Approachability in repeated games: Computational aspects and a Stackelberg variant

We consider a finite two-player zero-sum game with vector-valued rewards. We study the question of whether a given polyhedral set D is "approachable," that is, whether Player 1 (the "decision maker") can guarantee that the long-term average reward belongs to D, for any strategy of Player 2 (the "adversary"). We examine Blackwell's necessary and sufficient conditions for approachability, and show that the problem of checking these conditions is NP-hard, even in the special case where D is a singleton. We then consider a Stackelberg variant whereby, at each stage, the adversary gets to act after observing the decision maker's action. We provide necessary and sufficient conditions for approachability, and again establish that checking these conditions is NP-hard, even when D is a singleton. On the other hand, if the dimension of the reward vector is fixed, an approximate version of these conditions can be checked in polynomial time.