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The Coleman–Shapley index: being decisive within the coalition of the interested

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Abstract

The Coleman power of a collectivity to act (CPCA) is a popular statistic that reflects the ability of a committee to pass a proposal. Applying the Shapley value to that measure, we derive a new power index—the Coleman–Shapley index (CSI)—indicating each voter’s contribution to the CPCA. The CSI is characterized by four axioms: anonymity, the null voter property, the transfer property, and a property stipulating that the sum of the voters’ power equals the CPCA. Similar to the Shapley–Shubik index (SSI) and the Penrose–Banzhaf index (PBI), our new index reflects the expectation of being a pivotal voter. Here, the coalitional formation model underlying the CPCA and the PBI is combined with the ordering approach underlying the SSI. In contrast to the SSI, voters are ordered not according to their agreement with a potential bill, but according to their vested interest in it. Among the most interested voters, power is then measured in a way similar to the PBI. Although we advocate the CSI over the PBI so as to capture a voter’s influence on whether a proposal passes, our index gives new meaning to the PBI. The CSI is a decomposer of the PBI, splitting the PBI into a voter’s power as such and the voter’s impact on the power of the other voters by threatening to block any proposal. We apply our index to the EU Council and the UN Security Council.

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Notes

  1. We do without the assumption \(N\in {\mathcal {W}}\) frequently made in the literature. This extends the domain of our axioms. However, our results would also hold true if the axioms were restricted to the smaller domain.

  2. Working on a fixed set \(N,\) we omit it in the following and, e.g., write \({\text {CPCA}}\left( {\mathcal {W}}\right)\) instead of \({\text {CPCA}}\left( N, {\mathcal {W}}\right)\).

  3. Mann and Shapley (1964), Felsenthal and Machover (1996), Kurz and Napel (2018) and Bernardi and Freixas (2018) generalize this definition/interpretation of the Shapley–Shubik index by considering more or less general probability distributions on roll call votes, which consist of a rank ordering of the voters and information on their voting behavior, yea or nay.

  4. Equation (3) is not only satisfied by the CSI. Yet, CSI is the only index that itself is decomposable. Indeed, decomposability (i.e., requiring that there exists an index that decomposes) and 2CPCA-Efficiency are characteristic of the CSI.

  5. Note that although 2Efficiency requires invariance of the merging voters’ power, it is silent about the other players’ power, such that the proportions of power might change.

  6. We use the EU population sizes on 01.01.2017 rounded up to the nearest ten (http://ec.europa.eu/eurostat). Detailed numbers can be found in online supplementary materials.

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Acknowledgements

We are grateful to the anonymous referees and the editors as well as to participants of several seminars, workshops, and conferences for helpful comments on our paper, particularly to Sergiu Hart and Annick Laruelle. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—388390901 and 288880950.

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Authors and Affiliations

  1. HHL Leipzig Graduate School of Management, Jahnallee 59, 04109, Leipzig, Germany

    André Casajus

  2. Dr. Hops Craft Beer Bar, Eichendorffstr. 7, 04277, Leipzig, Germany

    André Casajus

  3. ESMT European School of Management and Technology, Schlossplatz 1, 10178, Berlin, Germany

    Frank Huettner

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  1. André Casajus
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  2. Frank Huettner
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Correspondence to Frank Huettner.

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Casajus, A., Huettner, F. The Coleman–Shapley index: being decisive within the coalition of the interested. Public Choice 181, 275–289 (2019). https://doi.org/10.1007/s11127-019-00654-y

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