Dynamic Games and Applications

, Volume 3, Issue 1, pp 38–67 | Cite as

Transforming Monitoring Structures with Resilient Encoders—Application to Repeated Games

Article

Abstract

An important feature of a dynamic game is its monitoring structure namely, what the players effectively see from the played actions. We consider games with arbitrary monitoring structures. One of the purposes of this paper is to know to what extent an encoder, who perfectly observes the played actions and sends a complementary public signal to the players, can establish perfect monitoring for all the players. To reach this goal, the main technical problem to be solved at the encoder is to design a source encoder which compresses the action profile in the most concise manner possible. A special feature of this encoder is that the multi-dimensional signal (namely, the action profiles) to be encoded is assumed to comprise a component whose probability distribution is not known to the encoder and the decoder has a side information (the private signals received by the players when the encoder is off). This new framework appears to be both of game-theoretical and information-theoretical interest. In particular, it is useful for designing certain types of encoders that are resilient to single deviations and provide an equilibrium utility region in the proposed setting; it provides a new type of constraints to compress an information source (i.e., a random variable). Regarding the first aspect, we apply the derived result to the repeated prisoner’s dilemma.

Keywords

Arbitrarily varying source Dynamic games Folk theorem Games with imperfect monitoring Information constraint Observation structure Source coding 

References

  1. 1.
    Ahlswede R (1979) Coloring hypergraphs: a new approach to multi-user source coding, part 1. J Comb Inf Syst Sci 4(1):76–115 MathSciNetMATHGoogle Scholar
  2. 2.
    Ahlswede R (1980) Coloring hypergraphs: a new approach to multi-user source coding, part 2. J Comb Inf Syst Sci 5(3):220–268 MathSciNetMATHGoogle Scholar
  3. 3.
    Akyildiz IF, Su W, Sankarasubramaniam Y, Cayirci E (2002) Wireless sensor networks: a survey. Comput Netw 38(4):393–422 CrossRefGoogle Scholar
  4. 4.
    Aumann RJ (1981) Survey of repeated game, pp 11–42 Google Scholar
  5. 5.
    Aumann RJ (1981) Survey of repeated game. In: Essays in game theory and mathematical economics in honor of Oskar Morgenstern. Wissenschaftsverlag, Bibliographisches Institut, Mannheim, pp 11–42 Google Scholar
  6. 6.
    Bavly G, Neyman A (2003) Online concealed correlation by boundedly rational players. Discussion paper series. The Center for the Study of Rationality, Hebrew University, Jerusalem Google Scholar
  7. 7.
    Bondy J, Murty U (1976) Graph theory with applications. Elsevier Science, Amsterdam MATHGoogle Scholar
  8. 8.
    Cover TM, Thomas JA (2006) Elements of information theory, 2nd edn. Wiley-Interscience, New York MATHGoogle Scholar
  9. 9.
    Csiszár I, Körner J (1981) Information theory: coding theorems for discrete memoryless systems Google Scholar
  10. 10.
    DaSilva MJ, Taffin A, Lasaulce S, Buljore S (2011) Closed loop transmit diversity enhancements for UMTS narrowband and wideband TD-CDMA. In: IEEE proc of the 53rd vehicular technology conference, vol 3, pp 1963–1967 Google Scholar
  11. 11.
    El Gamal AA, van der Meulen E (1981) A proof of Marton’s coding theorem for the discrete memoryless broadcast channel. IEEE Trans Inf Theory 27(1):120–122 MATHCrossRefGoogle Scholar
  12. 12.
    Fudenberg D, Levine D, Maskin E (1994) The folk theorem with imperfect public information. Econometrica 62(5):997–1039 MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Gallager RG (1976) Source coding with side information and universal coding. In: IEEE international symposium on information theory, Renneby, Sweden Google Scholar
  14. 14.
    Gossner O, Tomala T (2007) Secret correlation in repeated games with imperfect monitoring. Math Oper Res 32(2):413–424 MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Gossner O, Hernandez P, Neyman A (2006) Optimal use of communication resources. Econometrica 74(6):1603–1636 MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Han TS (2003) Information-spectrum methods in information theory. Springer, Berlin MATHGoogle Scholar
  17. 17.
    Hörner J, Olszewski W (2006) The folk theorem for games with private almost-perfect monitoring. Econometrica 74(6):1499–1544 MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kowalewski F (2000) Joint predistortion and transmit diversity. In: IEEE proc of the global telecommunications conference, vol 1, pp 245–249 Google Scholar
  19. 19.
    Lehrer E (1988) Repeated games with stationary bounded recall strategies. J Econ Theory 46:130–144 MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Lehrer E (1991) Internal correlation in repeated games. Int J Game Theory 19(4):431–456 MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    LeTreust M, Lasaulce S (2010) A repeated game formulation of energy-efficient decentralized power control. IEEE Trans Wirel Commun 9(9):2860–2869 CrossRefGoogle Scholar
  22. 22.
    LeTreust M, Lasaulce S (2011) The price of re-establishing almost perfect monitoring in games with arbitrary monitoring structures. In: ACM proc of the 4th international workshop on game theory in communication networks (GAMECOMM11), Cachan (Paris), France Google Scholar
  23. 23.
    LeTreust M, Lasaulce S (2011) Resilient source coding. In: IEEE proc of the international conference on network games, control and optimization (NETGCOOP11), Paris, France Google Scholar
  24. 24.
    Peretz R (2011) Correlation through bounded recall strategies. Discussion paper series. The Center for the Study of Rationality, Hebrew University, Jerusalem Google Scholar
  25. 25.
    Peyton Y (2004) Strategic learning and its limits Google Scholar
  26. 26.
    Renault J, Tomala T (2011) General properties of long-run supergames. Dyn Games Appl 1(2):319–350 MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423 MathSciNetMATHGoogle Scholar
  28. 28.
    Shannon CE (1956) The zero error capacity of a noisy channel. IRE Trans Inf Theory 2:8–19 MathSciNetCrossRefGoogle Scholar
  29. 29.
    Slepian D, Wolf JK (1973) Noiseless coding of correlated information sources. IEEE Trans Inf Theory 19:471–480 MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Sorin S (1992) Repeated games with complete information. In: Handbook of game theory with economic applications, vol 1. Elsevier Science, Amsterdam Google Scholar
  31. 31.
    Witsenhausen H (1976) The zero-error side information problem and chromatic numbers (corresp). IEEE Trans Inf Theory 22(5):592–593 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Université Paris-Est Marne-la-ValléeInstitut d’électronique et d’informatique Gaspard-MongeMarne-la-ValléeFrance
  2. 2.Laboratoire des Signaux et Systèmes, CNRSUniversité Paris Sud XI, SupélecParisFrance

Personalised recommendations