Decision Making with Clustered Majority Judgment

Emanuele D’ajello

, Davide Formica

, Elio Masciari

, Gaia Mattia

, Arianna Anniciello

Cristina Moscariello

, Stefano Quintarelli

and Davide Zaccarella

University of Napoli Federico II, Napoli, Italy

Copernicani, Milano, Italy

Keywords:

Decision Making, Social Choice, Cluster, Majority Judgement, K-Medoids.

Abstract:

In order to make a decision process that can best represent the will of a group of people who express them-

selves about something, like the election of a president or any other situation where people make judgements

about more than two possibilities, this paper wants to propose the usage of unsupervised learning techniques,

in particular cluster techniques, to extend a single-winner voting system Majority Judgement to a multi-winner

system which aggregate the preferences of subsets of voters. After an introduction about Majority Judgement,

the algorithm used for its clustered version is presented. In the end, a case study will be reported to highlight

the differences with the classic Majority Judgment, since sometimes it could be preferable based on the con-

tingencies of the particular election, especially when there is a desire not to neglect minority groups with the

same preferences.

1 INTRODUCTION

In this work we ﬁrst describe general behiavour and

advantages of Majority Judgement.

Limits of this model are shown in the case of mul-

tiwinner elections as it can lead to scenarios in which

minorities, albeit numerous, are not adequately repre-

sented.

For this reason our aim is to implement a clus-

tered version of this algorithm, in order to mitigate

these disadvantages: it creates clusters taking into

account the similarity between the expressed prefer-

ences and then for, each of these created groups, Ma-

jority Judgement rule is applied to return a ranking

over the set of candidates. These traits make the al-

gorithm available for applications in different areas

of interest in which a decisional process is involved.

Different voting rules provides different results. Their

use depends on the main characteristics we would like

to have during a decision process. For example, we

could be more interested in avoiding tactical strategy,

while accepting some limits about how representa-

tive judgements are. We want to explore an example

of this trade-off and then describe a ’more inclusive’

strategy, using clustering applied to majority judge-

ment. Consider three agents who express their binary

judgement (”Yes” or ”No”) for four statements A, B,

A ∧ B and A ←→ B, comparing outcomes from two

different rules. Premised-based rule ﬁrst take major-

ity decisions on A and B and then infers conclusions

on the other two propositions.

As shown in the table 1, results are quite different

considering the used role.

We now focus on Agent 2 case: he’s represented

in just one of the propositions (A), and his judgement

doesn’t agree with the outcome, in all the other cases.

Here appears clear that Agent 2 could think about

manipulating the outcome, pretending a disagreement

for A. As consequence, the premised model reacts

by providing as ﬁnal outcome on 3 agents’ votation

a ”No” for both A ∧ B and A ←→ B, as originally

expressed by Agent 2.

Table 1: Three agent case of voting.

A B A ∧ B A ←→ B

Agent 1 Yes Yes Yes Yes

Agent 2 Yes No No No

Agent 3 No Yes No No

Premised rule Yes Yes Yes Yes

Majority Yes Yes No No

In such a way, by strategically voting, Agent 2

could manipulate ﬁnal results. This is the major draw-

back of using premised voting as rule.

On the other hand, we can highlight a paradoxal

aspect if we consider the majority rule: outcomes of

the latest two propositions are inconsistent with ”Yes”

value assigned to both A and B.

This is known as discursive dilemma and deals

134

D’ajello, E., Formica, D., Masciari, E., Mattia, G., Anniciello, A., Moscariello, C., Quintarelli, S. and Zaccarella, D.

Decision Making with Clustered Majority Judgment.

DOI: 10.5220/0011524600003335

In Proceedings of the 14th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2022) - Volume 3: KMIS, pages 134-140

ISBN: 978-989-758-614-9; ISSN: 2184-3228

with inconsistency problem in judgement aggregation

based on majority rule (G. Bellec, 2020).

Both premised and majority rule present draw-

backs, but the latter has one important feature: it

doesn’t suffer from deﬁciency shown by the ﬁrst, so

that, if an Agent care about the number of proposi-

tions agreeing with his own judgement, then it is al-

ways in his best interest to report his true preference.

For this reason we focus our attention on majority rule

as a transparent asset in decisional process, while try-

ing to deal with its intrinsic problems related to judge-

ment aggregation (Kleinberg, 2002).

Our attempt is not aimed to solve above-

mentioned dilemma, rather joining a more reﬁned

majority rule (Majority Judgement) with cluster ap-

proach’s advantages in aggregating similar patterns.

2 STRATEGIES OF DECISION

MAKING

2.1 Collective Decision Process and

Majority Judgement

Business meetings are often perceived as useless

and unproductive. Moreover, strong difﬁculties arise

when there is an important decision to take: when

complexity and effort increase, it’s more likely to cre-

ate ’clusters’ representing opposite opinions. More

often, ﬁnal decision becomes leader’s task. The

biggest difﬁculty is to decide for the best alternative:

in many contexts, more than maximizing the number

of people in agreement with the taken decision, it’s

about making all different groups of people feel in-

cluded in the decision process. This can be achieved

both with leadership, that makes subordinates feel im-

portant, and with an inclusive criterion we identify in

this paper as the clustered version of Majority Judge-

ment.

Social choice theory studies methods to consoli-

date the different views of many individuals into a sin-

gle outcome. The main applications of social choice

theory are voting and jury decisions(Brandt et al.,

2016).

During voting, electors in a democracy choose one

candidate among a list of many candidates, while in

a jury decision the individual judges evaluate com-

petitors in a competition, ranking them. Social choice

theory’s fundamental problem is to ﬁnd a social deci-

sion function that elaborates the preference of judges

or voters converging into a jury or electoral decisions

while adhering to the main principles of fair voting

procedures such as non-dictatorship, universality, in-

dependence of irrelevant alternative. Arrow’s impos-

sibility theorem shows that the fundamental problem

has no acceptable solution in the traditional model

(Arrow, 2012). In (Seraﬁni, 2019) Condorcet and

Borda methods and limits, Arrow’s impossibility the-

orem and Majority Judgement are illustrated. The

results of the general elections have shown that vot-

ing systems can run into the Arrow’s paradox. A fa-

mous example is the 2000 US presidential election.

The presence of a minor candidate, Ralph Nader, who

had no chance of winning, made Bush the winner in

Gore’s place. Given the political positions of Nader

and Gore, it is very likely that Nader votes they would

have gone to Gore if Nader hadn’t shown up. It is

also likely that Nader supporters preferred Gore to

Bush. The American ﬁrst past the post electoral sys-

tem, whereby only one vote can be cast and the can-

didate who gets the most votes wins, has not allowed

voters to fully express their preferences.

Majority Judgement (MJ) is a voting technique

proposed by two mathematicians in 2007, Michel

Balinski and Rida Laraki, aiming to overcome tradi-

tional voting methods’ paradoxes and inconsistencies.

In (Balinski and Laraki, 2007), Balinski and Laraki

brieﬂy describe MJ, moving from a social choice the-

ory analysis which highlights traditional voting meth-

ods failures. Arrow’s impossibility theorem shows

that the fundamental problem has no acceptable so-

lution in the traditional model. Practice, otherwise,

suggests a different input formulation, traditionally

expressed as a preference ranking. Measuring and

voting is used during sports competitions such as ice-

skating, gymnastics, or wine competitions.

Measuring occurs when a common language is de-

ﬁned, either if this is a quantitative or qualitative lan-

guage. In this perspective, Arrow’s theorem can be

interpreted as follows: in absence of a common lan-

guage a coherent collective decision cannot be made.

Hence the need for a voting method where voters eval-

uate candidates in terms of a common language rather

than simply ranking them. MJ makes it possible,

since this method asks for electors/judges to express

a judgment on all the candidates/competitors, using

a known common language. Theorems and experi-

ments conﬁrm that, while there is no method which

can completely overcome strategic voting, majority

judgment strongly resists manipulation. Balinski and

Laraki present MJ as a method both for evaluation and

ranking of competitors, candidates or alternatives. In

(Balinski and Laraki, 2014) authors underline that the

assumption of traditional methods that electors don’t

really make a personal ranking of candidates, is false

and the reason behind the inadequacy of traditional

voting models. Forcing electors to rank candidates

Decision Making with Clustered Majority Judgment

135

leads to incoherence, impossibility and incompatibil-

ity. Balinski and Laraki (Balinski and Laraki, 2011)

present the case of the French presidential elections

of 2002 and the results experiment related to the MJ

conducted on the occasion of the French presidential

elections of 2007, that is a perfect example of Arrow’s

paradox: the winner depends on the presence or ab-

sence of candidates, including those who have abso-

lutely no chance of winning.

2.2 Social Theory’s Requirements

To introduce social choice theory formally, consider

a simple decision problem: a collective choice be-

tween two alternatives. The ﬁrst involves impos-

ing some ‘procedural’ requirements on the relation-

ship between individual votes and social decisions

and showing that majority rule is the only aggrega-

tion rule satisfying them. May (1952) (May, 1952)

(Caroprese and Zumpano, 2020) introduced four such

requirements for majority voting rule must satisﬁes:

• Universal Domain: the domain of admissible in-

puts of the aggregation rule consists of all logi-

cally possible proﬁles of votes < v

, v

, ...,v

where each v

∈ [−1, 1] (to cope with any level of

‘pluralism’ in its inputs);

• Anonimity: applying any kind of permutation on

individual preferences does not affect the outcome

(to treat all voters equally), i.e.,

f (v

, v

, ..., v

) = f (w

, w

, ..., w

) (1)

• Neutrality: each alternative has the same weight

and for any admissible proﬁle < v − 1, v

, ..., v

if the votes for the two alternatives are reversed,

the social decision is reversed too (to treat all al-

ternatives equally), i.e.

f (−v

, −v

, ..., −v

) = − f (v

, v

, ..., v

) (2)

• Positive Responsiveness: For any admissible

proﬁle < v

, v

, ..., v

>, if some voters change

their votes in favour of one alternative (say the

ﬁrst) and all other votes remain the same, the so-

cial decision does not change in the opposite di-

rection; if the social decision was a tie prior to

the change, the tie is broken in the direction of

the change, i.e., if w

> v

for some i and w

= v

for all other j] and f (v

, v

, ..., v

) = 0 or 1, then

f (w

, w

, ..., w

) = 1.

A multi-winner election (V,C,F,k) is deﬁned by a set

of voters V expressing preferences over a number of

candidates C, and then a voting rule F returns a sub-

set of size k winning candidates. A voting rule can

perform its role on different types of ordered prefer-

ences, even though the most common refers to a pre-

ﬁxed linear order on the alternatives. In most of cases,

these are chosen a priori.

Formally we denote set of judgements performed

by the i-th voter as proﬁle preferences P

. Each proﬁle

contains information about the grade of candidates by

voters. The voting rule F associates with every proﬁle

P a non-empty subset of winning candidates.

In multi-winner elections more precise traits are

required, compared to the ones stated in May’s theory

(Fabre, 2018). Indeed:

• Representation: for each subset of voters

∈ V (with

≥

k

(3)

at least one successful candidate is elected from

that partition;

• Proportionality: for each subset of voters

∈ V (with

≥

k

(4)

number of elected candidate is proportional to the

subset’s size.

An implicit assumption so far has been that pref-

erences are ordinal and not interpersonally compa-

rable: preference orderings contain no information

about each individual’s strength or about how to com-

pare different individuals’ preferences with one an-

other. Statements such as ‘Individual 1 prefers alter-

native x more than Individual 2 prefers alternative y’

or ‘Individual l prefers a switch from x to y more than

Individual 2 prefers a switch from x* to y*’ are con-

sidered meaningless. In voting contexts, this assump-

tion may be plausible, but in welfare-evaluation con-

texts - when a social planner seeks to rank different

social alternatives in an order of social welfare - the

use of richer information may be justiﬁed.

2.3 Single-winner Majority Judgement

In order to describe the majority judgement, we need

to use a table that refers to ranking for all the candi-

dates C, by using tuples (Balinski, 2006). Suppose

having six possible choices we may use the words:

excellent, very good, good, discrete, bad, very bad.

So each candidate is described by a bounded set of

vote.

It is a single winner system, found comparing recur-

sively median grade between candidates: ﬁrst, grades

are ordered in columns from the highest to the lowest

according to the order relation, then the middle col-

umn (lower middle if number of grades are even) with

the highest grade between candidates’row is selected.

KMIS 2022 - 14th International Conference on Knowledge Management and Information Systems

136

If there’s a tie, algorithm keeps on discarding grades

equal in value to the shared median, until one of the

tied candidate is found to have the highest median.

It’s possible to generalize this system to a multi-

winner strategy. The use of a particular cluster in such

a contest is crucial, so we ﬁrst describe the possible

options and then we motivate our choice, explaining

how K-medoids work.

3 CLUSTERS

3.1 Categories of Clusters

We can state that different types of cluster share the

ability to divide data into groups with some common

features. We can distinguish:

1. Connectivity Models: distance between data

points is computed and according to this they show

similarity.

Two approaches are equally valid: bottom-up

where each observation constitutes a group and then

pairs of clusters are merged; top-down, where obser-

vation are included in one cluster and then it’s segre-

gated; this kind of model is not ﬂexible as there is no

chance to modify cluster once created;

2. Distribution Models: in this case, probabilities

are computed. They refer to the belonging of a partic-

ular distribution once the cluster is created. Applying

distribution methods sometimes can be risky as they

are prone to overﬁt data if a precise constraint on com-

plexity is given;

3. Density Models: areas of higher density are identi-

ﬁed and local cluster are there created, while remain-

ing data can be grouped into arbitrary shaped region,

with no assumption about da ta distribution; for their

ﬂexibility, these models are ﬁt to handle noise better

than organizing data on ﬁxed required body.

Since we would like to model clusters that satisfy re-

quirements expressed before, based on pretty ﬁxed

structure with no assumption about distribution fol-

lowed by data, it seems more accurate considering a

different class of clustering algorithm known as cen-

troid models.

3.2 K-Medoids

For our goal, namely selecting winners from a group

of candidates, K-medoids clustering are used, because

medoids are the representative objects that are consid-

ered, in order to have a result that belongs to the group

of candidates: it is based on the most centrally located

object in a cluster, so it is less sensitive to outliers

in comparison with the K-means clustering, which is

not the best model in our case since it could result in

something that is not present in the candidate list due

to the fact that is an average-based method rather than

median. In fact, the medoid is a data point (unlike

the centroid) which has the least total distance to the

other members of its cluster (Fazzinga et al., 2013).

Another advantage for this choice is that the mean

of the data points is a measure that gets highly af-

fected by the extreme points; so, in K-Means algo-

rithm, the centroid may get shifted to a wrong po-

sition and hence result in incorrect clustering if the

data has outliers because then other points will move

away from. On the contrary, the K-Medoids algo-

rithm is the most central element of the cluster, such

that its distance from other points is minimum. Thus,

K-Medoids algorithm is more robust to outliers and

noise than K-Means algorithm (Ceci et al., 2015).

The used K-medoid algorithm is part of the python

sklearn library (Pedregosa et al., 2011), which is

oriented to machine learning. This library supports

partitioning around medoids (PAM) (Leonard Kauf-

man, 2015) proposed by Kaufman and Rousseeuw

(1990). The workﬂow of PAM is described below

(Hae-Sang Park, 2008).

The PAM procedure consists of two phases: BUILD

and SWAP:

• In the BUILD phase, primary clustering is per-

formed, during which k objects are successively

selected as medoids.

• The SWAP phase is an iterative process in which

the algorithm makes attempts to improve some of

the medoids. At each iteration of the algorithm,

a pair is selected (medoid and non-medoid) such

that replacing the medoid with a non-medoid ob-

ject gives the best value of the objective function

(the sum of the distances from each object to the

nearest medoid). The procedure for changing the

set of medoids is repeated as long as there is a

possibility of improving the value of the objective

function.

Suppose that n objects having p variables each should

be grouped into k (k < n) clusters, where k is known.

Let us deﬁne j-th variable of object i as X

i j

(i = 1, ..., n;

j = 1, ..., p). As a dissimilarity measure is used the

Euclidean distance, that is deﬁned, between object i

and object j, by:

i j

∑

a=1

− X

)

(5)

where i and j range from 1 to n. The medoids is se-

lected in this way:

• calculate the Euclidean distance between every

pair of all objects;

Decision Making with Clustered Majority Judgment

137

• calculate v

∑

i=1

i j

∑

l=1

;

• sort all v

for j = 1, ..., n in ascending order and

select the ﬁrst k object that have smallest initial

medoids value;

• from each object to the nearest medoid we can ob-

tain the initial cluster result;

• calculate the sum of distances from all objects to

their medoids;

• update the current medoid in each cluster by re-

placing with the new medoid, selected minimiz-

ing the total distance from a certain object to other

objects in its cluster;

• assign each object to the nearest medoid and ob-

tain the cluster result;

• calculate the sum of distance from all objects to

their medoids, so if the sum is equal to the pre-

vious one, then stop the algorithm; otherwise, go

back to the update step.

In our case, prior knowledge about the number of win-

ners is required, and identiﬁed clusters are restricted

in minimum size that is number of voters on the num-

ber of candidates (

3.3 Clustered Majority Judgement

For each cluster majority judgement is applied, and a

ﬁnal ranking of candidates is returned (Andrea Loreg-

gia, 2020). Given k the number of candidates to be

elected, algorithm seeks the optimal number of clus-

ter to create.

This ranges from 1 to k and has to satisfy an im-

portant additional requirement: once selected a num-

ber of clusters, if a tie occurs and so k’ vacant seats are

left, algorithm is repeated k’ times until tie’s broken.

In case there’s no broken tie, ﬁxed number of cluster

is changed.

In order to explain how the algorithm deals with

polarization problem, most relevant steps are de-

scribed in pseudocode and in annotated strides:

1. set the number of winners as maximum number of

clusters;

2. cluster are created decreasing the maximum num-

ber of clusters until the optimal number is not

achieved. This number is bound by the size

of cluster, that satisﬁes the following proportion:

number of voters : number of winners = number

of voters in one cluster : one winner;

3. the function winners calculates the median for ev-

ery created cluster;

4. check that winners from cluster are different be-

tween each other ; in case it’s not true (condi-

tion=”ko” on pseudocode) algorithm goes back to

step 2 with a maximum number of cluster equal

to number of vacant seats and the proceedings are

held until all seats have been ﬁlled.

Algorithm 1.

Require: k ≥ 0

Ensure: n winners = (n

, ..., n

), k > 1

k ← number winners

max cluster ← k

condition ← ”ko”

while condition = ”ko” do

cluster list ← cluster(vote list)

for all list cluster do

winners per cluster ←

compute winners(cluster)

all winners ←

list o f all winners(winners per cluster)

end for

list winner distinct =

list o f all distinct winners(all winners)

option remaining ← number winners −

len(list winner distinct)

if option remaining = 0 then

condition =

′

else

k ← option remaining

condition ←

′

end if

end while

3.4 Case Study: Using Clustered

Majority Judgement to Maximize

Agreement

In this section, we describe an interesting compar-

isons of majority judgement (MJ) and clustered ma-

jority judgement (CMJ).

In order to test our algorithm, we asked a group

of people about their preferences on food. If the pre-

sented dish is not considered at least acceptable by

people, it would be discarded. So the aim is to max-

imize the number of not discarded dishes, with two

common choices for all the participants. Input param-

eters of Clustered Majority Judgement test are Excel-

lent, Very Good, Good, Acceptable, Poor, To Reject,

No Opinion and the number of winners is set a priori

equal to 2. 63 voters took part into this experiment

and the algorithm form two clusters, exactly like the

number of winners.

KMIS 2022 - 14th International Conference on Knowledge Management and Information Systems

138

Table 2: CMJ results.

Cluster Cluster size Winner

Cluster 1 33 Bovine meat

Cluster 2

30 Tuna

Table 3: Top 2 of single-winner Majority Judgement ap-

plied to voters.

Ranking MJ Candidate

1 Bovine meat

2 Chicken

We can compare CMJ results with single-winner

MJ ranking, comparing Table 3 3 and Table 2 2.

Expressed judgements are very polarizing and the

two formed cluster seems in opposition between each

other, since the most preferred dishes for one are the

most negatively judged by the other. For this rea-

son, we notice both for Majority Judgement and Clus-

tered Majority Judgement the tendency to avoid the

favourite dishes, focusing on the moderate ones.

In case of Majority Judgement, the solution is

Bovine meat and Chicken, where both alternatives

are considered not acceptable for cluster 2 and for

this reason, 29 dishes would be discarded. With clus-

tered approach, the solution takes into account cluster

2’s preferences, and provides a lower number of dis-

carded dishes.

4 CONCLUSIONS

In section 1, we dealt with logical issues involved in

voting rules and judgement aggregation, highlighting

majority rule’s resistance to strategical vote.

In section 2, a more ﬁned model of majority rule,

Majority Judgement, has been presented as an option

to better estimate the most shared candidate.

In section 3, the related works have been shown

and in section 4, all possible categories of cluster-

ing approach has been reported in order to choose the

ﬁttest one for our generalization of Majority Judge-

ment as a multi-winner strategy. After that, a case

study is reported, with a particular attention to the

comparison between MJ and CMJ results.

The CMJ, as shown, represents the optimal com-

promise in case of polarized groups (clusters), so, in

this situations, this method could be the preferable

choice in order to take into account minorities judge-

ments.

In spite of non-deterministic nature of K-Medoids,

Clustered Majority Judgement is thought to be used

in high populated disputes. For these reasons, we

feel conﬁdent about clustering’s role of taking into

account all different perspectives could be shown in

such situation.

Moreover, our implementation is not strictly

linked to political ﬁeld, as is clearly shown in the

case studies (except the ﬁrst one), mostly because it

requires only some ﬁxed parameters: number of win-

ners, number of grades and grades themselves.

An important future challenge could be speeding

up the algorithm or making a more ﬂexible structure,

even though all the constraints already explained in

previous sections need to be satisﬁed.

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