Assuming a 'spectrum' or ordering of the players of a coalitional game, as in a political spectrum in a parliamentary situation, we consider a variation of the Shapley value in which coalitions may only be formed if they are connected with respect to the spectrum. This results in a naturally asymmetric power index in which positioning along the spectrum is critical. We present both a characterization of this value by means of properties and combinatoric formulae for calculating it. In simple majority games, the greatest power accrues to 'moderate' players who are located neither at the extremes of the spectrum nor in its center. In supermajority games, power increasingly accrues towards the extremes, and in unanimity games all power is held by the players at the extreme of the spectrum.
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We wish to thank the referees and the editor for their comments and suggestions which substantially improved a previous version of the document. M. Álvarez-Mozos acknowledges the financial support of the Spanish Ministry of Economy and Competitiveness through Project MTM2011-27731-C03-02 . Z. Hellman acknowledges support by the European Research Council under the European Commissionʼs Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 249159 and Israel Science Foundation grants 538/11 and 212/09 . E. Winter is grateful to the German–Israeli Foundation for Research for its financial support.
- Coalitional games
- Political spectrum
- Restricted cooperation
- Shapley value