Algorithms for computing the Shapley value of cooperative games on lattices

We study algorithms to compute the Shapley value for a cooperative game on a lattice LΣ=(FΣ,⊆) where FΣ is the family of closed sets given by an implicational system Σ on a set N of players. The first algorithm is based on the generation of the maximal chains of the lattice LΣ and computes the Shapley value in O(|N|3.|Σ|.|Ch|) time complexity using polynomial space, where Ch is the set of maximal chains of LΣ. The second algorithm proceeds by building the lattice LΣ and computes the Shapley value in O(|N|3.|Σ|.|FΣ|) time and space complexity. Our main contribution is to show that the Shapley value of weighted graph games on a product of chains with the same fixed length is computable in polynomial time. We do this by partitioning the set of feasible coalitions relevant to the computation of the Shapley value into equivalence classes in such a way that we need to consider only one element of each class in the computation.