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Social Choice and Welfare

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21.09.2017 | Original Paper

The price of ‘one person, one vote’

verfasst von: Yaron Azrieli

Erschienen in: Social Choice and Welfare | Ausgabe 2/2018

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Abstract

I study the design of binary voting rules in environments where agents are heterogenous either in the stakes that they have in the decision or in the quality of information they possess regarding the correct course of action. The price of ‘one person, one vote’ is defined as the reduction in welfare resulting from constraining the voting rule to treat the agents symmetrically. I analyze how this price depends on the parameters of the environment, particularly on the level of heterogeneity in the society. The results shed light on the tradeoff between equality and efficiency in the context of constitutional design.

Vorheriger Artikel The supercovering relation, the pairwise winner, and more missing links between Borda and Condorcet

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According to Cicero, wealthier and older citizens had greater voting power to make the decisions of the assembly more in line with the will of those who are more experienced and had more to lose. See Abbott (1963, pp. 21, 75) for a description of the structure of the Centuriate Assembly before and after the reform.

While I term the constraint on the voting rule ‘one person, one vote’, the voters in the model can also be representatives of groups.

This document is available at http://www.fda.gov/downloads/RegulatoryInformation/Guidances/UCM125641.pdf.

Thus, in the private values model I assume that it is possible to measure intensity of preferences and make comparisons of this measure across agents. One possible interpretation of preference intensity in this context is in terms of willingness to pay: agents have quasi-linear preferences over pairs of an alternative and money, and the intensity of preferences is the maximal willingness to pay for their favorite alternative to be chosen. This is the view taken in the vast literature on costly voting, see Sect. 4 for references. In the context of international organizations, as described above, the intensity may reflect the size of a state or some measure of its relative importance in the world economy.

See Atkinson (1970) and Dasgupta et al. (1973) for a more modern treatment of inequality measures in general and the Pigou–Dalton principle in particular. This principle is at the core of inequality measurement: while many measures of inequality that completely rank distributions have been suggested in the literature, as far as I know they are all consistent with the Pigou–Dalton partial order.

Similar characterizations of welfare-maximizing voting weights have been obtained by Fleurbaey (2008) and Barberà and Jackson (2006).

To simplify notation I write a[n] and b[n] instead of a([n]) and b([n]), respectively.

The optimal welfare \(W^{*}\) is not everywhere differentiable and in the formal proof I use a different argument based on convexity. Nevertheless, the above argument provides the intuition behind the result.

The fact that \(\rho \) increases under these operations is an immediate consequence of Proposition 1 and the homogeneity of degree 0 of \(\rho \) as a function of a. However, since \(\tau \) is not homogeneous of degree 0 we cannot use Proposition 1 to conclude that it increases, and a different argument is needed to establish the result.

This type of worst-case analysis, which is most common in the Computer Science literature, has been used in several papers to measure the welfare loss due to constraints imposed on mechanisms, see for example Schmitz and Tröger (2006) in the context of voting, Neeman (2003) in auction design, and Pycia (2014) in matching markets. Note that for a worst-case analysis it is more informative to use the relative loss \(\rho \), since the absolute loss \(\tau \) is unbounded.

Note that the case \(0<\alpha _i<0.5\) is also covered by our analysis (by switching the labels of the signals h and l).

The fact that the optimal voting rule is a weighted majority in which agent i’s weight is the log likelihood ratio associated with his signal is well-known. For early references see Nitzan and Parush (1982) and Shapley and Grofman (1984).

One way to see this formally is to fix some \(k<n/2\) and consider a bijection \(\varphi \) from the collection of coalitions of size k to the collection of coalitions of size \(n-k\), such that \(S\subseteq \varphi (S)\) for all S (such bijections can be explicitly constructed). Since \(\alpha _i>0.5\) for all i, it immediately follows that \(\alpha (S){\bar{\alpha }}(S^c) < \alpha (\varphi (S)) {\bar{\alpha }}(\varphi (S)^c)\). Summing up over all coalitions S of size k gives \(\sum _{\{S : |S|=k\}} \alpha (S){\bar{\alpha }}(S^c) < \sum _{\{S : |S|=k\}} {\bar{\alpha }}(S)\alpha (S^c)\), and hence that \(f(k)={\bar{L}}\) is optimal. For \(k>n/2\) a similar argument applies.

For this assertion to be true it is important that the expectation of \({\tilde{a}}_i\) is independent of the coalition S of agents who prefer alternative A. Correlation between the \({\tilde{a}}_i\)’s is allowed.

This can be interpreted as another reason for the use of anonymous mechanisms: if the voting rule cannot be changed frequently (i.e., the same rule should be used for many decisions), and if no agent systematically has higher stakes than others, then anonymous rules are optimal. I thank an anonymous referee for suggesting this argument.

See also the related result in Schmitz and Tröger (2006, Appendix B).

I thank an anonymous referee for suggesting this extension.

With different \(p_i\)’s the optimal anonymous rule is still qualified majority, but it is no longer necessarily true that when \(a=b\) simple majority is optimal. See Azrieli and Kim (2014, Theorem 4) for details.

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Titel: The price of ‘one person, one vote’
verfasst von: Yaron Azrieli
Publikationsdatum: 21.09.2017
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 2/2018
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-017-1087-z

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