On the Inverse problem for Semivalues of cooperative TU games

The Semivalues were introduced axiomatically by Dubey et al [7], as weighted values of cooperative games. For transferable utility games (TU games), they obtained a formula for computing the Semivalue associated with a given weight vector. Among the Semivalues are the well known Shapley value, Banzhaf value, and many other values, for different weight vectors. Let G be the space of cooperative TU games with the set of players N and SE : G −→ R be a Semivalue associated with a given weight vector p; n = |N |. The inverse problem for this Semivalue may be stated as: find out all games (N, ν) ∈ G , such that SE(N, ν) = L, where L ∈ R is an a priori given vector. The inverse problem has been solved for the Shapley value in an earlier paper by Dragan [3]; in the present paper, we solve it for any Semivalue. The potential approach by Hart et al [8], [9], has been used in the first case, while now we use the potential due to Calvo et al [2]. An algorithm called a dynamic algorithm is a byproduct of the results. AMS Subject Classification: 90D12