IJPAM: Volume 22, No. 4 (2005)


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 $G^N$ be the space of cooperative TU games with the set of players $N$ and $SE:
G^N\longrightarrow R^n$ be a semivalue associated with a given weight vector $p^n$; $n=\vert N\vert$. 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.

Received: July 8, 2005

AMS Subject Classification: 90D12

Key Words and Phrases: TU games, semivalues, Shapley value, potential, potential basis

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 22
Issue: 4