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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.

Suggested Citation

  • Lehrer, Ehud & Solan, Eilon, 2009. "Approachability with bounded memory," Games and Economic Behavior, Elsevier, vol. 66(2), pages 995-1004, July.
  • Handle: RePEc:eee:gamebe:v:66:y:2009:i:2:p:995-1004
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    References listed on IDEAS

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    1. Neyman, Abraham, 1985. "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma," Economics Letters, Elsevier, vol. 19(3), pages 227-229.
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    4. Fudenberg, Drew & Levine, David K., 1999. "Conditional Universal Consistency," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 104-130, October.
    5. Lehrer, Ehud, 2003. "A wide range no-regret theorem," Games and Economic Behavior, Elsevier, vol. 42(1), pages 101-115, January.
    6. Sergiu Hart & Andreu Mas-Colell, 2000. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Econometrica, Econometric Society, vol. 68(5), pages 1127-1150, September.
    7. Watson Joel, 1994. "Cooperation in the Infinitely Repeated Prisoners' Dilemma with Perturbations," Games and Economic Behavior, Elsevier, vol. 7(2), pages 260-285, September.
    8. Lehrer, Ehud, 1988. "Repeated games with stationary bounded recall strategies," Journal of Economic Theory, Elsevier, vol. 46(1), pages 130-144, October.
    9. Rubinstein, Ariel, 1986. "Finite automata play the repeated prisoner's dilemma," Journal of Economic Theory, Elsevier, vol. 39(1), pages 83-96, June.
    10. 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.
    11. Rustichini, Aldo, 1999. "Minimizing Regret: The General Case," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 224-243, October.
    12. Jeheil Phillippe, 1995. "Limited Horizon Forecast in Repeated Alternate Games," Journal of Economic Theory, Elsevier, vol. 67(2), pages 497-519, December.
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    Citations

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    Cited by:

    1. Rene Saran & Roberto Serrano, 2012. "Regret Matching with Finite Memory," Dynamic Games and Applications, Springer, vol. 2(1), pages 160-175, March.
    2. Andriy Zapechelnyuk, 2009. "Limit Behavior of No-regret Dynamics," Discussion Papers 21, Kyiv School of Economics.
    3. Lagziel, David & Lehrer, Ehud, 2015. "Approachability with delayed information," Journal of Economic Theory, Elsevier, vol. 157(C), pages 425-444.
    4. repec:eee:gamebe:v:109:y:2018:i:c:p:271-293 is not listed on IDEAS
    5. repec:eee:jetheo:v:169:y:2017:i:c:p:145-169 is not listed on IDEAS
    6. Schlag, Karl H. & Zapechelnyuk, Andriy, 2017. "Dynamic benchmark targeting," Journal of Economic Theory, Elsevier, vol. 169(C), pages 145-169.
    7. 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.
    8. Karl Schlag & Andriy Zapechelnyuk, 2009. "Decision Making in Uncertain and Changing Environments," Discussion Papers 19, Kyiv School of Economics.
    9. Dario Bauso & Hamidou Tembine & Tamer Başar, 2016. "Robust Mean Field Games," Dynamic Games and Applications, Springer, vol. 6(3), pages 277-303, September.
    10. 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.

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