IJPAM: Volume 22, No. 4 (2005)

ON THE INVERSE PROBLEM FOR SEMIVALUES
OF COOPERATIVE TU GAMES

Irinel Dragan
Department of Mathematics
The University of Texas at Arlington
411 S. Nedderman Dr., 443, Arlington
Texas, 76019-0408, USA
e-mail: dragan@uta.edu

Abstract.The semivalues were introduced axiomatically by Dubey et al [#!P.Dubey!#], 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 be the space of cooperative TU games with the set of players and $SE: G^N\longrightarrow R^n$ be a semivalue associated with a given weight vector ; $n=\vert N\vert$ . The inverse problem for this semivalue may be stated as: find out all games $(N,\nu)\in G^N$ , such that $SE(N,\nu)=L$ , where $L\in R^n$ is an a priori given vector. The inverse problem has been solved for the Shapley value in an earlier paper by Dragan [#!I.Dragan1!#]; in the present paper, we solve it for any semivalue. The potential approach by Hart et al [#!S.Hart1!#], [#!S.Hart2!#], has been used in the first case, while now we use the potential due to Calvo et al [#!E.Calvo!#]. An algorithm called a dynamic algorithm is a byproduct of the results.