Abstract
In this note we study a truncated additive normalization of the Banzhaf value. We are able to show that it corresponds to the least square nucleolus (LS-nucleolus), which was originally introduced as the solution of a constrained optimization problem [4]. Thus, the main result provides an explicit expression that eases the computation and contributes to the understanding of the LS-nucleolus. Lastly, the result is extended to the broader family of individually rational least square values [6].
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Notes
We will write \(S\cup i\) instead \(S\cup \{i\}\) and \(S{\setminus }i\) instead \(S{\setminus } \{i\}\) to simplify the notation.
Given a finite set \(S\), we denote by lowercase \(s\) its number of elements.
A value on \(\mathcal {G}\), \(\mathsf {f}\), satisfies strategic equivalence if for every \((N,v)\in \mathcal {G}\), \(\alpha >0\), and \(\beta \in \mathbb {R}^N\), \(\mathsf {f}(N,\alpha v+\beta )=\alpha \mathsf {f}(N,v)+\beta \), where \((N,\alpha v+\beta )\) is defined for every \(S\subseteq N\) by \((\alpha v+\beta )(S)=\alpha v(S)+\beta (S)\).
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Acknowledgments
Authors acknowledge the financial support of Ministerio de Ciencia e Innovación through projects MTM2011-27731-C02 and MTM2011-27731-C03, and of Generalitat de Catalunya through project 2014SGR40. Last but not least, we would like to thank the associated editor and the referees for their comments and suggestions which helped improve a previous version of the manuscript. Finally, the usual disclaimer applies.
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Alonso-Meijide, J.M., Álvarez-Mozos, M. & Fiestras-Janeiro, M.G. The least square nucleolus is a normalized Banzhaf value. Optim Lett 9, 1393–1399 (2015). https://doi.org/10.1007/s11590-014-0840-9
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DOI: https://doi.org/10.1007/s11590-014-0840-9