A note on the Shapley value for characteristic functions on bipartitions

The Shapley value is a well-known allocation concept in cooperative game theory. It assigns to each player a part of the worth generated by the coalition of all players. The Shapley value is fair in the sense that it satisfies several desirable properties and axioms. We consider a cooperative game with a bipartition that indicates which players are participating. The paper provides a simple analytical solution for the Shapley value when the worth of a coalition only depends on the number of participating coalition players. The computational complexity is only linear in the number of players, which contrasts with the usual exponential increase. This efficient result remains true when we introduce (i) randomization of the bipartition, and (ii) randomly draw an appropriate characteristic/worth function. We illustrate our result with an example that is related to the problem of allocating systemic risk among banks.