# American Institute of Mathematical Sciences

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

## Approachability, regret and calibration: Implications and equivalences

 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.
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
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