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

Abstract

We focus on bicooperative games, a variation of the classic cooperative games and, in particular, on the Banzhaf and the Shapley bisemivalues defined on these games. We consider three special classes of players: weak null, necessary defender and necessary detractor players. By introducing new properties related to this kind of players, we provide new axiomatic characterizations of the Banzhaf and the Shapley bisemivalues giving, in both cases, a set of independent properties that univocally determine them. We also provide a computational procedure to calculate the allocations given by the Shapley bisemivalue in terms of the generalized multilinear extension of the game.

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Notes

  1. 1.

    The term “multilinear” means that, for each \(i\in N\), the function is linear in \(x_i\), that is, of the form \(f_v(x_1,x_2,\dots ,x_n)=g_i(x_1,x_2,\dots ,\overset{\wedge }{x_i},\dots ,x_n)x_i + h_i(x_1,x_2,\dots ,\overset{\wedge }{x_i},\dots ,x_n)\).

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Acknowledgements

The authors wish to thank the managing editor for encouraging them to improve the paper, and two anonymous reviewers for their interesting comments and helpful suggestions, most of which have been incorporated into the text.

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Correspondence to María Albina Puente.

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This research project was partially supported by funds from the Spanish Ministry of Science and Innovation grant PID2019-104987GB-I00.

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Domènech, M., Giménez, J.M. & Puente, M.A. Weak null, necessary defender and necessary detractor players: characterizations of the Banzhaf and the Shapley bisemivalues. Ann Oper Res (2021). https://doi.org/10.1007/s10479-021-04153-6

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Keywords

  • Cooperative game
  • Bicooperative game
  • Banzhaf bisemivalue
  • Shapley bisemivalue

Mathematics Subject Classification

  • 91A12