nash equilibria and correlated equilibria. Int J Game Theory 32:443–453zbMATHMathSciNetCrossRefGoogle Scholar
  50. Papadimitriou CH (2005) Computing correlated equilibria in multiplayer games. In: Proceedings of the 37th ACM symposium on theory of computing. STOC, Baltimore, pp 49–56Google Scholar
  51. Ray I (1996) Coalition-proof correlated equilibrium: a definition. Games Econ Behav 17:56–79CrossRefGoogle Scholar
  52. Renault J, Tomala T (2004) Communication equilibrium payoffs in repeated games with imperfect monitoring. Games Econ Behav 49:313–344zbMATHMathSciNetCrossRefGoogle Scholar
  53. Solan E (2001) Characterization of correlated equilibrium in stochastic games. Int J Game Theory 30:259–277zbMATHMathSciNetCrossRefGoogle Scholar
  54. Solan E, Vieille N (2002) Correlated equilibrium in stochastic games. Game Econ Behav 38:362–399zbMATHMathSciNetCrossRefGoogle Scholar
  55. Urbano A, Vila J (2002) Computational complexity and communication: coordination in two-player games. Econometrica 70:1893–1927zbMATHMathSciNetCrossRefGoogle Scholar
  56. Urbano A, Vila J (2004a) Computationally restricted unmediated talk under incomplete information. J Econ Theory 23:283–320zbMATHMathSciNetCrossRefGoogle Scholar
  57. Urbano A, Vila J (2004b) Unmediated communication in repeated games with imperfect monitoring. Games Econ Behav 46:143–173zbMATHMathSciNetCrossRefGoogle Scholar
  58. Vida P (2007) From communication equilibria to correlated equilibria. Mimeo, University of Vienna, ViennaGoogle Scholar
  59. Viossat Y (2008) Is having a unique equilibrium robust? J Math Econ 44:1152–1160zbMATHMathSciNetCrossRefGoogle Scholar
  60. Viossat Y (2006) The geometry of Nash equilibria and correlated equilibria and a generalization of zero-sum games. Mimeo, S-WoPEc working paper 641. Stockholm School of Economics, StockholmGoogle Scholar
  61. Viossat Y (2007) The replicator dynamics does not lead to correlated equilibria. Games Econ Behav 59:397–407zbMATHMathSciNetCrossRefGoogle Scholar

Books and Reviews

  1. Forges F (1994) Non-zero sum repeated games and information transmission. In: Megiddo N (ed) Essays in game theory in honor of Michael Maschler. Springer, Berlin, pp 65–95CrossRefGoogle Scholar
  2. Mertens JF (1994) Correlated- and communication equilibria. In: Mertens JF, Sorin S (eds) Game theoretic methods in general equilibrium analysis. Kluwer, Dordrecht, pp 243–248CrossRefGoogle Scholar
  3. Myerson R (1985) Bayesian equilibrium and incentive compatibility. In: Hurwicz L, Schmeidler D, Sonnenschein H (eds) Social goals and social organization. Cambridge University Press, Cambridge, pp 229–259Google Scholar
  4. Myerson R (1994) Communication, correlated equilibria and incentive compatibility. In: Aumann R, Hart S (eds) Handbook of game theory, vol 2. Elsevier, Amsterdam, pp 827–847Google Scholar
  5. Sorin S (1997) Communication, correlation and cooperation. In: Mas Colell A, Hart S (eds) Cooperation: game theoretic approaches. Springer, Berlin, pp 198–218Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.CeremadeUniversité Paris-DauphineParisFrance