## Abstract

Representatives from differently sized constituencies take political decisions by a weighted voting rule and adopt the ideal point of the weighted median amongst them. Preferences of each representative are supposed to coincide with the constituency’s median voter. The paper investigates how each constituency’s population size should be mapped to a voting weight for its delegate when the objective is to maximize the total expected utility generated by the collective decisions. Depending on the considered utility functions, this is equivalent to approximating the sample mean or median voter of the population by a weighted median of sub-sample medians. Monte Carlo simulations indicate that utilitarian welfare is maximized by a *square root rule* if the ideal points of voters are all independent and identically distributed. However, if citizens are risk-neutral and their preferences are sufficiently positively correlated within constituencies, i.e., if heterogeneity between constituencies dominates heterogeneity within, then a *linear rule* performs better.

We are grateful for constructive discussion at the Leverhulme Trust’s *Voting Power in Practice Symposium* 2011.

## Access this chapter

Tax calculation will be finalised at checkout

Purchases are for personal use only

### Similar content being viewed by others

## Notes

- 1.
See, e.g., Cho and Duggan (2009).

- 2.
Historically, most attention has been devoted to giving each citizen an equally effective voice in elections (cf.

*Reynolds v. Sims*, 377 U.S. 533, 1964). In two-tier voting systems, this calls for an a priori equal chance of each voter to indirectly determine the policy outcome. For binary policy spaces, Penrose (1946) has shown that individual powers are approximately equalized if voting weights of the representatives are chosen such that their Penrose–Banzhaf voting powers (Penrose 1946; Banzhaf 1965) are proportional to the square root of the corresponding population sizes. An extension to convex policy spaces is provided by Maaser and Napel (2007) and Kurz et al. (2014). - 3.
Dubey and Shapley (1979) provide a generalization of this result to the domain of all simple games.

- 4.
Felsenthal and Machover refer to this allocation rule as the

*second square root rule*in order to distinguish it from Penrose’s (1946) (first) square root rule, which requires representatives’*voting powers*—rather than their weights—to be proportional to the square roots of their constituencies’ population sizes. - 5.
This situation is known in the social choice literature as a

*referendum paradox*(see, e.g., Nurmi 1998). - 6.
Also see Felsenthal and Machover (1998, pp. 70ff).

- 7.
To be precise, Penrose’s square root rule is nested only asymptotically, namely when \(\mathfrak{C}\) involves a great number

*r*of constituencies with a regular size distribution. See Lindner and Machover (2004) and Chang et al. (2006) on the vanishing difference between voting weights and voting powers as*r*→*∞*. - 8.
Note that even though total utility from the decisions which result from the considered two-tier process typically falls short of the global maximum achieved under a direct democracy, representative democracy has a number of advantages. These presumably also generate utility for citizens which is not considered in our model.

- 9.
See Beisbart and Bovens (2013) for a related investigation in a binary voting model. They ask the worst-case question: which

*number*of equipopulous districts*maximizes*the mean majority deficit? - 10.
The problem of finding the optimal value of

*α*bears some resemblance to choosing an appropriate power-law transformation in order to improve the symmetry of a skewed empirical distribution (see, e.g., Yeo and Johnson 2000). - 11.
Since the considered number of voters in each constituency \(\mathcal{C}_{j}\) is large (\(n_{j} \gg 50\)), the respective population and constituency medians will approximately have normal distributions irrespective of the specific

*F*which one considers. For the sake of completeness, let it still be mentioned that individual ideal points were drawn from a standard uniform distribution**U**(0, 1) in our simulations. The MATLAB source code is available upon e-mail request. - 12.
In particular, variation in population sizes

*n*_{ j }∼**N**(2000, 200) is rather small. This results in an objective function that is essentially flat for a large range of values of*α*. - 13.
Specifically, we draw

*μ*_{ j }from a uniform distribution**U**(−*a*,*a*) with variance*σ*_{ ext }^{2}, and then obtain \(\nu ^{i} =\mu _{j}+\varepsilon\) with \(\varepsilon \sim \mathbf{U}(0, 1)\). - 14.
We have used 2010 population data measured in 1,000 individuals for computational reasons. This corresponds with the “block model” in Barberà and Jackson (2006), which supposes that a constituency can be subdivided into equally sized “blocks” whose members have perfectly correlated preferences within blocks, but are independent across blocks.

## References

Arnold, B. C., Balakrishnan, N., & Nagaraja, H. N. (1992).

*A first course in order statistics*. New York: Wiley.Banks, J. S., & Duggan, J. (2000). A bargaining model of collective choice.

*American Political Science Review*,*94*(1), 73–88.Banzhaf, J. F. (1965). Weighted voting doesn’t work: A mathematical analysis.

*Rutgers Law Review*,*19*(2), 317–343.Barberà, S., & Jackson, M. O. (2006). On the weights of nations: Assigning voting weights in a heterogeneous union.

*Journal of Political Economy*,*114*(2), 317–339.Beisbart, C., & Bovens, L. (2007). Welfarist evaluations of decision rules for boards of representatives.

*Social Choice and Welfare*,*29*(4), 581–608.Beisbart, C., & Bovens, L. (2013). Minimizing the threat of a positive majority deficit in two-tier voting systems with equipopulous units.

*Public Choice*,*145*(1–2), 75–94. doi:10.1007/s11127-011-9810-2.Beisbart, C., Bovens, L., & Hartmann, S. (2005). A utilitarian assessment of alternative decision rules in the Council of Ministers.

*European Union Politics*,*6*(4), 395–419.Chang, P.-L., Chua, V. C., & Machover, M. (2006). L S Penrose’s limit theorem: Tests by simulation.

*Mathematical Social Sciences*,*51*(1), 90–106.Cho, S., & Duggan, J. (2009). Bargaining foundations of the median voter theorem.

*Journal of Economic Theory*,*144*(2), 851–868.Cramér, H. (1946).

*Mathematical methods of statistics*. Princeton landmarks in mathematics. Princeton: Princeton University Press [Reprinted 1999].Dubey, P., & Shapley, L. (1979). Mathematical properties of the Banzhaf power index.

*Mathematics of Operations Research*,*4*(2), 99–131.Feix, M. R., Lepelley, D., Merlin, V., Rouet, J.-L., & Vidu, L. (2008).

*Majority efficient representation of the citizens in a federal union*. Mimeo: Université de la Réunion, Université de Caen, and Université d’Orléans.Felsenthal, D., & Machover, M. (1998).

*The measurement of voting power – theory and practice, problems and paradoxes*. Cheltenham: Edward Elgar.Felsenthal, D., & Machover, M. (1999). Minimizing the mean majority deficit: The second square-root rule.

*Mathematical Social Sciences*,*37*(1), 25–37.Gelman, A., Katz, J. N., & Tuerlinckx, F. (2002). The mathematics and statistics of voting power.

*Statistical Science*,*17*(4), 420–435.Gerber, E. R., & Lewis, J. B. (2004). Beyond the median: Voter preferences, district heterogeneity, and political representation.

*Journal of Political Economy*,*112*(6), 1364–1383.Harsanyi, J. C. (1955). Cardinal welfare, individual ethics, and interpersonal comparison of utility.

*Journal of Political Economy*,*63*(4), 309–321.Kaniovski, S. (2008). The exact bias of the Banzhaf measure of power when votes are neither equiprobable nor independent.

*Social Choice and Welfare*,*31*(2), 281–300.Kirsch, W. (2007). On Penrose’s square-root law and beyond.

*Homo Oeconomicus*,*24*(3/4), 357–380.Koriyama Y., Laslier, J.-F., Macé, A. & Treibich R. (2013).

*Optimal Apportionment. Journal of Political Economy*,*121*(3), 584–608.Kurz, S., Maaser, N., & Napel, S. (2014).

*On the democratic weights of nations*. Mimeo: University of Bayreuth.Leech, D., & Leech, R. (2009). Reforming IMF and World Bank governance: In search of simplicity, transparency and democratic legitimacy in the voting rules. Warwick Economic Research Papers 914.

Lindner, I., & Machover, M. (2004). L. S. Penrose’s limit theorem: Proof of some special cases.

*Mathematical Social Sciences*,*47*(1), 37–49.Maaser, N., & Napel, S. (2007). Equal representation in two-tier voting systems.

*Social Choice and Welfare*,*28*(3), 401–420.Maaser, N., & Napel, S. (2012). A note on the direct democracy deficit in two-tier voting.

*Mathematical Social Sciences*,*62*(2), 174–180.Nurmi, H. (1998).

*Rational behaviour and design of institutions*. Cheltenham: Edward Elgar.Penrose, L. S. (1946). The elementary statistics of majority voting.

*Journal of the Royal Statistical Society*,*109*(1), 53–57.Rae, D. W. (1969). Decision rules and individual values in constitutional choice.

*American Political Science Review*,*63*(1), 40–56.Schwertman, N. C., Gilks, A. J., & Cameron, J. (1990). A simple noncalculus proof that the median minimizes the sum of the absolute deviations.

*American Statistician*,*44*(1), 38–39.Yeo, I.-K., & Johnson, R. A. (2000). A new family of power transformations to improve normality or symmetry.

*Biometrika*,*87*(4), 954–959.

## Author information

### Authors and Affiliations

### Corresponding author

## Editor information

### Editors and Affiliations

## Rights and permissions

## Copyright information

© 2014 Springer International Publishing Switzerland

## About this chapter

### Cite this chapter

Maaser, N., Napel, S. (2014). The Mean Voter, the Median Voter, and Welfare-Maximizing Voting Weights. In: Fara, R., Leech, D., Salles, M. (eds) Voting Power and Procedures. Studies in Choice and Welfare. Springer, Cham. https://doi.org/10.1007/978-3-319-05158-1_10

### Download citation

DOI: https://doi.org/10.1007/978-3-319-05158-1_10

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-319-05157-4

Online ISBN: 978-3-319-05158-1

eBook Packages: Business and EconomicsEconomics and Finance (R0)