Repeated games over networks with vector payoffs: the notion of attainability

We introduce the concept of strongly attainable sets of payoffs in two-player repeated games with vector payoffs in continuous time. A set of payoffs is called strongly attainable if player 1 has a strategy guaranteeing, even in the worst case, that the distance between the set and the cumulative payoff shrinks with time to zero. We characterize when any vector is strongly attainable and illustrate the motivation of our study on a multi-inventory application.