Values and potential of games with cooperation structure
For traditional cooperative games in which all coalitions are possible, there exist several natural ways of dividing the total payoff \(v(N)\), e.g., the Shapley value in which \(i\)-th player gets an average of what he brings to all possible coalitions: \(\Phi_i(N,v)=\sum_{S:i\in S\subseteq N} (s-1)!\cdot (n-s)!\cdot [v(S)-v(S-\{i\})]/n!\), where \(s\) is the number of players in \(S\), and \(n\) is the total number of players. In some real-life situations, not all coalitions are possible. The class of all possible coalitions is called a cooperative structure. The author generalizes the known values (including the Shapley value) to games with cooperative structure, shows how known algorithms for computing these values can be generalized to such games, and which properties of the values are preserved under this generalization. The proofs are based on convex geometry, a discrete analogue of convex sets.
scientific article; zbMATH DE number 1429797
scientific article