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

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. 1

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]? 2

Outline • Model • Analytical problems • Monte Carlo simulation • Results • Concluding remarks 3

Model Union level Constituency level 4

Model Union level Constituency level 5

Model Union level Constituency level 6

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 w j 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 n 1 , …, n m , find weights w 1 , …, w n s.t. each voter has equal chance of determining x * 7

First analysis • Each voter in constituency j has chance 1/ n j to be its median ! ⇒ Pr( λ j = λ P : m ) = c · n j for all j (with c >0) • Assuming i.i.d. voters , different n j 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 µ j = F -1 (0.5), σ j =[2 f ( µ j )·sqrt( n j )] -1 ⇒ Larger constituencies are a priori more central in the electoral college and more likely to be pivotal under a 50%-quota. 8

Analytical problems • Already for unweighted voting, i.e. P ≡ ( m +1)/2, we run into trouble: • Asymptotic approximation with only n 1 varying and n 2 =…= n m seems possible, but for general n 1 , …, n m ? 9

Monte Carlo simulation • Probability π j := Pr( j = ( P )) is the expected value of random variable H j ( λ 1 , …, λ m ) which is 1 if j = ( P ) and 0 otherwise • H j ’s expected value can be approximated by the empirical average of many independent draws of H j • Weight vectors are constructed from given population sizes by w j = n j α • For fixed weights ( w 1 , …, w m ) and populations ( n 1 , …, n m ), we draw λ 1 , …, λ m from the beta distributions corresponding to i.i.d. U [0,1] voters in all constituencies and average H 1 , …, H m over 10 million draws  We search for the α which yields smallest cumulative (individual) quadratic deviation of π j from the ideal egalitarian probability π j *= n j / ∑ n k ( j =1, …, m ) 10

EU Council of Ministers • Using EU25 population data, α *=0.5 with 50%-quota would give almost equal representation: 0.225 0.200 ideal probabilities 0.175 á probabilities for = 0.3 0.150 á probabilities for = 0.5 0.125 á probabilities for = 0.7 0.100 0.075 0.050 0.025 0.000 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 population in millions 11

US Electoral College • Again, α *=0.5 comes very close to equal representation: 0.18 ideal probabilities 0.16 á probabilities for = 0.3 0.14 á probabilities for = 0.4 0.12 á probabilities for = 0.5 0.10 á probabilities for = 0.6 0.08 á probabilities for = 0.7 0.06 0.04 0.02 0.00 0 5 10 15 20 25 30 35 population in millions 12

US Electoral College • Cumulative individual quadratic deviation from equal representation in the US Electoral College: cum. ind. quad. dev. 1,20E-09 1.2 E-09 1,00E-09 1.0 E-09 8,00E-10 8.0 E-10 6,00E-10 6.0 E-10 4,00E-10 4.0 E-10 2,00E-10 2.0 E-10 0.0 E+ 00 0,00E+00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 á 13

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 14

EU Council of Ministers • Nice weights and quota of 50%: 0.18 estimated probabilities (quota 50.00%) 0.16 ideal probabilities 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 Czech Rep. Hungary Sweden Cyprus Germany France U.K. Italy Spain Poland Netherlands Greece Belgium Portugal Austria Denmark Slovakia Finland Ireland Lithuania Latvia Slovenia Estonia Luxembourg Malta 0.020 0.015 0.001 0.005 0.000 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 15

EU Council of Ministers • Nice weights and quota of 72.2%: estimated probabilities (quota 72.27%) 0.18 0.16 ideal probabilities 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 Czech Rep. Hungary Sweden Cyprus Germany France U.K. Italy Spain Poland Netherlands Greece Belgium Portugal Austria Denmark Slovakia Finland Ireland Lithuania Latvia Slovenia Estonia Luxembourg Malta 0.02 0.01 0.00 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 16

Results: uniformly distributed n j • We look, first, at m ranging from 10 to 50 with randomly generated constituency sizes n 1 , …, n m and, second, at two prominent real-world population configurations • With i.i.d. n j from U [0.5·10 6 , 99.5·10 6 ], optimal α is: 17

Results: normally distributed n j • If constituencies are created for efficiency reasons, sizes possibly are distributed around some ‘ideal size’ • With i.i.d. n j from N (10 6 ; 200 000), optimal α is: 18

Results: normally distributed n j • For moderately many similar constituencies, weighted voting may allow only quite high (and flat) inequality of representation: cum. ind. quad. dev . 3.5 E-10 3,50E-10 3.0 E-10 3,00E-10 #const = 30 #const = 50 2.5 E-10 2,50E-10 #const = 100 2.0 E-10 2,00E-10 1.5 E-10 1,50E-10 1.0 E-10 1,00E-10 5.0 E-11 5,00E-11 0.0 E+ 00 0,00E+00 0.0 0.2 0.4 0.6 0.8 1.0 á 19

Results: Pareto distributed n j • More realistically, with i.i.d. n j 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) 20