Attainability in Repeated Games with Vector Payoffs

We introduce the concept of attainable sets of payoffs in two-player repeated games with vector payoffs. A set of payoff vectors is called attainable by a player if there is a finite horizon T such that the player can guarantee that after time T the distance between the set and the cumulative payoff is arbitrarily small, regardless of the strategy Player 2 is using. We provide a necessary and sufficient condition for the attainability of a convex set, using the concept of B-sets. We then particularize the condition to the case in which the set is a singleton, and provide some equivalent conditions. We finally characterize when all vectors are attainable.