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Comparing Approval At-Large to Plurality At-Large in Multi-Member Districts

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Using Monte Carlo simulations, we compare approval at-large (AAL) voting to plurality at-large (PAL) voting in multi-member districts. While similar simulations have been run to measure the performance of single-winner elections, this work seeks to confirm and extend prior results to multiple-winner elections. We intend this evaluation of approval and plurality voting in multi-member districts to be the first step towards a more comprehensive evaluation of a wider selection of voting rules. Among the conclusions we reach, we find that approval at-large performs slightly worse than approval in single-winner elections according to several metrics, while plurality at-large surprisingly performs better than plurality in single-winner elections. We also find that approval at-large performs roughly the same regardless of the number of candidates, but improves as the number of winners increases.
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Comparing Approval At-Large to
Plurality At-Large in Multi-Member Districts
Jeremy A. Hansen
Abstract
Using Monte Carlo simulations, we compare approval at-large (AAL) voting to plu-
rality at-large (PAL) voting in multi-member districts. While similar simulations
have been run to measure the performance of single-winner elections, this work
seeks to confirm and extend prior results to multiple-winner elections. We intend
this evaluation of approval and plurality voting in multi-member districts to be the
first step towards a more comprehensive evaluation of a wider selection of voting
rules. Among the conclusions we reach, we find that approval at-large performs
slightly worse than approval in single-winner elections according to several metrics,
while plurality at-large surprisingly performs better than plurality in single-winner
elections. We also find that approval at-large performs roughly the same regardless
of the number of candidates, but improves as the number of winners increases.
1 Introduction
Multi-member districts (MMDs) are districts where voters elect multiple representatives and
are usually entitled to vote for a number of candidates equal to the number of seats avail-
able [7]. Northfield, Vermont, for example, is part of two MMDs: its State Representative
district has two seats and its State Senate district has three seats. Elections in such districts
in the United States are often decided by a plurality at-large (PAL) rule (sometimes referred
to as “bloc voting”), where each voter casts votes for a number of candidates equal to the
number of seats available, and the candidates who receive the most votes win the seats.
Approval voting is a voting rule that allows each voter to indicate their “approval” for
each candidate by voting for any number (or all!) candidates listed on the ballot [2, 4].
In an election with multiple winners who are selected using approval voting, we say that
the election is decided by the approval at-large (AAL) voting rule. It has been shown that
plurality and approval behave similarly when there are three candidates and two seats [5].
Approval voting is a particularly attractive alternative to PAL since it requires only a minor
change to the ballots, and the method for determining the winner is identical. For these
reasons, “upgrading” to AAL may be more appealing to the electorate than changing to
a system that relies on complicated ballots or an unintuitive procedure to determine the
winner.
It is our goal in this paper to determine whether approval is superior to plurality by
using Monte Carlo simulations of both in identical situations. The approach of running
Monte Carlo simulations to test the performance of voting rules was not pioneered by Smith
(who seems to have been significantly inspired by Merrill [10]), but his 2000 work certainly
provided the most comprehensive analysis of voting rules to date [13]. Using several measures
of utility (described in Section 2.4), we evaluate both AAL and PAL and conclude that
neither rule uniformly performs better than the other.
2 Definitions
Most of the definitions here follow the style of [14], but have been extended to more clearly
support multiple winners. We define Aas the set of all candidates, WAas the set of win-
ners of an election, and Nas the set of voters such that |N|=n. We define P= (P1, . . . , Pn)
as a profile, an n-tuple of ballots, such that PiArepresents the ballots cast by voter i
according to the voting rule. The profile, then, is the collection of all such ballots from every
voter in N.
2.1 Voting Rules
Definition 1. Aballoting method specifies what format each ballot Pishould take for a
particular voting rule.
Definition 2. Adecision rule specifies how the ballots in Pare combined to determine the
set of winners Wfor a voting rule.
Definition 3. The plurality at-large voting rule is defined by a balloting method such that
|Pi| ≤ |W| ∀ iN. In PAL, with |A|= 3, for example, there are four legal ballots:
Pi∈ {(1,0,0),(0,1,0),(0,0,1),(0,0,0)}.
For convenience, we define NP(a) as the number of voters who cast their vote for candidate
aA:
NP(a) = |{iN:aPi}|.
The winners are the |W|candidates who receive the most votes. More formally, the decision
rule is:
xW, yA\W, NP(x)NP(y).
Definition 4. The approval at-large voting rule is defined by a balloting method such that
|Pi| ≤ |N| ∀ iN. In AAL, with |A|= 3, for example, there are eight legal ballots:
Pi∈ {(1,0,0),(0,1,0),(0,0,1),(0,0,0),(1,1,0),(1,0,1),(0,1,1),(1,1,1)}
The winners are the |W|candidates who receive the most votes. The decision rule is the
same as in PAL1:
xW, yA\W, NP(x)NP(y).
2.2 Candidate Utilities
In order to concretely compare the two voting rules in simulation, we construct a large
number of reasonable scenarios, paying particular attention to the those found in [13] and
[15]. We focus first on how voters determine for which candidates they will cast their votes.
Definition 5. We define the candidate utility,Ui(x) as a real value assigned by voter i
to candidate x. A candidate with a higher utility is preferred to a candidate with a lower
utility.
Definition 6. The most straightforward method to produce interesting values for a simula-
tion is to select the values uniformly at random within an appropriate interval. This method,
which Smith called random uniform utilities [13] and Merrill called a random society [10],
sets the values of Ui(x) as follows:
Ui(x) = U(0,1).
1Indeed, Merrill identifies that plurality really refers to a decision rule, while the different sorts of voting
rules that use plurality decision rules are characterized by their balloting method. [10]
Definition 7. A slightly more realistic method is what Smith called issue-based utilities
and Merrill called a spatial model, where each candidate and each voter is said to have a
q-tuple of positions Quthat represent his or her stances on each of q.2Each position is
an abstraction describing the voter’s or candidate’s support (with a positive number) or
opposition (with a negative number) to some arbitrary issue. We expect voters to prefer
the candidates that are “closest” to their own positions.
Qu= (pu(1), . . . , pu(q)) uAN.
The collection of all such q-tuples is Q:
Q=[
uAN
Qu.
The candidate utility is then a function of the voter’s and candidate’s positions:
Ui(x) = f(Qi, Qx).
A number of reasonable functions to compare the pair of q-tuples (and to decide which
candidates are “closest” to each voter) are evident: dot product, Euclidean distance, and
Manhattan distance.3If we interpret Qiand Qxas vectors, we can compute the utility as
the dot product of the two vectors [13]4:
Ui(x) = hpi(1), . . . , pi(q)i·hpx(1), . . . , px(q)i=
q
X
j=1
pi(j)px(j).
If we interpret Qiand Qxas points in q-dimensional space, we compute the utility as
the Euclidean distance between those two points [15]. Since candidates whose positions are
further from the voter’s are less desirable than candidates with closer positions, the utility
is a negative number, so that higher distances result in lower numbers.
Ui(x) = v
u
u
t
q
X
j=1 pi(j)px(j)2
.
As with Euclidean distance, if we interpret Qiand Qxas points in q-dimensional space,
we can compute the utility as the Manhattan distance:
Ui(x) =
q
X
j=1 pi(j)px(j).
2.3 Casting votes
After the utilities for each voter-candidate pair are calculated, each voter imust cast a
ballot Pisubject to the balloting method in effect, as described in Section 2.1.
2Smith went on to add an “ignorance” term uniformly selected at random from [1,1], presumably
based upon Merrill’s conclusion that “...it appears that electoral behavior possesses both random and spatial
components.” [10].
3The squared Euclidean distance, Minkowski distance (with values of p /∈ {1,2}), and Shepsle utility
function are additional reasonable possibilities we do not consider here.
4After calculating the dot product, Smith normalized the utility so that it always falls within [0,1].
Though the rules are fixed, there may be more than one way for a voter to cast a ballot.
For example, a voter may simply select the candidates with the highest utilities, up to the
maximum number possible5. We refer to this as a full ballot :
xPi,yA\Pi:Ui(x)Ui(y).
Notice that with the AAL rule, casting a full ballot makes little sense. Alternatively,
voter imay select only candidates whose utilities exceed some approval threshold αi, and
cast what we refer to as a threshold ballot:
xPi,yA\Pi:Ui(x)Ui(y), Ui> αi.
The threshold ballot is sensible in both PAL, where |Pi| ≤ |W|and in AAL, where
|Pi| ≤ |A|, but it does invite the question of how the threshold should be calculated. In
order to determine each voter’s approval threshold, Merrill expected votes to be cast for any
candidate above mean utility of all possible candidates [10]. That threshold is calculated
as:
αi=1
|A|X
xA
Ui(x).
Another similar strategy introduced by Merrill for calculating the threshold [10] uses the
single most preferred and single least preferred candidates’ utilities:
zA, z /∈ {x, y}:Ui(x)Ui(z)Ui(y).
αi=Ui(x) + Ui(y)
2.
Yee assigned random values to αiin a log-normal distribution, though the parameters for
that distribution were not published [15].
αi=ln N(µ, σ2).
We also expect that there may be cases where a candidate’s utility is so low (even though
it exceeds the mean), that the voter will choose not to cast a vote for that candidate.
2.4 Outcome Utilities
We say that each voter iNhas a utility function Oi(W) describing how closely the
outcome (i.e. the selection of W) matched his or her individual preferences. Using these
individual outcome utilities, we can describe the utility O(W) of the outcome across all
voters.
In this section, we discuss several metrics for calculating the utility of a particular election
outcome, such as:
Social-utility efficiency6:Does the voting rule produce an outcome that, on average,
satisfies voters?
Egalitarian-utility efficiency: How favorable is the voting rule’s selected outcome to
the least-satisfied voter?
5Though it is certainly a legal ballot, we assume hereafter that no voter will cast an empty ballot, or a
ballot with no candidates selected
6Smith referred to a similar formulation as Bayesian regret [13].
Degree of proportionality: How well do the multiple members selected represent an
accurate cross-section of the voters?
Centrist versus extremist: Does the rule tend to result in the election of candidates
holding middle-of-the-road positions or positions that skew far to one side or another?
2.4.1 Social-utility efficiency
Definition 8. Following Merrill’s definition [10], we say that the social utility of candidate
xis the sum of each voter’s utility for that candidate:
Definition 9. For a particular voting rule, we define the social-utility efficiency (SUE) as
the ratio of the sum of the social utilities of all winners to the sum of the social utilities of
the candidates with the |W|highest social utilities. This ratio may be normalized to allow
for comparisons between different voting rules.
2.4.2 Egalitarian-utility efficiency
The previous measurements are utilitarian measures of the outcome in that they emphasize
improvements over the entire population of voters. However, there may be circumstances
where a subset of voters are particularly poorly represented by the election of candidates
that would otherwise be of high social utility.
Definition 10. We define the egalitarian utility of candidate xas the minimum utility of
a candidate across all voters:
min
iNUi(x).
Definition 11. We in turn define the egalitarian-utility efficiency (EUE) as the ratio of
the sum of the egalitarian utilities of all winners to the sum of the egalitarian utilities of the
candidates with the |W|highest egalitarian utilities. This ratio, like social-utility efficiency,
may be normalized.
min
iNUi(x).
2.4.3 Degree of proportionality
The degree of proportionality is usually discussed in terms of the fairness of party represen-
tation in elections where parties or cohorts of candidates receive votes rather than individual
candidates [1, 6, 8]. In at-large non-party elections, this notion requires some modification.
We define the degree of proportionality of such elections in two ways.
Definition 12. The social-utility proportionality (SUP) is the mean of the highest-utility
winner for each voter, the intuition being that as long as each voter has at least one high-
utility candidate among the winners, they are being well-represented.
Oi(W) = max
xWUi(x).
O(W) = 1
nX
iN
Oi(W).
Definition 13. The second option is to measure proportionality in an egalitarian fashion,
in terms of the voter with the lowest utility for the closest winning candidate. We call this
the egalitarian-utility proportionality (EUP):
Oi(W) = max
xWUi(x).
O(W) = min
iNOi(W).
2.4.4 Centrist versus extremist
When using issue-based utilities, a centrist candidate is one that is near the mean position
(in all qdimensions) of all voters, while an extremist candidate is one far from the mean.
Definition 14. The centrist tendency (CT) of a voting rule is the sum of the distances
(computed as Euclidean, Manhattan, dot product, and the like) from the mean voter position
Qto the winning candidates’ positions, divided by the sum of the distances from the mean
voter position to the closest candidates’ positions.
Q= 1
nX
iN
pi(1),..., 1
nX
iN
pi(q)!.
3 Simulation Methodology & Results
To begin, and to confirm the results of [15], we fix three candidates’ positions in two dimen-
sions (i.e. q= 2) to (0.02,0.02), (0.20,0.16), and (0.66,0.56). We then fix the mean
position7of 2,000 voters to the point (1.00,1.00) and proceed through several steps:
1. Generate the normally-distributed positions of all the voters, distributed around the
current mean position.
2. Using Euclidean distances for Ui(x), calculate the outcome of a single-winner election
using both approval8and plurality voting.
3. Depending on which candidate won, assign a color to the (x, y) position corresponding
to the mean voter position. The first, second, and third candidates are colored red,
green, and blue, respectively.
4. Calculate the centrist tendency, social-utility efficiency, and egalitarian-utility effi-
ciency of the outcome, all normalized to fall within [0,1].
5. Increment the x-coordinate by 0.02, until it reaches the value 1.00, at which point the
x-coordinate is reset to 1.00 and the y-coordinate is incremented by 0.02.
These steps continue 10,201 times until the mean voter position at (1.00,1.00) is simulated.
Using that data, we produce a 101 ×101 bitmap portraying the outcomes of the elections.
Figure 1 shows this graphical representation of the outcomes. Notice that the moderate
“red” candidate at (0.02,0.02) wins many more elections when approval voting is used.
Table 1 lists the tested rules’ centrist tendency, social-utility efficiency, and egalitarian-utility
efficiency.
7We set the standard deviation arbitrarily to 0.5.
8Voters cast threshold ballots with an αchosen per Section 2.3 and [15], with µ= 0.5 and σ= 0.3.
Figure 1: Illustration of three-candidate outcomes with plurality voting (left) and approval
voting (right). Each candidate’s position in two dimensions is indicated with a darkened
circle. Each of the 10,201 pixels in the 101×101 images are colored to represent the candidate
chosen when the mean voter position is located at that point in two dimensions.
Rule SUE EUE CT
Plurality 0.9538701 0.9156188 0.9515912
Approval 0.9989238 0.9544690 0.9985330
Table 1: Mean social-utility efficiency (SUE), egalitarian-utility efficiency (EUE), and cen-
trist tendency (CT) measurements in three-candidate one-winner elections with 2000 voters
and fixed candidate positions.
For the remaining simulations, we run 10,000 iterations of the following, with three, four,
five, seven, and ten candidates:
1. Generate the normally-distributed positions of the candidates, distributed around
(0,0) with a standard deviation of 0.5 in each dimension.
2. Generate the normally-distributed positions of 2,000 voters, distributed around (0,0)
with a standard deviation of 0.5 in each dimension.
3. Using Euclidean distances for Ui(x), calculate the outcome of a single-winner election
using both approval and plurality voting.
4. Calculate the centrist tendency, social-utility efficiency, and egalitarian-utility effi-
ciency9of the outcome, all normalized to fall within [0,1].
Table 2 lists the results of the second run, for one-winner elections. Table 3 shows the
results of two-winner elections, Table 4 shows three-winner elections, and Table 5 shows
four-winner elections.
9Due to technical difficulties, we were note able to produce usable results for either social-utility propor-
tionality or egalitarian-utility proportionality.
Rule # Cand SUE EUE CT
Plurality 30.9373570 0.8795737 0.9316064
Approval 0.9946266 0.9187953 0.9934602
Plurality 40.9323401 0.8714852 0.9171037
Approval 0.9971207 0.9325357 0.9956288
Plurality 50.9209787 0.8587347 0.8963229
Approval 0.9974364 0.9365752 0.9956487
Plurality 70.9027765 0.8295677 0.8589720
Approval 0.9975346 0.9403506 0.9950697
Table 2: Mean social-utility efficiency (SUE), egalitarian-utility efficiency (EUE), and cen-
trist tendency (CT) measurements in three-candidate, four-candidate, five-candidate, and
seven-candidate one-winner elections with 2,000 voters and random candidate positions.
Rule # Cand SUE EUE CT
Plurality 30.9822055 0.9166944 0.9824740
Approval 0.9560856 0.8895889 0.9845721
Plurality 40.9887856 0.9278008 0.9901555
Approval 0.9744467 0.9089423 0.9885096
Plurality 50.9814709 0.9250207 0.9811238
Approval 0.9792827 0.9112150 0.9909740
Plurality 70.9653297 0.9067825 0.9575639
Approval 0.9759419 0.9068702 0.9905049
Plurality 10 0.9365907 0.8666353 0.9086101
Approval 0.9780376 0.9077905 0.9912491
Table 3: Mean social-utility efficiency (SUE), egalitarian-utility efficiency (EUE), and cen-
trist tendency (CT) measurements in three-, four-, five-, seven-, and ten-candidate two-
winner elections with 2,000 voters and random candidate positions.
Rule # Cand SUE EUE CT
Plurality 40.9889651 0.9313973 0.9902398
Approval 0.9812539 0.9160924 0.9927429
Plurality 50.9902220 0.9334472 0.9918170
Approval 0.9803292 0.9154303 0.9906426
Plurality 70.9897037 0.9375014 0.9898484
Approval 0.9839368 0.9215040 0.9925393
Plurality 10 0.9760985 0.9208488 0.9709877
Approval 0.9802377 0.9141811 0.9916466
Table 4: Mean social-utility efficiency (SUE), egalitarian-utility efficiency (EUE), and cen-
trist tendency (CT) measurements in four-, five-, seven-, and ten-candidate three-winner
elections with 2,000 voters and random candidate positions.
Rule # Cand SUE EUE CT
Plurality 50.9893449 0.9355236 0.9923575
Approval 0.9790500 0.9168343 0.9908143
Plurality 70.9933003 0.9428174 0.9937888
Approval 0.9873247 0.9305083 0.9937106
Plurality 10 0.9920172 0.9414007 0.9912474
Approval 0.9865669 0.9256595 0.9936499
Table 5: Mean social-utility efficiency (SUE), egalitarian-utility efficiency (EUE), and cen-
trist tendency (CT) measurements in five-, seven-, and ten-candidate four-winner elections
with 2,000 voters and random candidate positions.
4 Conclusions
Surprisingly, AAL did not fare as well across all scenarios as we had expected, but we reach
a handful of interesting conclusions.
Conclusion 1. In single-winner elections, approval performs roughly the same in terms of
SUE, EUE, and CT, regardless of the number of candidates. This confirms Merrill’s result
for SUE in [11] and extends it to EUE and CT.
Conclusion 2. In multiple-winner elections, AAL performs slightly worse than approval in
single-winner elections in terms of SUE, EUE, and CT. Again, we see this result regardless
of the number of candidates. In multiple-winner elections, however, AAL’s SUE, EUE, and
CT improve as the number of winners increases.
Conclusion 3. In two-winner elections with three, four, or five candidates, in all but one
three-winner elections, and in all four-winner elections, AAL performs slightly worse than
PAL in terms of SUE and EUE.10
Conclusion 4. Except for a handful of scenarios11 , AAL outperforms PAL in terms of CT.
5 Future Directions
Our results identify some clear patterns but provoke some follow-up questions. In future
work, we intend to investigate the anomaly between AAL’s and PAL’s CT values for some
of the scenarios.
As stated in the abstract, we hope to eventually extend our simulations of multiple-
winner elections to include other voting rules such as range voting, single transferable vote,
ranked pairs, runoff, and Borda. In conjunction with more voting rules, we are interested
in the effects of insincere or strategic voting [12, 14] on our selected metrics. Other metrics
like those in [9, 10, 13], and non-uniform distributions of voter positions like multimodal
and arcsine distributions may be interesting in future analyses.
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10When there are seven or ten candidates in a two-winner election, AAL performs slightly better than
PAL. Ten-candidate, three-winner AAL performs slightly better than PAL in the same circumstances.
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candidate four-winner elections.
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Jeremy A. Hansen
Department of Computing
Norwich University
Northfield, VT USA
Email: jeremyhansen@acm.org
ResearchGate has not been able to resolve any citations for this publication.
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A variety of electoral systems for single-winner, multicandidate elections are evaluated according to their tendency to (a) select the Condorcet candidate--the candidate who could beat each of the others in a two-way race--if one exists, and (b) select a candidate with high social (average) utility. The proportion of Condorcet candidates selected and a measure of social-utility efficiency under either random society or spatial model assumptions are estimated for seven electoral systems using Monte Carlo techniques. For the spatial model simulations, the candidates and voters are generated from multivariate normal distributions. Numbers of candidates and voters are varied, along with the number of spatial dimensions, the correlation structure, and the relative dispersion of candidates to voters. The Borda, Black, and Coombs methods perform well on both criteria, in sharp contrast to the performance of the plurality method. Approval voting, plurality with runoff, and the Hare system (preferential voting) provide mixed, but generally intermediate results. Finally, the results of the spatial model simulations suggest a multicandidate equilibrium for winning-oriented candidates (under plurality, runoff, and Hare) that is not convergent to the median.
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This book addresses a significant area of applied social-choice theory--the evaluation of voting procedures designed to select a single winner from a field of three or more candidates. Such procedures can differ strikingly in the election outcomes they produce, the opportunities for manipulation that they create, and the nature of the candidates--centrist or extremist--whom they advantage. The author uses computer simulations based on models of voting behavior and reconstructions of historical elections to assess the likelihood that each multicandidate voting system meets political objectives.Alternative procedures abound: the single-vote plurality method, ubiquitous in the United States, Canada, and Britain; runoff, used in certain primaries; the Borda count, based on rank scores submitted by each voter; approval voting, which permits each voter to support several candidates equally; and the Hare system of successive eliminations, to name a few. This work concludes that single-vote plurality is most often at odds with the majoritarian principle of Condorcet. Those methods most likely to choose the Condorcet candidate under sincere voting are generally the most vulnerable to manipulation. Approval voting and the Hare and runoff methods emerge from the analyses as the most reliable.Originally published in 1988.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.