Abstract
This article proposes a value which can be considered an extension of the Banzhaf value for cooperative games. The proposed value is defined on the class of j-cooperative games, i.e., games in which players choose among a finite set of ordered actions and the result depends only on these elections. If the output is binary, only two options are available, then j-cooperative games become j-simple games. The restriction of the value to j-simple games leads to a power index that can be considered an extension of the Banzhaf power index for simple games. The paper provides an axiomatic characterization for the value and the index which is closely related to the first axiomatization of the Banzhaf value and Banzhaf power index in the respective contexts of cooperative and simple games.
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Notes
S precedes T in the lexicographic order if for the smallest \(p \in N\) such that \(p \in S_i\) and \(p \in T_h\) with \(i \ne h\), it holds \(i<h\).
By solving the linear system of equations it can be seen that these coefficients are \(c_S = \sum \nolimits _{T {\subseteq }^j S} (-1)^{\sum \nolimits _{i=1}^j(t_i-s_i)} v(T)\).
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Acknowledgements
This research was partially supported by funds from the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER funds) under grant MTM2015-66818-P (MINECO/FEDER).
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Freixas, J. The Banzhaf Value for Cooperative and Simple Multichoice Games. Group Decis Negot 29, 61–74 (2020). https://doi.org/10.1007/s10726-019-09651-4
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DOI: https://doi.org/10.1007/s10726-019-09651-4