Cooperative games with externalities and probabilistic coalitional beliefs

We revisit cooperative games with externalities, i.e. cooperative games where the payoff of a coalition depends on the partition of the entire set of players. We define the worth of a coalition assuming that its members have probabilistic beliefs over the coalitional behavior of the outsiders, i.e., they assign various probability distributions on the set of partitions that the outsiders can form. We apply this framework to symmetric aggregative games and derive conditions on coalitional beliefs that guarantee the non-emptiness of the core of the induced cooperative games.