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

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  • Hart, Sergiu
  • Mansour, Yishay

Abstract

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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Bibliographic Info

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

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Web page: http://www.elsevier.com/locate/inca/622836

Related research

Keywords: Uncoupled dynamics Nash equilibrium Communication complexity Correlated equilibrium Speed of convergence;

References

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  1. Fabrizio Germano & Gábor Lugosi, 2004. "Global Nash convergence of Foster and Young's regret testing," Economics Working Papers 788, Department of Economics and Business, Universitat Pompeu Fabra.
  2. S. Hart & A. Mas-Collel, 2010. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Levine's Working Paper Archive 572, David K. Levine.
  3. Sergiu Hart, 2004. "Adaptive Heuristics," Discussion Paper Series dp372, The Center for the Study of Rationality, Hebrew University, Jerusalem.
  4. Drew Fudenberg & David K. Levine, 1996. "The Theory of Learning in Games," Levine's Working Paper Archive 624, David K. Levine.
  5. Sergiu Hart & Andreu Mas-Colell, 1999. "A general class of adaptative strategies," Economics Working Papers 373, Department of Economics and Business, Universitat Pompeu Fabra.
  6. 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.
  7. Jordan J. S., 1993. "Three Problems in Learning Mixed-Strategy Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 368-386, July.
  8. Kalai, Ehud & Lehrer, Ehud, 1993. "Rational Learning Leads to Nash Equilibrium," Econometrica, Econometric Society, vol. 61(5), pages 1019-45, September.
  9. Amotz Cahn, 2004. "General procedures leading to correlated equilibria," International Journal of Game Theory, Springer, vol. 33(1), pages 21-40, January.
  10. 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.
  11. Hart, Sergiu & Mas-Colell, Andreu, 2006. "Stochastic uncoupled dynamics and Nash equilibrium," Games and Economic Behavior, Elsevier, vol. 57(2), pages 286-303, November.
  12. Aumann, Robert J., 1974. "Subjectivity and correlation in randomized strategies," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 67-96, March.
  13. Foster, Dean P. & Young, H. Peyton, 2003. "Learning, hypothesis testing, and Nash equilibrium," Games and Economic Behavior, Elsevier, vol. 45(1), pages 73-96, October.
  14. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
  15. Young, H. Peyton, 2004. "Strategic Learning and its Limits," OUP Catalogue, Oxford University Press, number 9780199269181, September.
  16. Jordan, J. S., 1991. "Bayesian learning in normal form games," Games and Economic Behavior, Elsevier, vol. 3(1), pages 60-81, February.
  17. 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.
  18. 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.
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Citations

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Cited by:
  1. Babichenko, Yakov, 2012. "Completely uncoupled dynamics and Nash equilibria," Games and Economic Behavior, Elsevier, vol. 76(1), pages 1-14.
  2. Yakov Babichenko, 2014. "How long to Pareto efficiency?," International Journal of Game Theory, Springer, vol. 43(1), pages 13-24, February.
  3. Sergiu Hart & Noam Nisan, 2013. "The Query Complexity of Correlated Equilibria," Levine's Working Paper Archive 786969000000000819, David K. Levine.
  4. Itai Arieli & H Peyton Young, 2011. "Stochastic Learning Dynamics and Speed of Convergence in Population Games," Economics Series Working Papers 570, University of Oxford, Department of Economics.
  5. Pradelski, Bary S.R. & Young, H. Peyton, 2012. "Learning efficient Nash equilibria in distributed systems," Games and Economic Behavior, Elsevier, vol. 75(2), pages 882-897.
  6. Babichenko, Yakov, 2013. "Best-reply dynamics in large binary-choice anonymous games," Games and Economic Behavior, Elsevier, vol. 81(C), pages 130-144.
  7. Babaioff, Moshe & Blumrosen, Liad & Schapira, Michael, 2013. "The communication burden of payment determination," Games and Economic Behavior, Elsevier, vol. 77(1), pages 153-167.

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