Equal Representation in Two-tier Voting Systems Nicola Maaser and - - PowerPoint PPT Presentation

Equal Representation in Two-tier Voting Systems

Nicola Maaser and Stefan Napel

Economics Department University of Hamburg, Germany maaser@econ.uni-hamburg.de

1

Introduction

• History and efficiency considerations often call for two-tier

electoral systems:

• 1. People’s preferences are aggregated in constituencies
• 2. Constituencies’ preferences are aggregated in an electoral college
• Question:

How should constituencies’ voting weights in the college be chosen s.t. a priori all individuals have identical influence?

• Allocating weights proportional to population sizes seems

straightforward, but:

• In general, voting power is not linear in voting weight, e.g. EU

Council of Ministers 1958.

• Power measures as the Penrose-Banzhaf- or the Shapley-

Shubik-Index are designed to capture the non-trivial relationship between weight and power.

2

Penrose‘s square root rule

• Penrose’s square root rule (1946):

Choose weights s.t. constituencies’ Penrose-Banzhaf index is proportional to square root of population

• For most practical reasons (especially, if the number of

constituencies is “large”), a simpler rule suffices: weight = sqrt(Population)

• The rule requires decisions x∈{0,1} and (in expectation)

equi-probable independent 0 or 1-votes

• What if the world is not dichotomous but, e.g., x∈[0,1]?

3

Outline

• Model
• Analytical problems
• Monte Carlo simulation
• Results
• Concluding remarks

4

Model

Constituency level Union level

5

Model

Constituency level Union level

6

Model

Constituency level Union level

7

Model

• Voters are partitioned into m constituencies and have single-

peaked preferences with a priori uniformly distributed ideal points λ ∈ X ≡ [0,1]

• Constituency j’s representative is chosen to match the median

voter in his constituency

• Each constituency j has weight wj in the electoral college;

a 50%-quota q is used

• Pivotal constituency (P) is defined by P ≡ min{r : ∑k=1

r w(k) >q}

[permutation (·) orders constituencies from left to right]

• (P) gets its will, i.e. x*=λ(P)
• Problem of equal representation:

Given population sizes n1, …, nm, find weights w1, …,wn s.t. each voter has equal chance of determining x*

8

First analysis

• Each voter in constituency j has chance 1/nj to be its median

⇒ Pr(λj =λP:m) = c·nj for all j (with c>0)

• Assuming i.i.d. voters, different nj imply different a priori

distributions of medians

• With density f and c.d.f. F for individual voters’ ideal points,

representatives’ ideal points are asymptotically normal with µj =F-1(0.5), σj =[2f(µj)·sqrt(nj)]-1 ⇒ Larger constituencies are a priori more central in the electoral college and more likely to be pivotal under a 50%-quota.

!

9

Analytical problems

• Already for unweighted voting, i.e. P ≡ (m+1)/2, we run into

trouble:

• Asymptotic approximation with only n1 varying and n2=…=nm

seems possible, but for general n1, …, nm?

10

Monte Carlo simulation

• Probability πj := Pr(j = (P)) is the expected value of random

variable Hj (λ1, …, λm) which is 1 if j = (P) and 0 otherwise

• Hj’s expected value can be approximated by the empirical

average of many independent draws of Hj

• Weight vectors are constructed from given population sizes by

wj =nj

α

• For fixed weights (w1, …, wm) and populations (n1, …, nm), we

draw λ1, …, λm from the beta distributions corresponding to i.i.d. U[0,1] voters in all constituencies and average H1, …, Hm over 10 million draws

• We search for the α which yields smallest cumulative (individual)

quadratic deviation of πj from the ideal egalitarian probability π

j*=nj / ∑nk (j =1, …, m)

11

EU Council of Ministers

• Using EU25 population data, α*=0.5 with 50%-quota would

give almost equal representation:

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 population in millions ideal probabilities probabilities for = 0.3 probabilities for = 0.5 probabilities for = 0.7 0.225 0.200 0.175 0.150 0.125 0.100 0.075 0.050 0.025 0.000

á á á

12

US Electoral College

• Again, α*=0.5 comes very close to equal representation:

5 10 15 20 25 30 35 population in millions

ideal probabilities probabilities for = 0.3 probabilities for = 0.4 probabilities for = 0.5 probabilities for = 0.6 probabilities for = 0.7 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

á á á á á

13

US Electoral College

• Cumulative individual quadratic deviation from equal

representation in the US Electoral College:

0,00E+00 2,00E-10 4,00E-10 6,00E-10 8,00E-10 1,00E-09 1,20E-09 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

á

• cum. ind. quad. dev.

1.2 E-09 1.0 E-09 8.0 E-10 6.0 E-10 4.0 E-10 2.0 E-10 0.0 E+ 00

14

Concluding remarks

• While analytical proof of this looks out of reach, assigning

weights proportional to square root of population provides a quite stable and satisfying answer to our question

• Thus, Penrose’s square root rule is much more robust than

suggested in the literature; unexpectedly, it extends from binary decisions to rich (one- dimensional convex) policy spaces, from simple games to spatial voting

• Future research:

– A better reference point than voting weight – Effects of supermajority rule

15

EU Council of Ministers

Germany France U.K. Italy Spain Poland Netherlands Greece Belgium Portugal Czech Rep. Hungary Sweden Austria Denmark Slovakia Finland Ireland Lithuania Latvia Slovenia Estonia Cyprus Luxembourg Malta estimated probabilities (quota 50.00%) ideal probabilities 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.020 0.015 0.001 0.005 0.000

• 0.005
• 0.010
• 0.015
• 0.020
• 0.025
• 0.030
• Nice weights and quota of 50%:

16

EU Council of Ministers

Germany France U.K. Italy Spain Poland Netherlands Greece Belgium Portugal Czech Rep. Hungary Sweden Austria Denmark Slovakia Finland Ireland Lithuania Latvia Slovenia Estonia Cyprus Luxembourg Malta estimated probabilities (quota 72.27%) ideal probabilities 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.02 0.01 0.00

• 0.01
• 0.02
• 0.03
• 0.04
• 0.05
• 0.06
• Nice weights and quota of 72.2%:

17

Results: uniformly distributed nj

• We look, first, at m ranging from 10 to 50 with randomly

generated constituency sizes n1, …, nm and, second, at two prominent real-world population configurations

• With i.i.d. nj from U[0.5·106, 99.5·106], optimal α is:

18

Results: normally distributed nj

• If constituencies are created for efficiency reasons, sizes

possibly are distributed around some ‘ideal size’

• With i.i.d. nj from N(106; 200 000), optimal α is:

19

Results: normally distributed nj

0,00E+00 5,00E-11 1,00E-10 1,50E-10 2,00E-10 2,50E-10 3,00E-10 3,50E-10 #const = 30 #const = 50 #const = 100

á

• cum. ind. quad. dev.

0.0 0.2 0.4 0.6 0.8 1.0 3.5 E-10 3.0 E-10 2.5 E-10 2.0 E-10 1.5 E-10 1.0 E-10 5.0 E-11 0.0 E+ 00

• For moderately many similar constituencies, weighted voting

may allow only quite high (and flat) inequality of representation:

20

Results: Pareto distributed nj

• More realistically, with i.i.d. nj from a Pareto distribution with

skewness parameter k, optimal α is: → General finding: As long as m ≥15, α*=0.5 comes close to equal representation; it does best amongst all considered rules for large m

(and for small m if the electorate’s partition is not too equal nor oceanic)