Quantifying strategic voting under
different Preference Aggregation
Rules and Ranking Correlation
Measures
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Quantifying strategic voting under
different Preference Aggregation
Rules and Ranking Correlation
Measures
Rutger R.W. van Woerkom 11297654
Bachelor thesis Credits: 18 EC
Bachelor Kunstmatige Intelligentie
University of Amsterdam Faculty of Science Science Park 904 1098 XH Amsterdam Supervisor S. Botan MSc
Institute for Logic, Language and Computation Faculty of Science
University of Amsterdam Science Park 107 1098 XG Amsterdam
Abstract
Modern day democratic electoral systems are all susceptible to strategic voting. Especially with voting methods (preference aggregation rules) such as the still widely used Plurality rule. It important that insight is gained into strategic vot-ing in democratic electoral systems, to potentially increase their effectiveness and fairness.
We created a computational election model that generates simulated election data, with which we quantify the manipulability of elections with varying combina-tions of preference aggregation rules and ranking correlation measures. The manip-ulability of electoral systems is quantified using the Plurality rule, Copeland’s rule or the Borda count as their preference aggregation rule. Additionally, the effect of different ranking correlation measures, on the manipulability of these preference aggregation rules is examined. To more accurately resemble the way human voters compare different ballots, we introduce a new type of ranking correlation measure: exponential vector correlation.
We conclude that the Plurality rule consistently is the least strategyproof of the three preference aggregation rules that were considered in this thesis. Furthermore, we explain why exponential vector correlation might be better suited in election models than more classical ranking correlation measures (commonly known as distance measures) such as Kendall’s tau and Spearman’s rho.
1
Introduction
The 2016 Pennsylvania senatorial election ended with a very close outcome. For the republican voters that partook in the election, the choice was apparent: Toomey ac-quired 100% of votes in the republican primary election. However, the democratic votes coincided much less, with McGinty and Sestak acquiring 42.50% and 32.57% of votes, respectively, in the democratic primary election. Democratic voters thus were left with two sensible options: either voting for their sincerely preferred can-didate, thereby risking maintaining the split, or cooperating by voting for the same democratic candidate, generating a much more promising candidate. Despite the democratic voters agreeing on a single candidate to back, the Pennsylvania general election resulted in republican candidate Toomey winning with a 1.43% majority over democratic candidate McGinty. This election presented a classic scenario in which strategic voting could have impacted the outcome decisively. While at first, no single democratic candidate seemed to form competition against the republican Toomey, the outcome of the general election was rather close.
Strategic voting is the act of constructing a ballot, potentially misrepresent-ing sincere preferences, with the goal to increase the probability of a specifically desired outcome of an election. In any election that comprises more than two alter-natives, and that is not dictatorial, it cannot be guaranteed that strategic voting will not occur (Gibbard, 1973; Satterthwaite, 1975). In fact, Kawai and Watanabe (2013) reported proportions of up to nearly 85% of voters who voted strategically in the Japanese general election. These high percentages suggest that voters often misrepresent their opinions, leading to an outcome of an election that is not rep-resentative of the true preferences of a population of voters. Ideally, voters would never be incentivized to vote strategically; voters would always vote sincerely and the outcome of an election would be perfectly optimized to the preferences of the voters. That is, the distance between the outcome of an election and the preference of each voter would be minimized. Analyzing the proportions of strategic voting under different types of electoral systems may help to distinguish more effective systems from less effective ones, to ultimately bring us closer to a perfect electoral system.
A crucial part of any election, especially one where strategic voting plays a role, is the comparison of different ballots. A human voter, either directly or indirectly, compares different potential rankings of alternatives to construct a ballot that best represents their sincere preferences. Human voters often exhibit a strong preference to a particular outcome of an election, to some degree caring about the popularity of all alternatives and not just the winner. It is important that human aspects of voting in electoral systems are incorporated, in order to test them for their effectiveness.
the manipulability of elections is quantified through simulations with a computa-tional framework. The goal of this is to substantiate ideas about the effectiveness and strategyproofness of different electoral systems. Additionally, these PARs are paired with different ranking correlation measures (RCMs), that simulate the com-parison between two ballots by a voter. With this, some insight can be gained into how strong the effects of different comparison methods are on the outcomes of elections.
The effect of different preference aggregation rules (PARs) on the manipulabil-ity of elections is quantified through simulations with a computational framework. This computational framework, or election model, can simulate elections with both social choice functions (SCFs) and social welfare functions (SWFs). SCFs only out-put the winner of an election, neglecting the ranking of the rest of the alternatives. However, in order to quantify the effect of different methods of comparing complete ballots, PARs must output an aggregated ranking of all alternatives participating in an election. To do this, the Plurality rule, Copeland’s method and the Borda Count are modified and used as SWFs, that return an aggregated ranking of al-ternatives as the outcome of an election. The computational framework includes implementations of some RCMs; classic distance measures such as Kendall’s tau, Spearman’s rho are used as RCMs, taking two rankings of alternatives as input and returning a correlation score that indicates how much the rankings are alike. However, RCMs based on more universal distance measures such as Kendall’s tau and Spearman’s rho are not the most well suited for simulating voters in an elec-tion per se. Hence, a new type of correlaelec-tion measure is introduced, namely the exponential vector correlation (EVC). We created this RCM with the intent to more closely resemble how a human voter would compare two different rankings of alternatives.
2
Theoretical components
An election consists of a set of alternatives A = {a1, ..., am}, where m > 1, and a
set of voters V = {v1, ..., vn}, where n > 1. To discover to what extent strategic
voting can play a part in an election, we created an election model. The main focus of this model is to quantify the ways in which voters can manipulate the outcome of the election by casting their vote. A classical theoretical framework for vot-ing, especially one that handles strategic votvot-ing, consists of a number of different components. Each component is explained separately and comprehensively in this section.
2.1
Ballots
During an election, the alternatives of set A are ranked based on the preferences of the voters of set V . Voters can specify their relative preferences of the alternatives, i.e. cast a vote, by submitting a ballot. The vector of ballots in an election, named a profile, can be defined as P = hR1, ..., Rni where Ri is the ballot of voter
i. These ballots comprise a strict ranking of alternatives, based on the voter’s preference. Thus, Ri is a linear ordering i of A, where i is transitive, complete,
reflexive and antisymmetric (Brandt et al., 2016). Ri denotes the strict ranking
of alternatives [a1, a2, ..., am], where a1 is strictly preferred over a2, which in turn
is strictly preferred over a3, and so forth. Voters may not always submit a ballot
that reflects their true preferences. A voter may be inclined to cast an insincere vote, ranking the alternatives in a less preferred order. Insincere voting can lead to a manipulated outcome of an election.
2.2
Preference Aggregation Rules
A preference aggregation rule (PAR) is a function that takes as input a vector of ballots, and gives as output either a single alternative or a ranking of alternatives. Two main categories of PARs can be distinguished; the first type of PAR is a social choice function (SCF). An SCF takes a vector of ballots as input and returns a single alternative that is deemed the winner of that election. Examples of SCFs are: the Plurality rule, Copeland’s method, the Borda count and Slater’s rule. SCFs can determine the winning alternative of an election, however they do not provide any information about the ranking of the rest of the alternatives. As one of the main focus points of this thesis is quantifying the manipulation of the entire outcome, the Plurality rule, Copeland’s method and the Borda count are converted to a second type of PAR: social welfare functions (SWFs). SWFs return a ranking of alternatives as opposed to just the set of winners of the election. With this, voters can compare their entire ballots with the entire outcomes of elections.
One of the biggest advantages of Copeland’s method, the Borda count and the Plurality rule is that they are relatively easily implementable in a computational model. Although they are based on the majority graph, Copeland’s method and the Borda count use a point system, solving majority cycles through tie-breaking. Nonetheless, Many PARs are more complex to implement in a computational model. Aggregation with PARs like Slater’s rule (Slater, 1961), is in fact NP-hard: nearly impossible to solve for large numbers of alternatives. Slater’s rule, defined in more detail by Brandt et al. (2016), aggregates a ranking of alternatives based on the majority graph. What separates Slater’s rule however, is that it does not work by awarding alternatives with points. Instead, Slater’s rule creates a ranking of the alternatives by inverting as few edges as possible in the majority
graph. In doing so, Slater’s rule directly resolves majority cycles.
The Plurality rule, Copeland’s method and the Borda count are implemented in our framework, each one reacting differently to strategic voting. These PARs will now be presented.
2.2.1 Plurality rule
A simple and commonly used voting rule: the Plurality rule, is primarily added to our framework to represent most real elections. Additionally, a comparison can be drawn between the behavior of this common and simple rule, and more complex rules such as Copeland’s rule and the Borda count.
For every ballot Ri, αi represents the alternative x ∈ A in the profile P that
is ranked at first place in Ri. The Pscore of an alternative is simply the number of
times it appears in first place in the ballots of a profile, formally defined as: Pscore(x, P ) = |(αi ∈ P | αi = x) for i = 1, ..., n|
Now, for any profile P , the Plurality winner is the alternative with the highest Pscore. The Plurality rule can be defined as an SCF, formally as:
P lurality(P ) = argmax
x∈A
Pscore(x, P )
P lurality(P ) can return more than one winner. However, tie-breaking is ap-plied to determine a singular winner from the set.
Table 1: Voting profile P1
R1..11 R12..21 R22..30
11 10 9
a1 a2 a3
a2 a3 a2
a3 a1 a1
Consider Profile P1 as seen in Table 1. Alternative a1, a2 and a3 have Pscore
of 11, 10 and 9 respectively. Hence, according to the Plurality rule, the winner of the election is alternative a.
In order for the Plurality rule to function as an SWF, a modified definition can be used. This SWF is simply a vector of alternatives ordered from highest to lowest Pscore, where ties between alternatives are broken as specified in section
P luralitySW F(P ) = [x1, ..., xm | Pscore(x1, P ) > ... > Pscore(xm, P )]
Using the same profile, P1, to compute the vector P luralitySW F(P1), we get
the aggregated ranking: [a1, a2, a3].
In this example, while 19 of the total 30 voters placed a1 at the bottom of
their rankings and only 11 voters placed a1 on top, alternative a1 still wins the
election. The Plurality rule only regards the alternatives ranked at first place, while the rest of the voters’ ranking are ignored. Due to this indifference towards part of the preferences of voters, favoring unpopular alternatives participating in elections, the Plurality rule is often criticized; the justification for the Plurality rule is grounded in non-substantial arguments (Blais and Carty, 1988).
2.2.2 Copeland’s method
Copeland’s method orders the alternatives that participate in an election by their number of victories in pairwise majority contests. It is based on the unweighted majority graph, since it does not account for the margins of victories. The major-ity relation, denoted by >P, expresses the preference for one of two alternatives
over the other of the majority of the population of voters. Let x and y be different alternatives: x >P y expresses that a majority of voters prefer alternative x over alternative y. For every such pairwise contest, the alternative that is more fre-quently preferred over the other is considered the winner, and receives 1 point. A point gets subtracted from the point total of the losing alternative of the contest. Occasionally, a pairwise majority contest may end in a draw: for alternatives x and y in some pairwise majority contest, alternative x is preferred above alter-native y precisely as frequently as y is preferred above x. If such a draw occurs, both alternatives gain .5 points. The Copeland set is the set of alternatives that acquired the single highest number of points. The Copeland score of an alternative x (Brandt et al., 2016) is formally defined as:
Copeland(x, P ) = |{y ∈ A|x >P y}| − |{y ∈ A|y >P x}|
Furthermore, the Copeland set of a profile P is defined as the set of alternatives with the highest Copeland scores:
CO(P ) = argmax
x∈A
Copeland(x, P )
As the Copeland set can contain multiple winners, tie-breaking is used to de-termine a singular winner from the set.
Again, consider Profile P1 as seen in Table 1. To compute the Copeland set of
alternative a1and a2, a1 is preferred over a2 only in R1..11, receiving 11 points, with
a2 in R12..21 and R22..30 receiving (10 + 9 =) 19 points. Alternative a1 loses against
a2, therefore a point is deducted of its point total. Comparing alternative a1 and
a3, alternative a1 again loses this contest, losing another point. The Copeland
score of a1: Copeland(a1) = −2. Computing the Copeland score for alternative
a2 and a3, we get Copeland(a2) = 2 and Copeland(a3) = 0. The Copeland set
CO(P1) then consists of only alternative a2.
For Copeland’s method to be used as an SWF, a modified definition can be used. This SWF is simply a vector of alternatives ordered from highest to lowest Copeland(x, P ), where ties between alternatives are broken as specified in section 2.2.4:
COSW F(P ) = [x1, ..., xm | Copeland(x1, P ) > ... > Copeland(xm, P )]
Using the same example profile P1, we get a the aggregated ranking COSW F(P1) =
[a2, a3, a1].
Copeland’s method takes the entire ballots of the voters into account, while the Plurality rule only regards the alternatives that are placed first on the ballots. Consequently, Copeland’s method is generally deemed a more compromising voting method than the Plurality rule.
2.2.3 Borda count
The Borda count is similar in behavior to Copeland’s method, as it also derives a winner from a profile based on pairwise majority contests between all the alterna-tives. However, as opposed to Copeland’s method, the Borda takes the margins of the victories of the contests into account. This is done by assigning weights to each alternative on the ballot. Hereby, it is based on the weighted majority graph. With the asymmetric Borda count applied to an election with m alternatives, a score vector W = hm − 1, m − 2, ..., 0i is applied to the ballots. For instance, W applied to ballot [a1, a2, a3] awards a1 with 2 points, a2 with 1 point and a3 with
0 points. Thus, the points of alternatives that are ranked higher on a ballot weigh more than the points of lower ranked alternatives. The asymmetric Borda count (Brandt et al., 2016) for any specific alternative in a profile with N total voters can be defined as:
Borda(x, P ) = X
y∈A
|{i ∈ N : x i y}|
The Borda set of a profile P can then be defined as the set of alternatives with the highest Borda counts:
BO(P ) = argmax
x∈A
Borda(x, P )
Since there can be multiple winners, tie-breaking is used to determine a singular winner from the Borda set.
To compute the Borda set for P1 as seen in Table 1, we first determine the
Borda counts for every alternative. Since P1 comprises three alternatives the score
vector equals: W = h2, 1, 0i. On ballots R1..11, alternative a1 collects (2 · 11 =)22
points. On ballots R12..21 and R22..30, a1 collects a total of 0 points, since it is
ranked lowest. Thus, the Borda count of alternative a1 amounts to 22 points total.
Calculating the Borda count for alternatives a2 and a3 we get totals of 40 and 28
respectively. Alternative a2 is again deemed the winner of the election.
To use the Borda count an SWF, a modified definition can be used. This SWF is simply a vector of alternatives ordered from highest to lowest Borda(x, P ), where ties between alternatives are broken as specified in section 2.2.4:
BOSW F(P ) = [x1, ..., xm | Borda(x1, P ) > ... > Borda(xm, P )]
Using the same example profile P1, we get a the aggregated ranking BOSW F(P1) =
[a2, a3, a1].
The Borda count, much like Copeland’s score, takes into account the entire ballots of the voters. It differentiates itself from Copeland’s score however, by expressing the margins by which alternatives are preferred over each other. With this, the Borda count encapsulates more of the information given in the profile.
2.2.4 Tie Breaking
Often times the SCFs implemented in this framework may produce a tied outcome. Furthermore, ties between alternatives in SWFs need to be broken in order to produce a strict aggregated ranking. To solve this problem, numerical tie-breaking is implemented in the framework.1 For any voting method, the winner of a pair
of alternatives with equal scores is the alternative that is denoted with the lowest number as its subscript.
For profile P2, as seen in Table 2, the Copeland scores of alternative a1, a2 and
a3 are 1.5, 1.5 and -2 respectively. Alternative a1 and a2 are tied, which means
that the Copeland set of the profile CO(P2) consists of both alternative a1 and
a2. Thus, with the numerical tie-breaking, the winner of the election is alternative
a1.
This tie-breaking mechanic is also what allows us to convert the previously considered SCFs into SWFs. The SWFs compute the score for every alternative in
Table 2: Voting profile P2 R1..10 R11..20 10 10 a1 a2 a2 a1 a3 a3
the election and apply tie-breaking for every pair of tied alternatives to construct a final strict ranking. Suppose the Borda counts for every alternative a1, a2 and
a3 in an election are 36, 27 and 27 respectively. It is not immediately clear what
ranking the SWF should output in this scenario, since two of the alternatives are tied. To solve this, the SWF will use tie-breaking to determine a strict ranking of all tied alternatives; in this case the SWF will rank alternative a2 above a3. The
final ranking that the SWF outputs is then equal to [a1, a2, a3].
2.3
Ranking Correlation Measures
In the context of strategic voting; in order to decide on which outcome of an election is most preferred by a voter, rankings must be compared to each other. Let a1, a2, a3 ∈ A be the alternatives participating in some election that uses F
as its preference aggregation rule. Voter vi ∈ V , a strategic voter, is looking
to cast a vote that will lead to his most preferred outcome. Suppose if voter vi were to cast his vote sincerely, the outcome F (P1) of the election would be
[a1, a3, a2]. However, voter vi knows that by casting an insincere vote, the outcome
of the election can in this case be manipulated. By casting his insincere vote, the resulting outcome F (P2) of the election would be [a2, a1, a3]. The voter vi must
now compare both outcomes F (P1) and F (P2) and decide which one is closer to
their sincerely preferred outcome, in order to decide on their final ballot.
An important part of modelling strategic aspects voting is comparing ballot rankings. Ranking correlation measures2 can be used to estimate the correlation
between two different ranked collections of data. A multitude of options is available when choosing an RCM. However, a ranking correlation measure in the case of this thesis must be selected such that it resembles human decision making as close as possible. Thus, a good ranking correlation measure will fit within the reasoning of a rational agent.
Note that for both the Kendall’s tau and the Spearman’s rho RCMs, which
we will consider next, the two ballots that are compared must first be converted to arrays of integers that represent the rank of each alternative on the first of the two ballots. Alternatives [x, y, z] in the first ballot R1 are replaced with numbers
increasing from 1 to get the first array of relative ranks: rel1 = [1, 2, 3].
Subse-quently, the alternatives [y, z, x] on the second ballot get replaced by the numbers that the same alternatives on the first ballots were assigned with. This results in a second array of relative ranks: rel2 = [2, 3, 1].
2.3.1 Kendall’s tau
Kendall’s τ (tau) (Kendall, 1938), is an RCM that measures the concordance be-tween two rankings. This is closely related to the minimum number of sequential conversions that need to be made to convert a ballot into the ballot that it is com-pared with (Brandt et al., 2016).3 In the context of this thesis, Kendall’s τ is used
to measure the correlation between two ranked ballots. To do this, the numbers of concordant and discordant pairs of observations need to be counted. Firstly, the ranked ballots are converted to arrays of their relative ranks as demonstrated in the beginning of this section.
An observation consists of the relative ranks of two alternatives [k, l], where k ∈ rel1 and l ∈ rel2 are positioned at relatively the same place in the array.
A pair of observations [k, l] and [k0, l0], is concordant if k0 > k and l0 > l or if k0 < k and l0 < l. A pair is discordant if k0 > k and l0 < l or k0 < k and l0 > l. Once the number of concordant pairs (C) and the number of discordant pairs (D) have been counted, Kendall’s τ can be computed. The difference between the number of concordant pairs and discordant pairs is divided by the total number of pairs of observations, where m denotes the number of alternatives on each ballot:
τ = nC − D
2(m − 1)
Let R1 = [a1, a2, a3] and R2 = [a3, a1, a2] be two ranked ballots, which
con-verted into arrays of their relative ranks gives rel1 = [1, 2, 3] and rel2 = [3, 1, 2]
To compute Kendall’s τ , the number of concordant and discordant pairs of obser-vations of R1 and R2 are first counted. The first pair of observations consists of
the relative ranks assigned to the first alternatives on R1 and R2: [1, 3], and the
second alternatives on R1 and R2: [2, 1]. This pair of observations is discordant,
increasing D by 1. Counting for every pair of observations, we get C = 1, D = 2 and m = 3. Kendall’s τ then equals: 31−2
2(3−1)
= −.33.
3In this thesis, Kendall’s tau is implemented in a more general way, to more closely resemble
the code that was used. For a more simple and intuitive definition of Kendall’s tau, see Brandt et al. (2016)
While Kendall’s tau offers a good impression of the correlation between two ballots, using it assumes a potentially unwanted property of the ballots. Namely, Kendall’s tau weighs every alternative on a ballot equally. In a real voting scenario however, it can not be assumed that voters care about the rank of every alternative on the ballot equally.
2.3.2 Spearman’s rho
Spearman’s ρ (rho) (Spearman, 1904), is an RCM that measures the distance between two rankings. It is closely related to the difference in the rankings of each alternative between two ballots.4 To compute Spearman’s ρ for two ranked
ballots, the ballots are first converted to arrays of relative ranks as demonstrated in the introduction to this section. Now, the distance between the two ballots can be computed by subtracting the second array of relative ranks from the first (rel1− rel2): [1, 2, 3] − [2, 3, 1] = [−1, −1, 2], denoted as d. This resulting array d,
together with the number of alternatives on each ballot m, can then be used to compute Spearman’s ρ.
ρ = 1 − 6 ·P d
2
m(m2− 1)
Let R1 = [a1, a2, a3] and R2 = [a3, a1, a2] be two ranked ballots. To compute
Spearman’s ρ, these ballots are converted two the arrays of relative ranks [1, 2, 3] and [3, 1, 2] respectively. The distances between these are d = [−1, −1, 2]. The sum of the squared distances amounts to ((−1)2 + (−1)2 + 22) = 6. With the
number of alternatives m = 3, Spearman’s ρ then equals 1 − 3·(36·62−1) = .5
Even though Spearman’s rho often suggests more extreme correlation, it be-haves similarly to Kendall’s tau. Spearman’s rho also weighs every alternative on the ballots equally.
2.3.3 Exponential Vector Correlation
To counteract the unwanted property as presented in sections 2.3.1 and 2.3.2, where every alternative on the ballot is weighed equally, we have created the Exponential Vector Correlation (EVC) measure. With EVC, ballots are converted into vectors, of which every element is scaled individually. Consequently, the top ranked alternative on a ballot can have more impact on the correlation measure than the lower ranked alternatives.
4Just like with Kendall’s tau, Spearman’s rho is implemented in a more general way in this
thesis in order to more closely resemble the code that was used. For a more simple and intuitive definition of Spearman’s rho, see Brandt et al. (2016)
Let R1 = [a1, a2, a3] and R2 = [a2, a1, a3] be ranked ballots in some election.
The first ballot, R1, can be converted into an exponential vector V1 like so, where
m is equal to the number of alternatives on ballots R1 and R2:
Vec(m, β) = hβm, βm−1, . . . , β1i
Increasing the parameter β will increase the difference between elements of the vector V(R, β), thus laying more emphasis on the higher ranked alternatives. In this thesis, a parameter value of β = 2 will be used.
Converting the first ballot R1 into an exponential vector gives V1 = Vec(R1, β =
2) = h8, 4, 2i. Now, the second exponential vector can be constructed by assigning the integer values 8, 4 and 2 to the same alternatives on ballot 2: V2 = h4, 8, 2i.
To complete the calculation of EVC value for R1 and R2, the norm of the
distance between vectors V1 and V2 is inverted:
EV C(V1, V2) =
1 ||V1− V2||
For vectors V1 = h8, 4, 2i and V2 = h4, 8, 2i, EV C(V1, V2) = 1/||h4, −4, 0i|| ≈
0.18.
In real elections, voters often care more about the position of some alterna-tives than they do about others. Because of this, RCMs like Kendall’s tau and Spearman’s rho may not be fully suited for the comparison of ballots. EVC on the other hand, distinguishes itself from the other two RCMs, as it allows for a differentiation between the importance of the ranks of alternatives. Specifically, with EVC, alternatives that are more preferred by a voter have more impact on the correlation measure than the less preferred alternatives do.
Suppose two ballots R1 = [a1, a2, a3, a4] and R2 = [a2, a1, a3, a4] are compared
by a voter in an election. The only difference between the ballots is that the first two alternatives are swapped. Using Kendall’s tau to compare R1 with R2, we get
a correlation score of .66, indicating that R1 and R2 match fairly closely. Now,
using EVC as the RCM, we get a correlation score of only roughly .18, which indicates that the voter heavily differentiates between R1 and R2. Thus, using
EVC, the voter lays much more emphasis on the ranking of the alternatives higher on the ballot, in this case resulting in a much lower correlation score relative to Kendall’s tau.
2.4
Strategic Voting
In hopes of beneficially altering the outcome of an election, voters may choose to submit insincere ballots. Within the context of this thesis, strategic voting is the voter’s act of submitting an insincere ballot with the belief that it might alter the
outcome of the election beneficially to them. A more beneficial outcome is one that closer represents the sincerely preferred outcome of the voter; an outcome with a higher correlation measure to their sincerely preferred outcome.
Strategic voting is only possible if the election, under a given PAR f and an RCM R, is single voter manipulable in at least one case. When looking at complete ballots5, the election is single voter manipulable if for any pair of profiles P, P0 and
the sincere ballot of voter i, Rs
i: R(f (P 0), Rs
i) > R(f (P ), Rsi), where P
0 equals a
manipulated alternative of P caused by the insincere ballot of voter i. Table 3: Voting profile P3
R1..2 R3..4 R5..7 R8..9 R10
2 2 3 2 1
a1 a2 a3 a3 a1
a2 a1 a1 a2 a3
a3 a3 a2 a1 a2
To demonstrate what strategic voting might look like: look at P3 as seen in
Table 3. Computing Copeland’s score for each alternative of this profile, we get: {a1 : 1.5, a2 : −2.0, a3 : 1.5}. Consequently, COSW F(P3) = [a1, a3, a2], where a1
wins the election. For the voters who submitted their sincere ballots R5, R6 and
R7, this outcome is sub-optimal. Namely, the winner of the election is alternative
a1, while they prefer alternative a3 over a1. In this particular case, the outcome
could have been successfully manipulated to their benefit by any of the three voters in R5..7. Strategically, their most optimal ballot to submit would have been
[a3, a2, a1]. If one of the voters in R5..7would have submitted this ballot, the profile
of the election would look like P03, in Table 4.
The resulting Copeland scores would be: {a1 : 1.0, a2 : −0.5, a3 : 1.5}. The
ag-gregated ranking COSW F(P3) would then be [a3, a1, a2], leaving the most preferred
alternative on the original ballots R5..7 as the winner of the election.
This way of manipulating the outcome of an election, by submitting an insincere ballot, is possible in many different scenarios. That is, elections with many more voters and alternatives than in the example scenario, and elections that make use of other PARs.
5In the context of this thesis, we mainly look at the effect of different SWFs (which are paired
with an RCM in the election model). For a definition of single voter manipulability with SCFs, see Brandt et al. (2016).
Table 4: Voting profile P03 R01..2 R03..4 R5..60 R07..9 R010 2 2 2 3 1 a1 a2 a3 a3 a1 a2 a1 a1 a2 a3 a3 a3 a2 a1 a2
3
Methods
3.1
The Election Model Framework
To test for the possibilities of strategic voting under different circumstances, all theoretical aspects of section 2 are combined in the framework of the election model. For any preset election scenario, the model can be used to gather every possible way the election can be manipulated. Any instance of the model will contain the following elements:
1. A (finite) set of voters. 2. A (finite) set of alternatives.
3. A vector of ballots, one for each voter. That is, a strict ordering of all the alternatives in the election in the form of a list.
4. A profile; a list of every ballot that exists in the election, with a counter that shows how many voters have submitted that ballot.
5. A selected preference aggregation rule (Copeland’s rule by default)
6. A selected ranking correlation measure (Exponential Vector Correlation by default)
The election model was implemented with Python 3.7.
3.2
Preference initialization
Upon initiating the election model, a set of voters is created, with each their preferred ranking of alternatives. These preferences are generated according to the impartial culture model, which assumes that the voters’ preferences are drawn from a uniform distribution. Each voter’s preferences are represented by a strict ranking of all alternatives. The impartial culture model for generating voter preferences is
widely deemed an unrealistic model compared to real elections, with a relatively high probability of generating majority cycles (Tsetlin et al., 2003). Nevertheless, this method of generating preferences offers sufficient and non-biased data.
For a given set of alternatives, the model generates preferences for every voter by randomly reordering the alternatives. A strict ranking in the form of a Python list is then created, where for every alternative in the list except the first, the alternative to the left is strictly preferred over the selected alternative.
3.3
Implementation of PARs
Every preference aggregation rule that is presented in section 2, is implemented as a standalone function. After initiation of the model, it can be assigned with a PAR to use for computing the outcome of the election.
Every implemented PAR function takes the profile of the election as input. It then calculates a score for every alternative and outputs them in a Python dictio-nary. This dictionary contains the alternatives as keys, with their corresponding scores as values. For example:
par_copeland ( p r o f i l e ) = { ’ a1 ’ : 0 . 0 , ’ a2 ’ : 2 . 0 , ’ a3 ’ : −2.0} This Python dictionary is later converted to a Python list with the function ’par_to_ranking’, sorting the alternatives from highest to lowest score. In this case:
par_to_ranking ( par_copeland ( p r o f i l e ) ) = [ a2 , a1 , a3 ]
This final ranking of alternatives represents the aggregated preference of the population of voters in the model as a whole.
3.4
Implementation of RCMs
For every voter in an election, it is computed in which ways that voter could manipulate the outcome of the election such that it is beneficial to them. To do this, the sincerely preferred ranking of alternatives of the voter is compared with every hypothetical outcome. Similar to the implementation of the PARs, every RCM is an independently implemented function. This function takes as input the ranking that is sincerely preferred by a voter, and an additional ranking to compare it with. The output of the function is a correlation score, of which the boundaries depend on the correlation measure used. If a sincerely preferred ranking is compared with a multitude of other rankings, the ranking that produces the highest ranking correlation score is considered to be the most favorable.
Let ranking [a1, a2, a3] be the sincerely most preferred ranking of a voter. Say the hypothetical outcome of an election, when this sincere ballot is submitted, is the ranking: [a2, a1, a3]. Using EVC to compare it with the sincere ranking:
rcm_evc([a1, a2, a3], [a2, a1, a3]) ≈ 0.35
Now, say that the voter can successfully manipulate, leading to the alternative hypothetical outcome: [a1, a3, a2] Again, using EVC to compare it with the sincere ranking:
rcm_evc([a1, a2, a3], [a2, a1, a3]) ≈ 0.71
The second hypothetical outcome: [a1, a3, a2], has a higher correlation with the sincere preference of the voter, which is why the model counts this as one way strategic voting can occur with this election profile.
3.5
Running the model
Once the election model is initiated with a specific number of alternatives and voters, and a PAR and RCM are selected, data can be generated. The data that was generated by the model in this thesis, comprises neutral_manipulations, neutral_manipulations_win, strategies and strategies_win. For every alternative in the profile of the election, the model collects these four types of data. All permutations of each voter’s sincere ballot are first generated, after which for each permutation is tested if they lead to one of the four types of modifications of the outcome.
The neutral_manipulations of a specific profile is the total number of ways in which the outcome of an election can be altered by insincerely voting, accumu-lated over every voter in the election. That is, the number of ways in which the aggregated preferences, outputted by the PARs, can be rearranged by the insin-cere vote of a single voter. neutral_manipulations_win is calculated in the same way as manipulations, except it only counts the number of ways the winner of the election is altered.
The strategies of a profile is the total number of ways in which the outcome of the election can be altered by an insincere ballot, such that it is a more beneficial outcome to the voter of whom the ranking is altered. Similarly, strategies_win is calculated in the same way as strategies, except it only counts the number of ways the winner of the election is altered.
4
Results and interpretation
The election model was used to generate multiple instances of simulated election data. Each instance was generated with a different combination of a Preference Aggregation Rule (PAR) and a Ranking Correlation Measure (RCM). The vertical
axis of each graph is labeled as Average total manipulations, which indicates the frequency of either the neutral manipulations, the neutral manipulations_win, the strategies or the strategies_win. Every graph in this section shows the average manipulations of altered outcomes of 1000 iterations of the simulations.
Firstly, let us look at some three-dimensional graphs that show the data gener-ated by the election model as a whole. For each of the PARs, a three-dimensional graph is generated showing the number of manipulations and strategies for differ-ent numbers of voters and alternatives. The three figures that are displayed in this section show data generated using exponential vector correlation (EVC) as the RCM. In the Appendix, three-dimensional graphs can be found for data that was generated with other RCMs.
Figure 1: Borda Count average frequencies, using EVC
As can be seen in figures 1, 2 and 3, for a fixed number of voters, the average frequencies of manipulations increase exponentially as the number of alternatives increases. Not only is this the case for EVC as the RCM, but also for every other RCM as can be seen in the Appendix. This can be explained by the fact that as the number of alternatives grows, a voter can rearrange their ballots in exponentially more ways. Thus, if manipulation of the outcome is possible, a voter has exponentially more ways to rearrange their ballot and manipulate.
Oppositely, for a fixed number of alternatives, the number of manipulations exponentially decreases as the number of voters increases. As more ballots are
Figure 2: Copeland’s Score average frequencies, using EVC
submitted, the impact of each ballot on the aggregated outcome becomes less. Hence, there is a smaller probability for an individual voter to be in a pivotal position, thus having the power to manipulate the outcome of the election.
4.1
Impartial culture
The preference data was generated with the impartial culture model in every in-stance of the election model. Although it forms a good medium to experiment with, it remains uncertain if strong conclusions can ever be drawn from data generated with these preferences. Namely, the impartial culture model is a poor represen-tation of the distribution of preferences of real election profiles; real preference distributions are rarely uniform. Not only is a uniform distribution of preferences unrealistic, it also causes some unusual properties of the elections. Statistically, every preference occurs an equal number of times in the profile of every simulated election that uses the impartial culture model. This causes the scores among al-ternatives of any PAR used, to be very close to one another. In turn, more ties will have to be broken, and strategic voting is effective more often. Additionally, the impartial culture model maximizes the rate of majority cycles in the majority graph (Tsetlin et al., 2003), making it even more complex to find a representative aggregated ranking.
Figure 3: Plurality rule average frequencies, using EVC
4.2
Comparing Preference Aggregation Rules
To identify the difference in effect on the number of manipulations that the PARs have, three graphs are displayed. For each graph in figures 4, 5 and 6, data was generated with three different PARs, while using the same RCM.
Copeland’s method and the Borda Count account for closely related manipula-tion frequencies. Copeland’s method is based on the unweighted majority graph: it does not account for the margins between alternatives in pairwise contests. Contrarily, Borda count is based on the weighted majority graph, meaning that the margins between alternatives in pairwise contests have an impact on their scores. However, since the preferences of voters were generated uniformly, using the impartial culture model, these margins remain relatively small. A weighted majority graph with only small margins between alternatives closely resembles its unweighted version. Hence, Copeland’s score and the Borda count behave simi-larly.
The Plurality rule producing higher manipulation frequencies, indicates that it is consistently less strategyproof. Especially in a simulated election, where voters’ preferences are generated with the impartial culture model, the scores of alternatives under the Plurality rule will generally be closer to each other. Voters naturally have a higher probability of altering the outcome of an election where the scores of alternatives are relatively very similar.
Figure 4: Strategies at 5 alternatives, with different PARs, using Kendall’s tau Strikingly, a distinct pattern is noticeable in the frequency levels that Copeland’s rule produces. At even numbers of voters, the frequency levels under Copeland’s rule are relatively higher than at odd numbers of voters. This pattern in the data is most likely related to the fact that having a tie between two alternatives in the majority graph is impossible with an odd number of voters. The reason why only Copeland’s method is consistently influenced by this, suggests that this has to do with how Copeland’s rule handles ties in the majority graph. Under Copeland’s rule, the two alternatives that are tied each receive 0.5 points. This equal treat-ment of alternatives, coupled with the the fact that Copeland’s score is based on the unweighted majority graph, causes the final Copeland scores of the alterna-tives to be closer to each other. Naturally, there are more ways to manipulate the outcome if the scores are more alike.
4.3
Comparing Ranking Correlation Measures
Differences in effects of RCMs are, again, identified by three different graphs. The three graphs in figures 7, 8 and 9 each show data of three election model instances that were each generated with a different RCM, while using the same PAR.
Despite the evident pattern caused by Copeland’s method, Figures 7 and 8 show a similarity in behaviour. With both the Borda count and Copeland’s method, EVC appears to consistently produce the highest average manipulations values.
Figure 5: Strategies at 5 alternatives, with different PARs, using Spearman’s rho This suggests that voters more often view manipulated outcomes as better when using EVC as an RCM, compared to when using Kendall’s tau or Spearman’s rho. Conversely, the Plurality rule (figure 9) shows less distinction between the different RCMs used. This indicates that the outcomes produced by the Plurality rule are less influenced by the RCM used, which could be related to the big role that tie-breaking plays in the Plurality SWF.
4.4
Applicability of RCMs
The original applications of Kendall’s tau and Spearman’s rho were not in social choice theory. These RCMs are merely methods to compare two arbitrary rankings of any data. We’ve established that RCMs like Kendall’s tau and Spearman’s rho are by no means similar to the way a real voter would compare rankings. This is mainly due to their indifference to the differences in importance of ranks between alternatives: it is likely that a real voter would care more about the rank of some alternatives than the rank of others. In an attempt to more closely resemble the way human voters would compare rankings, EVC was developed. EVC partly solves the problem of indifference by assigning weights to each alternative in the ranking of a voter, where the higher the alternative is ranked, the greater their weight becomes. With this, voters care more about the ranking of their most preferred alternatives. Assuming that real voters often do this, EVC provides a
Figure 6: Strategies at 5 alternatives, with different PARs, using EVC more realistic method of comparing ballots.
Nonetheless, there are still some cases that EVC cannot handle. Say, a voter specifically cares about the rank of his least preferred alternative. With EVC, this alternative weighs little, since it is at the bottom of the voter’s ballot. This may lead to EVC producing an inaccurate correlation score.
Additionally, none of the RCMs that were used in this thesis can be used to express inconsistently varying margins of differences in preferences between alternatives. Say, a voter with ballot [a1, a2, a3], wants to express that they greatly
prefer alternative a1 over a2, whilst only slightly preferring a2 over a3. With
Kendall’s tau and Spearman’s rho, these margins are equal across the ballots. With EVC, going from most to least preferred alternative, these margins exponentially decrease consistently.
5
Conclusions
Based on the election model that was created in this thesis, we see a clear and somewhat expected relation between the numbers of alternatives and voters, and the average total manipulations in an election. As the number of alternatives increases, the average total manipulations increases. Contrarily, the average total manipulations decreases as the number of voters increases.
Figure 7: Strategies at 5 alternatives, with different RCMs, using the Borda Count Arguably more interesting, is the effect of different PARs, specifically on the total average strategies of elections. The Plurality rule consistently yields higher total average strategies than the more strategyproof Copeland’s rule and Borda count. This supports the statement that the Plurality rule, despite it being one of the more straight forward PARs, might not be the most representative of the true aggregated preference of a population of voters.
Furthermore, the effect of different ranking correlation measures on the out-come of elections was quantified. With Kendall’s tau and Spearman’s rho ex-hibiting similar effects on the average total strategies. However, Kendall’s tau and Spearman’s rho are not representative of the way in which a real voter would compare two ballots. In an attempt to better represent human voters, a new type of RCM was developed: Exponential Vector Correlation. Strikingly, EVC consis-tently shows higher average total strategies in elections that use Copeland’s score or the Borda count as its PAR. This indicates that voters are more inclined to strategically vote if they use EVC to compare ballots.
Future research
An obvious course of further development of this thesis is generating more data. Because of the exponential rise in computational complexity that comes with in-creasing the number of alternatives, data was only generated for elections with a
Figure 8: Strategies at 5 alternatives, with different RCMs, using Copeland’s Rule limited number of alternatives. It would be insightful to see the development of the data with more alternatives and more voters.
Additionally, only a small number of preference aggregation rules was used in this thesis. This is partly due to their relatively simple implementations. Each PAR implemented in the election model created in this thesis, assigns points to each alternative based on the profile of the election. It could be interesting to see the effect of a rule, like Slater’s rule, that does not use a point-based system. Slater’s rule for example, generates an aggregated ranking by arranging alterna-tives such that it resembles the majority graph as close as possible. The problem that arises here is that rules like Slater’s rule are NP-hard to implement in a computational model.
Lastly, a crucial assumption that was made in every instance of an election created by the election model, is that the distribution of preferences of alternatives is uniform. However, real life elections rarely exhibit preference distributions that correspond to the impartial culture model. Thus, it would be useful to construct models that use a more realistic distribution of preferences. These preference distributions could be induced from real election data.
Figure 9: Strategies at 5 alternatives, with different RCMs, using the Plurality rule
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Appendix
All 3d plots generated by the election model are presented here.
Figure 11: Copeland’s Score average frequencies, using EVC
Figure 13: Borda Count average frequencies, using Kendall’s tau
Figure 15: Plurality rule average frequencies, using Kendall’s tau
Figure 17: Copeland’s Score average frequencies, using Spearman’s rho
