American Institute of Mathematical Sciences

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April  2014, 1(2): 181-254. doi: 10.3934/jdg.2014.1.181

Approachability, regret and calibration: Implications and equivalences

Vianney Perchet 1,

1. 

Université Paris-Diderot, Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, 8 place FM/13, Paris, France

Received  January 2013 Revised  January 2014 Published  March 2014

Blackwell approachability, regret minimization and calibration are three criteria used to evaluate a strategy (or an algorithm) in sequential decision problems, described as repeated games between a player and Nature. Although they have at first sight not much in common, links between them have been discovered: for instance, both consistent and calibrated strategies can be constructed by following, in some auxiliary game, an approachability strategy.
    We gather seminal and recent results, develop and generalize Blackwell's elegant theory in several directions. The final objectives is to show how approachability can be used as a basic powerful tool to exhibit a new class of intuitive algorithms, based on simple geometric properties. In order to be complete, we also prove that approachability can be seen as a byproduct of the very existence of consistent or calibrated strategies.
Keywords: Blackwell's approachability, calibration., regret minimization.
Mathematics Subject Classification: Primary: 91A25, 68W27; Secondary: 91A2.
Citation: Vianney Perchet. Approachability, regret and calibration: Implications and equivalences. Journal of Dynamics & Games, 2014, 1 (2) : 181-254. doi: 10.3934/jdg.2014.1.181
References:
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show all references

References:
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[7]

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[9]

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[10]

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D. Blackwell, An analog of the minimax theorem for vector payoffs,, Pacific J. Math., 6 (1956), 1.  doi: 10.2140/pjm.1956.6.1.  Google Scholar

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[17]

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[18]

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[24]

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[25]

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[26]

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[27]

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[28]

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[30]

D. P. Foster and R. V. Vohra, Calibration: Respice, adspice, prospice,, Advances in Economics and Econometrics, 1 (2013), 423.  doi: 10.1017/CBO9781139060011.014.  Google Scholar

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[32]

D. Fudenberg and D. K. Levine, Conditional universal consistency,, Games Econom. Behav., 29 (1999), 104.  doi: 10.1006/game.1998.0705.  Google Scholar

[33]

D. Fudenberg and D. K. Levine, An easier way to calibrate,, Games Econom. Behav., 29 (1999), 131.  doi: 10.1006/game.1999.0726.  Google Scholar

[34]

F. Gul, D. Pearce and E. Stachetti, A bound on the proportion of pure strategy equilibria in generic games,, Math. Oper. Res., 18 (1993), 548.  doi: 10.1287/moor.18.3.548.  Google Scholar

[35]

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[36]

J. Hannan, Approximation to bayes risk in repeated play,, in Contributions to the Theory of Games, (1957), 97.   Google Scholar

[37]

S. Hart and A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium,, Econometrica, 68 (2000), 1127.  doi: 10.1111/1468-0262.00153.  Google Scholar

[38]

S. Hart and A. Mas-Colell, A general class of adaptive strategies,, J. Econom. Theory, 98 (2001), 26.  doi: 10.1006/jeth.2000.2746.  Google Scholar

[39]

S. Hart and A. Mas-Colell, Regret-based continuous-time dynamics,, Games Econom. Behav., 45 (2003), 375.  doi: 10.1016/S0899-8256(03)00178-7.  Google Scholar

[40]

E. Hazan and S. M. Kakade, (weak) calibration is computationaly hard,, J. Mach. Learn. Res.: Workshop Conf. Proc., 23 (2012), 1.   Google Scholar

[41]

J. Hofbauer and W. H. Sandholm, On the global convergence of stochastic fictitious play,, Econometrica, 70 (2002), 2265.  doi: 10.1111/1468-0262.00376.  Google Scholar

[42]

J. Hofbauer, S. Sorin and Y. Viossat, Time average replicator and best-reply dynamics,, Math. Oper. Res., 34 (2009), 263.  doi: 10.1287/moor.1080.0359.  Google Scholar

[43]

S. M. Kakade and D. P. Foster, Deterministic calibration and Nash equilibrium,, in Learning theory, (2004), 33.  doi: 10.1007/978-3-540-27819-1_3.  Google Scholar

[44]

O. Kallenberg and R. Sztencel, Some dimension-free features of vector-valued martingales,, Probability Theory and Related Fields, 88 (1991), 215.  doi: 10.1007/BF01212560.  Google Scholar

[45]

E. Kohlberg, Optimal strategies in repeated games with incomplete information,, Internat. J. Game Theory, 4 (1975), 7.  doi: 10.1007/BF01766399.  Google Scholar

[46]

J. Kwon, Hilbert Distance, Bounded Convex Functions, and Application to the Exponential Weight Algorithm,, Master's thesis, (2012).   Google Scholar

[47]

E. Lehrer, Any inspection is manipulable,, Econometrica, 69 (2001), 1333.  doi: 10.1111/1468-0262.00244.  Google Scholar

[48]

E. Lehrer, Approachability in infinite dimensional spaces,, Internat. J. Game Theory, 31 (2002), 253.  doi: 10.1007/s001820200115.  Google Scholar

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[50]

E. Lehrer and E. Solan, Excludability and bounded computational capacity,, Math. Oper. Res., 31 (2006), 637.  doi: 10.1287/moor.1060.0211.  Google Scholar

[51]

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