Abstract
This chapter investigates the robustness of square root rules for equal representation in two-tiered voting systems. When policy alternatives are non-binary and decisions are made by simple majority rule, Chap. 2 demonstrated that weight proportional to the square root of population size is approximately optimal, which may be interpreted as extending the scope of Penrose’s square root rule beyond the narrow limits of binary decision-making. However, in light of the normative character of this result, the simplifications used in the modeling of a complex real situation, such as, e.g., decision-making in the EU Council of Ministers, require special scrutiny.
Specifically, the aim of this chapter is to conduct a ‘sensitivity analysis’ regarding the square root rule, addressing the following questions How does a ‘simple’ voting rule that derives directly from constituency sizes perform compared to more sophisticated rules that use standard power indices as reference points? What is the fair voting rule under supermajority rules at the top tier? How does the fair voting rule react to heterogeneity across constituencies?
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Notes
- 1.
A subset of a partially ordered set (or poset) (P, < P ) – where P is a set, and < P is a partial order relation – is an antichain if any two elements of the subset are incomparable under < P . Applied to simple voting games, the power set 2N is partially ordered in respect to the inclusion ⊆ , and each set of minimum winning coalitions, characterizing a game, corresponds to an antichain.
- 2.
The minimum integer weight representations of these four games are (3; 2, 1, 1), (3; 1, 2, 1), (3; 1, 1, 2), and (2; 1, 1, 1). In Fig. 3.1, these games correspond to the four points in the interior of the simplex.
- 3.
In 15 of the configurations, population sizes were drawn from a uniform distribution, and in the other 15 from a Pareto distribution with κ = 1. 0.
- 4.
For special classes of weighted voting games, Lindner and Machover (2004) prove Penrose’s 1952 Limit Theorem with respect to the Penrose–Banzhaf index for q = 0. 5 and with respect to the Shapley–Shubik index for q ∈ (0, 1). Their conjecture that the Theorem holds ‘almost always’ under rather general conditions is corroborated in a simulation study by Chang et al. (2006).
- 5.
The Nelder–Mead algorithm does not rely on numerical or analytic gradients, which makes it particularly suitable to non-linear optimization problems like the present. In each step of the search, the probabilities \({\pi }_{j} \equiv \Pr \left (j = P\! :\! m\right )\) of representative j being pivotal in the top-tier committee are approximated by their empirical average over 10 million iterations. A MATLAB computer program is used for the computations. The source code is available upon request.
- 6.
Generally, α = 0. 5 is not exactly the best exponent among all power laws. Obviously, the best power law weights \({w}_{j} = {n}_{j}^{{\alpha }^{{_\ast}} }\) for a given configuration result in a lower deviation from egalitarian representation than simple square root weights, but they turn out to perform still worse than w β and w ϕ.
- 7.
The 12 configurations consist of 3 × 4 configurations with population sizes drawn from a uniform, a normal, and a Pareto (κ = 1. 0) distribution, respectively.
- 8.
One criticism advanced by Rae (1975) is that Pareto-optimality is compatible with a possibly outrageous distributional situation which is locked in under unanimity rule.
- 9.
On the other hand, supermajorities could serve to mitigate time inconsistencies if they make it more difficult to revise a policy.
- 10.
More precisely, the ‘social consensus’ which Caplin and Nalebuff (1988) presuppose amounts to the restriction that the density of voters’ ideal points is a logarithmically concave function, e.g., a uniform density over a convex set. Then, points exist which cannot be defeated by any other alternative under a majority requirement larger than or equal to \(1 - 1/e \approx 0.64\).
- 11.
The claim that supermajorities are minority-protecting rests on the assumption, uncovered by McGann (2004), that the status quo is more benign for the minority than government action to change it.
- 12.
The tests for 30-constituency unions reported at the end of Sect. 3.1 concern precisely the configurations which are used in the present simulations on quota variation.
- 13.
It is not possible to compare the columns in Table 3.3 with respect to the variance of constituency sizes because the variance of P(κ, \underline{x}) is infinity for κ ≤ 2.
- 14.
A ‘common belief’ is also represented by Straffin’s (1977) homogeneity assumption under which the probability of a voter ‘affecting the outcome’ coincides with the Shapley–Shubik index.
- 15.
Concerning the probabilistic interpretation of power measures (cf. Sect. 1.2.1), Braham and Steffen (2002) argue that the whole range of partial homogeneity assumptions is no less a priori than its two borderline cases, i.e., the Banzhaf index and the Shapley–Shubik index.
- 16.
If the distributions of individual ideal points are symmetric, \(\tilde{{\Delta }}_{j}\) could also refer to the mean of the distribution f j .
- 17.
Eurostat population numbers for EU27 countries as of 01/01/2007 are used as simulation input.
- 18.
The assumption that preferences are more heterogeneous in large populations is also made in Alesina and Spolaore (2003), and the trade-off between the costs of differences and the economies of scope in large jurisdictions determines nation size in their framework.
- 19.
The difference between these two quantities is very similar to the mean majority deficit which is also minimized under square root weights (see Felsenthal and Machover, 1998, pp. 72ff).
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Maaser, N.F. (2010). Robust Equal Representation. In: Decision-Making in Committees. Lecture Notes in Economics and Mathematical Systems, vol 635. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04153-2_3
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