Multiweighted Shapley Values and Random Order Values

Abstract

**Please note that the full text is embargoed** ABSTRACT: Recently, R. J. Weber introduced axiomatically in [14] the concept of random order value, an operator from [see pdf for notation], the space of TU games, to [see pdf for notation], satisfying linearity, dummy, efficiency and monotonicity axioms. In the present paper, we consider the concept of multiweighted Shapley value (MWSV), an operator from [see pdf for notation], satisfying linearity and carrier axioms. In the first section, we show that a MWSV is defined by its matrix representation relative to the unanimity basis for [see pdf for notation] and the standard basis for [see pdf for notation], if and only if this matrix possesses the so-called carrier property. Beside the Shapley value, the weighted Shapley value and the Kalai/Samet value considered in [5], the Owen coalition structure value introduced in [10] and the McLean weighted coalition structure value introduced in [7], are shown to be MWSVs, in the second section. A characterization of the MWSVs in terms of the coalitional form, and the fact that for a linear operator the carrier axiom is equivalent to the dummy and efficiency axioms, are enabling us to prove in the third section that any random order value is a MWSV. In [15], H. P. Young has introduced a coalitional and a strong monotonicity for such operators and a monotonicity axiom is used in [7] and [14] in the definitions of the McLean and the random order values. In the last section, we show that a MWSV may possess all three monotonicity properties or none of them. An algebraic characterization of monotonic MWSV is given. As such operators are random order values, we get an algebraic characterization alternative to that which can be extracted from Weber's work [14]. Finally, we show that this result follows also from Gilboa/Monderer work [3] on quasi-values.