Skip to main content
Top
Published in: Social Choice and Welfare 4/2023

08-07-2023 | Original Paper

Welfare ordering of voting weight allocations

Author: Kazuya Kikuchi

Published in: Social Choice and Welfare | Issue 4/2023

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This paper studies the allocation of voting weights in a committee representing groups of different sizes. We introduce a partial ordering of weight allocations based on stochastic comparison of social welfare. We show that when the number of groups is sufficiently large, this ordering asymptotically coincides with the total ordering induced by the cosine proportionality between the weights and the group sizes. A corollary is that a class of expectation-form objective functions, including expected welfare, the mean majority deficit and the probability of inversions, are asymptotically monotone in the cosine proportionality.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Footnotes
1
Koppel and Diskin (2009) provide an axiomatic analysis of cosine proportionality.
 
2
Sect. 5.1 discusses the alternative definition of social welfare as the sum of utilities when preference intensities may vary across individuals.
 
3
We say that a distribution F stochastically dominates another distribution G if \(F(x)\le G(x)\) for all x.
 
4
See, e.g., [Billingsley (2008), Theorem 27.2].
 
5
See Azzalini and Valle (1996) for basic properties of the skew normal distribution.
 
6
A sequence \(\{X^n\}\) of random variables is called uniformly integrable if for any \(\epsilon >0\), there exists \(x>0\) such that \(\textbf{E} (|X^n|\textbf{1}_{\{|X^n|>x\}}) <\epsilon \) for all n. See e.g. Williams (1991).
 
7
Hinich et al. (1975) derives a similar limit formula of p(a), assuming equally sized groups.
 
8
Penrose (1946) argued that the voting powers of groups should be proportional to the square roots of populations, but left ambiguous what allocation of voting weights would achieve this. In the limit as the number of groups goes to infinity, a result shown by Lindner and Machover (2004) implies that the weights proportional to the square roots of populations achieve the proposed allocation of voting power.
 
9
This is in contrast with the winner-take-all rule under which the square root weights are optimal. An intuitive reason for the difference between the rules is that social welfare is maximized by direct majority voting, which in our model is equivalent to the combination of the proportional representation rule and the proportional weights; in contrast, under the winner-take-all rule, no weight allocation is equivalent to direct voting, and the square-root weights are the best possible.
 
Literature
go back to reference Azzalini A, Valle AD (1996) The multivariate skew-normal distribution. Biometrika 83(4):715–726CrossRef Azzalini A, Valle AD (1996) The multivariate skew-normal distribution. Biometrika 83(4):715–726CrossRef
go back to reference Balinski ML, Young HP (2010) Fair representation: meeting the ideal of one man, one vote. Brookings Institution Press Balinski ML, Young HP (2010) Fair representation: meeting the ideal of one man, one vote. Brookings Institution Press
go back to reference Barberà S, Jackson MO (2006) On the weights of nations: assigning voting weights in a heterogeneous union. J Polit Econ 114:317–339CrossRef Barberà S, Jackson MO (2006) On the weights of nations: assigning voting weights in a heterogeneous union. J Polit Econ 114:317–339CrossRef
go back to reference Beisbart C, Bovens L (2007) Welferist evaluations of decision rules for boards of representatives. Soc Choice Welf 29:581–608CrossRef Beisbart C, Bovens L (2007) Welferist evaluations of decision rules for boards of representatives. Soc Choice Welf 29:581–608CrossRef
go back to reference Beisbart C, Bovens L (2008) A power analysis of amendment 36 in Colorado. Public Choice 134:231–246CrossRef Beisbart C, Bovens L (2008) A power analysis of amendment 36 in Colorado. Public Choice 134:231–246CrossRef
go back to reference Billingsley P (2008) Probability and measure. John Wiley & Sons Billingsley P (2008) Probability and measure. John Wiley & Sons
go back to reference De Mouzon O, Laurent T, Le Breton M, Lepelley D (2020) The theoretical Shapley-Shubik probability of an election inversion in a toy symmetric version of the US presidential electoral system. Soc Choice Welfare 54(2):363–395CrossRef De Mouzon O, Laurent T, Le Breton M, Lepelley D (2020) The theoretical Shapley-Shubik probability of an election inversion in a toy symmetric version of the US presidential electoral system. Soc Choice Welfare 54(2):363–395CrossRef
go back to reference Feix MR, Lepelley D, Merlin VR, Rouet J-L (2004) The probability of conflicts in a US presidential type election. Econ Theor 23(2):227–257CrossRef Feix MR, Lepelley D, Merlin VR, Rouet J-L (2004) The probability of conflicts in a US presidential type election. Econ Theor 23(2):227–257CrossRef
go back to reference Felsenthal DS, Machover M (1999) Minimizing the mean majority deficit: the second square-root rule. Math Soc Sci 37(1):25–37CrossRef Felsenthal DS, Machover M (1999) Minimizing the mean majority deficit: the second square-root rule. Math Soc Sci 37(1):25–37CrossRef
go back to reference Hinich MJ, Mickelsen R, Ordeshook PC (1975) The electoral college vs. a direct vote: policy bias, reversals, and indeterminate outcomes. J Math Sociol 4(1):3–35CrossRef Hinich MJ, Mickelsen R, Ordeshook PC (1975) The electoral college vs. a direct vote: policy bias, reversals, and indeterminate outcomes. J Math Sociol 4(1):3–35CrossRef
go back to reference Kaniovski S, Zaigraev A (2018) The probability of majority inversion in a two-stage voting system with three states. Theor Decis 84(4):525–546CrossRef Kaniovski S, Zaigraev A (2018) The probability of majority inversion in a two-stage voting system with three states. Theor Decis 84(4):525–546CrossRef
go back to reference Koppel M, Diskin A (2009) Measuring disproportionality, volatility and malapportionment: axiomatization and solutions. Soc Choice Welf 33(2):281CrossRef Koppel M, Diskin A (2009) Measuring disproportionality, volatility and malapportionment: axiomatization and solutions. Soc Choice Welf 33(2):281CrossRef
go back to reference Koriyama Y, Laslier J-F, Macé A, Treibich R (2013) Optimal apportionment. J Polit Econ 121:584–608CrossRef Koriyama Y, Laslier J-F, Macé A, Treibich R (2013) Optimal apportionment. J Polit Econ 121:584–608CrossRef
go back to reference Kurz S, Maaser N, Napel S (2017) On the democratic weights of nations. J Polit Econ 125:1599–1634CrossRef Kurz S, Maaser N, Napel S (2017) On the democratic weights of nations. J Polit Econ 125:1599–1634CrossRef
go back to reference Lindner I, Machover M (2004) L.S. Penrose’s limit theorem: proof of some special cases. Math Soc Sci 47(1):37–49CrossRef Lindner I, Machover M (2004) L.S. Penrose’s limit theorem: proof of some special cases. Math Soc Sci 47(1):37–49CrossRef
go back to reference Maaser N, Napel S (2012) A note on the direct democracy deficit in two-tier voting. Math Soc Sci 63(2):174–180CrossRef Maaser N, Napel S (2012) A note on the direct democracy deficit in two-tier voting. Math Soc Sci 63(2):174–180CrossRef
go back to reference Maaser N, Napel S (2014) The mean voter, the median voter, and welfare-maximizing voting weights. Voting Power and Procedures. Springer, pp 159–176CrossRef Maaser N, Napel S (2014) The mean voter, the median voter, and welfare-maximizing voting weights. Voting Power and Procedures. Springer, pp 159–176CrossRef
go back to reference May K (1948) Probabilities of certain election results. Amer Math Monthly 55:203–209CrossRef May K (1948) Probabilities of certain election results. Amer Math Monthly 55:203–209CrossRef
go back to reference Owen DB (1980) A table of normal integrals: a table. Commun Stat Simul Comput 9(4):389–419CrossRef Owen DB (1980) A table of normal integrals: a table. Commun Stat Simul Comput 9(4):389–419CrossRef
go back to reference Penrose LS (1946) The elementary statistics of majority voting. J Roy Stat Soc 109:53–57CrossRef Penrose LS (1946) The elementary statistics of majority voting. J Roy Stat Soc 109:53–57CrossRef
go back to reference Wada J (2010) Evaluating the unfairness of representation with the Nash social welfare function. J Theor Polit 22(4):445–467CrossRef Wada J (2010) Evaluating the unfairness of representation with the Nash social welfare function. J Theor Polit 22(4):445–467CrossRef
Metadata
Title
Welfare ordering of voting weight allocations
Author
Kazuya Kikuchi
Publication date
08-07-2023
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 4/2023
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-023-01474-2

Other articles of this Issue 4/2023

Social Choice and Welfare 4/2023 Go to the issue

Premium Partner