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Simple vs. Sophisticated Rules for the Allocation of Voting Weights

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Abstract

Representatives from differently sized constituencies form an assembly which takes political decisions by a weighted voting rule and adopts the ideal point of the weighted median amongst them. Preferences of each representative are supposed to coincide with the constituency’s median voter. Analytic results by Kurz et al. (J Polit Econ, 2017) for infinite chains of assemblies suggest that individual voters’ a priori influence on the collective decision can be equalized by allocating voting weight proportional to the square root of constituency sizes. This paper investigates numerically the performance of this simple square root rule and sophisticated variations, based on the Shapley value or the Penrose–Banzhaf power measure, when the number of constituencies is still “small”. Monte Carlo simulations indicate that power index-based rules are superior to simple rules.

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Notes

  1. The treaty defined voting weights and a quota. The Nice rules could be invoked in the EU until March 2017, when they were eventually replaced by the ‘double majority’ voting system agreed in the Treaty of Lisbon.

  2. Under the assumption that top-tier voters are stochastically identical the model would lead to a linear rule based on the Shapley–Shubik index. Note, however, that this assumption would be generally inconsistent with bottom-tier voters being stochastically identical.

  3. See Banks and Duggan (2000) for sophisticated non-cooperative support of policy outcomes inside or close to the core.

  4. Kurz et al.’s model also allows preferences within constituencies to be positively correlated, which leads them to different conclusions regarding the optimal rule for equal representation. Here, we consider their model without such correlations.

  5. For instance, there are only 117 structurally different weighted voting games with \(m=5\) constituencies even if all majority thresholds between 0 and 100% are permitted. This number (related to Dedekind’s problem in discrete mathematics) grows very fast, but obviously the set of distinct feasible influence distributions remains finite.

  6. A MATLAB computer program is used for all computations. The source code is available upon request.

  7. Generally, \(\alpha = 0.5\) is not exactly the best exponent among all power laws. Obviously, the best power law weights \(w_j = n_j^{\alpha ^{*}}\) for a given configuration result in a lower deviation from egalitarian representation than simple square root weights, but they turn out to perform still significantly worse than \(w_\beta\) and \(w_\phi\).

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Authors and Affiliations

  1. Economic Department, SOCIUM, University of Bremen, Bremen, Germany

    N. Maaser

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  1. N. Maaser
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Correspondence to N. Maaser.

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The paper has benefitted from critical comments and suggestions from three anonymous referees, and participants at the 5th World Congress of the Game Theory Society in Maastricht.

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Maaser, N. Simple vs. Sophisticated Rules for the Allocation of Voting Weights. Homo Oecon 34, 67–78 (2017). https://doi.org/10.1007/s41412-017-0036-5

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