Approachable sets of vector payoffs in stochastic games

The notion of an approachable set in a recurrent zero-sum vector-payoff game was introduced by Blackwell, who proved a sufficient condition for approachability in such games. This notion has been recently generalized to the case of arbitrary stochastic games with vector-payoffs, becoming dependent on the initial state. In this paper, we generalize Blackwell's condition to this wider context, giving a sufficient condition for approachability from all initial states. In addition, we show that this condition is also necessary for convex sets, thereby providing a complete characterization of all the approachable convex sets in a game.