Null players, outside options, and stability: The conditional Shapley value

We suggest a new component efficient solution for monotonic TU games with a coalition structure, the conditional Shapley value. Other than other such solutions, it satisfies the null player property. Nevertheless, it accounts for the players’ outside options in productive components of coalition structures. For all monotonic games, there exist coalition structures that are stable under the conditional Shapley value. For voting games, such stable coalition structures support Gamson’s theory of coalition formation (Gamson, 1961).