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Soc Choice Welf (2010) 34:3346 DOI 10.1007/s00355-009-0387-3
ORIGINAL PAPER
Power indices and minimal winning coalitions
Werner Kirsch Jessica Langner
Received: 30 July 2008 / Accepted: 17 March 2009 / Published online: 1 April 2009 Springer-Verlag 2009
Abstract The PenroseBanzhaf index and the ShapleyShubik index are the best-known and the most used tools to measure political power of voters in simple voting games. Most methods to calculate these power indices are based on counting winning coalitions, in particular those coalitions a voter is decisive for. We present a new combinatorial formula how to calculate both indices solely using the set of minimal winning coalitions.
1 Introduction
The theory of power indices is a systematic approach to measure political power in voting systems (cp. Taylor 1995; Felsenthal and Machover 1998a). Voting systems are also known as simple (voting) games in literature. The well-known PenroseBanzhaf index (Penrose 1946; Banzhaf 1965) and ShapleyShubik index (Shapley and Shubik 1954) rely on the concept of decisiveness of voters. On the other hand, the Deegan Packel index (Deegan and Packel 1978) and the HollerPackel index (Holler and Packel 1983) are based explicitly on the set of minimal winning coalitions (MWCs). MWCs are those coalitions each voter is decisive for. Particularly, a calculation of power indices is easy to handle in weighted voting systems. Here, voting weights are
W. KirschFakultt fr Mathematik und Informatik, FernUniversitt Hagen, 58095 Hagen, Germanye-mail: [email protected]
J. Langner (B)
Fakultt fr Mathematik, Ruhr-Universitt Bochum, 44780 Bochum, Germanye-mail: [email protected]
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assigned to each voter and a decision threshold is dened. A proposal is accepted if the sum of the voting weights in favor meets or exceeds the given threshold.
Usually, calculation methods are based on listing the set of winning coalitions. In this paper we develop a combinatorial approach how to determine power indices solely using the MWC-set. For illustration we use the examples of the PenroseBanzhaf index and the ShapleyShubik index. It is known that each voting system (whether it is weighted or not) has got a MWC-set and it is completely dened by it. More precisely, each set of voting rules can be quantied by a MWC-set. Thus, our approach makes it possible to calculate power indices for each potential MWC-set in a...