Some Properties for Bisemivalues on Bicooperative Games

Abstract

In this work, we focus on bicooperative games, a variation of the classic cooperative games, and investigate the conditions for the coefficients of the bisemivalues -a generalization of semivalues for cooperative games -necessary and / or sufficient in order to satisfy some properties, including among others, desirability

Weak null, necessary defender and necessary detractor players: characterizations of the Banzhaf and the Shapley bisemivalues

“…Lemma 2.4 ( Domènech et al [17]) Let ψ be a bisemivalue on BG N . The allocations to a player i ∈ N in the identity game δ (A,B) are given by…”

“…Following this idea, Domènech et al [16] defined the multilinear extension of a bicooperative game (MLE), that parallels the existing definition of the multilinear extension of a cooperative game given by Owen, and provided a method to compute (p, q)bisemivalues in general, and the Banzhaf bisemivalue in particular, by means of it. In a recent paper, Domènech et al [17] go further and give a computational procedure to calculate any bisemivalue, also by using the MLE of the game defined in [16].…”

“…However, the Shapley bisemivalue of a player is a weighted average of those contributions but, now, the weights depend on the size of the coalitions. In a recent paper, Domènech et al [17] investigated the conditions for the coefficients of the bisemivalues, necessary and / or sufficient, in order to satisfy some standard properties, considered for values on cooperative games (see, e.g., [10,15,21]). For more information about solution concepts on bicooperative games we refer the reader to [7] and [8].…”

“…Following this schema, Domènech et al in [16], defined a multilinear extension for each bicooperative game and gave a computational procedure to calculate (p, q)-bisemivalues in general, and the Banzhaf bisemivalue in particular. Moreover, in [17] a computational procedure to calculate bisemivalues in terms of the multilinear extension of the game was also given. The aim of this paper is to provide new axiomatic characterizations of the Banzhaf and Shapley bisemivalues on bicooperative games in a similar way to given in [3] for cooperative games.…”