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Approachability with bounded memory

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  • Lehrer, Ehud
  • Solan, Eilon

Abstract

We study Blackwell's approachability in repeated games with vector payoffs when the approaching player is restricted to use strategies with bounded memory: either strategies with bounded recall, or strategies that can be implemented by finite automata. Our main finding is that the following three statements are equivalent for closed sets. (i) The set is approachable with bounded recall strategies. (ii) The set is approachable with strategies that can be implemented with finite automata. (iii) The set contains a convex approachable set. Using our results we show that (i) there are almost-regret-free strategies with bounded memory, (ii) there is a strategy with bounded memory to choose the best among several experts, and (iii) Hart and Mas-Colell's adaptive learning procedure can be achieved using strategies with bounded memory.

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Bibliographic Info

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

Volume (Year): 66 (2009)
Issue (Month): 2 (July)
Pages: 995-1004

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Handle: RePEc:eee:gamebe:v:66:y:2009:i:2:p:995-1004

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

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Keywords: Approachability Repeated games Vector payoffs Bounded memory Bounded recall Automata No-regret Adaptive learning;

References

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  1. Kalai, Ehud & Stanford, William, 1988. "Finite Rationality and Interpersonal Complexity in Repeated Games," Econometrica, Econometric Society, vol. 56(2), pages 397-410, March.
  2. Sergiu Hart & Andreu Mas-Colell, 1997. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Game Theory and Information 9703006, EconWPA, revised 24 Mar 1997.
  3. Neyman, Abraham, 1985. "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma," Economics Letters, Elsevier, vol. 19(3), pages 227-229.
  4. Watson Joel, 1994. "Cooperation in the Infinitely Repeated Prisoners' Dilemma with Perturbations," Games and Economic Behavior, Elsevier, vol. 7(2), pages 260-285, September.
  5. Rustichini, Aldo, 1999. "Minimizing Regret: The General Case," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 224-243, October.
  6. Ariel Rubinstein, 1997. "Finite automata play the repeated prisioners dilemma," Levine's Working Paper Archive 1639, David K. Levine.
  7. Foster, Dean P. & Vohra, Rakesh, 1999. "Regret in the On-Line Decision Problem," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 7-35, October.
  8. Lehrer, Ehud, 1988. "Repeated games with stationary bounded recall strategies," Journal of Economic Theory, Elsevier, vol. 46(1), pages 130-144, October.
  9. Ehud Lehrer & Eilon Solan, 2003. "Excludability and Bounded Computational Capacity Strategies," Discussion Papers 1374, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  10. Lehrer, Ehud, 2003. "A wide range no-regret theorem," Games and Economic Behavior, Elsevier, vol. 42(1), pages 101-115, January.
  11. Jeheil Phillippe, 1995. "Limited Horizon Forecast in Repeated Alternate Games," Journal of Economic Theory, Elsevier, vol. 67(2), pages 497-519, December.
  12. Fudenberg, Drew & Levine, David, 1999. "Conditional Universal Consistency," Scholarly Articles 3204826, Harvard University Department of Economics.
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Cited by:
  1. Rene Saran & Roberto Serrano, 2010. "Regret matching with finite memory," Working Papers 2010-10, Instituto MadrileƱo de Estudios Avanzados (IMDEA) Ciencias Sociales.
  2. Karl Schlag & Andriy Zapechelnyuk, 2009. "Decision making in uncertain and changing environments," Economics Working Papers 1160, Department of Economics and Business, Universitat Pompeu Fabra.
  3. Saran Rene & Serrano Roberto, 2010. "Regret Matching with Finite Memory," Research Memorandum 033, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  4. Andriy Zapechelnyuk, 2009. "Limit Behavior of No-regret Dynamics," Discussion Papers 21, Kyiv School of Economics.
  5. Schlag, Karl & Zapechelnyuk, Andriy, 2012. "On the impossibility of achieving no regrets in repeated games," Journal of Economic Behavior & Organization, Elsevier, vol. 81(1), pages 153-158.
  6. Karl Schlag & Andriy Zapechelnyuk, 2010. "On the Impossibility of Regret Minimization in Repeated Games," Working Papers 676, Queen Mary, University of London, School of Economics and Finance.
  7. Rene Saran & Roberto Serrano, 2012. "Regret Matching with Finite Memory," Dynamic Games and Applications, Springer, vol. 2(1), pages 160-175, March.

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