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How long to equilibrium? The communication complexity of uncoupled equilibrium procedures

  • Hart, Sergiu
  • Mansour, Yishay

We study the question of how long it takes players to reach a Nash equilibrium in uncoupled setups, where each player initially knows only his own payoff function. We derive lower bounds on the communication complexity of reaching a Nash equilibrium, i.e., on the number of bits that need to be transmitted, and thus also on the required number of steps. Specifically, we show lower bounds that are exponential in the number of players in each one of the following cases: (1) reaching a pure Nash equilibrium; (2) reaching a pure Nash equilibrium in a Bayesian setting; and (3) reaching a mixed Nash equilibrium. We then show that, in contrast, the communication complexity of reaching a correlated equilibrium is polynomial in the number of players.

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Article provided by Elsevier in its journal Games and Economic Behavior.

Volume (Year): 69 (2010)
Issue (Month): 1 (May)
Pages: 107-126

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Handle: RePEc:eee:gamebe:v:69:y:2010:i:1:p:107-126
Contact details of provider: Web page: http://www.elsevier.com/locate/inca/622836

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  1. Sergiu Hart & Andreu Mas-Colell, 2004. "Stochastic Uncoupled Dynamics and Nash Equilibrium," Discussion Paper Series dp371, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
  2. Jordan J. S., 1993. "Three Problems in Learning Mixed-Strategy Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 368-386, July.
  3. Aumann, Robert J., 1974. "Subjectivity and correlation in randomized strategies," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 67-96, March.
  4. E. Kalai & E. Lehrer, 2010. "Rational Learning Leads to Nash Equilibrium," Levine's Working Paper Archive 529, David K. Levine.
  5. Germano, Fabrizio & Lugosi, Gabor, 2007. "Global Nash convergence of Foster and Young's regret testing," Games and Economic Behavior, Elsevier, vol. 60(1), pages 135-154, July.
  6. Drew Fudenberg & David K. Levine, 1998. "The Theory of Learning in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061945, June.
  7. Sergiu Hart & Andreu Mas-Colell, 2003. "Uncoupled Dynamics Do Not Lead to Nash Equilibrium," American Economic Review, American Economic Association, vol. 93(5), pages 1830-1836, December.
  8. Jordan, J. S., 1991. "Bayesian learning in normal form games," Games and Economic Behavior, Elsevier, vol. 3(1), pages 60-81, February.
  9. S. Hart & A. Mas-Collel, 2010. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Levine's Working Paper Archive 572, David K. Levine.
  10. Peyton Young, 2002. "Learning Hypothesis Testing and Nash Equilibrium," Economics Working Paper Archive 474, The Johns Hopkins University,Department of Economics.
  11. Hart, Sergiu & Mas-Colell, Andreu, 2001. "A General Class of Adaptive Strategies," Journal of Economic Theory, Elsevier, vol. 98(1), pages 26-54, May.
  12. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
  13. Stoltz, Gilles & Lugosi, Gabor, 2007. "Learning correlated equilibria in games with compact sets of strategies," Games and Economic Behavior, Elsevier, vol. 59(1), pages 187-208, April.
  14. Sergiu Hart, 2004. "Adaptive Heuristics," Levine's Bibliography 122247000000000471, UCLA Department of Economics.
  15. Foster, Dean P. & Young, H. Peyton, 2006. "Regret testing: learning to play Nash equilibrium without knowing you have an opponent," Theoretical Economics, Econometric Society, vol. 1(3), pages 341-367, September.
  16. Amotz Cahn, 2004. "General procedures leading to correlated equilibria," International Journal of Game Theory, Springer, vol. 33(1), pages 21-40, January.
  17. Foster, Dean P. & Vohra, Rakesh V., 1997. "Calibrated Learning and Correlated Equilibrium," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 40-55, October.
  18. Young, H. Peyton, 2004. "Strategic Learning and its Limits," OUP Catalogue, Oxford University Press, number 9780199269181, March.
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