Chapter summary

This chapter is devoted to a theory of repeated games with vector payoffs, known as the theory of approachability, developed by Blackwell in 1956. Blackwell considered two-player repeated games in which the outcome is an m-dimensional vector of attributes, and the goal of each player is to control the average vector of attributes. The goal can be either to approach a given target set S ⊆ ℝm, that is, to ensure that the distance between the vector of average attributes and the target set S converges to 0, or to exclude the target set S, that is, to ensure that the distance between the vector of average attributes and S remains bounded away from 0. If a player can approach the target set we say that the set is approachable by the player, whereas if the player can exclude the target set we say that it is excludable by that player. Clearly, a set cannot be both approachable by one player and excludable by the other player.

We provide a geometric condition that ensures that a set is approachable by a player, and show that any convex set is either approachable by one player or excludable by the other player.

Two applications of the theory of approachability are provided: it is used, respectively, to construct an optimal strategy for the uninformed player in two-player zero-sum repeated games with incomplete information on one side, and to construct a no-regret strategy in sequential decision problems with experts.