Clustered Majority Judgement

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

Keywords:

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

Abstract:

In order to overcome the classical methods of judgement, in the literature there is a lot of material about differ-

ent methodology and their intrinsic limitations. One of the most relevant modern model to deal with votation

system dynamics is the Majority Judgement.

It was created with the aim of reducing polarization of the electorate in modern democracies and not to alienate

minorities, thanks to its use of a highest median rule, producing more informative results than the existing al-

ternatives. Nonetheless, as shown in the literature, in the case of multiwinner elections it can lead to scenarios

in which minorities, albeit numerous, are not adequately represented.

For this reason our aim is to implement a clustered version of this algorithm, in order to mitigate these disad-

vantages: it creates clusters taking into account the similarity between the expressed judgements and then for,

each of these created groups, Majority Judgement rule is applied to return a ranking over the set of candidates.

These traits make the algorithm available for applications in different areas of interest in which a decisional

process is involved.

1 INTRODUCTION

Voting rules are different and behave differently ac-

cording to their limitations or sometimes paradoxal

traits. Asking for a more inclusive democracy also

represents a modern citizens’ quest, but what does ex-

actly it mean? First of all, we want to underline why a

majority voting system embodies the best option be-

tween the classical judgement methods.

Consider three agents who express their binary

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

∧ B and A ←→ B. Premised-based rule take majority

decisions on A and B and then infers conclusions on

the other two propositions.

As shown in the table 1, results are quite different

based on the used rule.

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

just one of the single proposition (A), and his judge-

ment doesn’t agree with the outcome, in the other

cases. So, a huge liability of this model could appear:

Agent 2 could think about manipulating the outcome,

pretending a disagreement for A. The premised model

reacts by providing as ﬁnal outcome on 3 agents’

votation a ”No” for both A ∧ B and A ←→ B, as orig-

inally expressed by Agent 2.

In such a way, a strategical approach on voting

could lead to a deviation effect, providing as result

the best tricker’s choice.

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

Looking at the table, we can also highlight another

paradoxal aspect: considering majority-based out-

come, the latest two propositions are inconsistent with

”Yes” value assigned to both A and B.

This is known as discursive dilemma and deals

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).

512

dâ

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

Clustered Majority Judgement.

DOI: 10.5220/0011319400003269

In Proceedings of the 11th International Conference on Data Science, Technology and Applications (DATA 2022), pages 512-519

ISBN: 978-989-758-583-8; ISSN: 2184-285X

Our attempt is not aimed to solve above-mentioned

dilemma, rather joining a more reﬁned majority rule

(Majority Judgement) with cluster approach’s advan-

tages in aggregating similar patterns.

2 MAJORITY JUDGEMENT

2.1 A Brief Overview of Collective

Decision Making

Many business magazines such as Harvard Business

Review and Millionaire.it focus on business meetings,

highlighting how inefﬁciently these meetings are con-

ducted. Business meetings are often perceived as use-

less and unproductive for many reasons: scope meet-

ings are quite vaguely deﬁned, a very large number

of people are involved and there is no such ﬁgure as a

mediator, making decision processes increase in com-

plexity and effort. (Streibel, 2003) provides a guide

on how to plan and conduct a meeting in order to

make it effective. In this work are deﬁned differ-

ent kinds of meeting and the main techniques which

can be used to meet each kind of meeting’s speciﬁc

target. Talking about business meetings and man-

agement, leadership is a key variable. Tannenbaum

and Schmidt (Tannenbaum and Schmidt, 2009) in-

troduce model about leadership styles. The model’s

parameters take into account the leader’s main fea-

tures, his subordinates’ features and the general con-

text. Hence one can easily tell a good leader needs

to know and adopt different leadership styles depend-

ing on different situations. The managerial grid pro-

posed in (Blake et al., 1964) categorize leaders upon

how focused they are across a production-oriented

(completing tasks) and a people-oriented (support-

ing individuals) dimension. In 1973 Vroom and Yet-

ton (Vroom and Yetton, 1973) presented some lead-

ership styles that Vroom and Jabo revised later in

1988. Besides these descriptive decision theories,

which provide theoretical models, many group deci-

sion making techniques have been studied to provide

a means to choose between the alternatives proposed

in a meeting. The most cited decision making tech-

niques in (Verzuh et al., 2021) are based on brain-

storming and mind mapping processes and focus on

priority ranking techniques. The latter are based on

voting techniques, such as Nominal Group Technique,

Paired Comparison Analysis and Grid Analysis. Dur-

ing board of direction’s meetings, voting is often used

to converge into a ﬁnal decision.

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 (e.g sport competition, wine

competitions, etc.), ranking them. Social choice the-

ory’s fundamental problem is to ﬁnd a social decision

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.

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), published in 2007,

Balinski and Laraki brieﬂy describe MJ, moving from

a social choice theory analysis which highlights tradi-

tional voting methods failures. Hence the need for

a voting method where voters evaluate candidates in

terms of a common language rather than simply rank-

ing 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 experiments conﬁrm that,

while there is no method which can completely over-

come strategic voting, majority judgment strongly re-

sists manipulation. Balinski and Laraki present MJ

as a method both for evaluation and ranking of com-

petitors, candidates or alternatives. In (Balinski and

Laraki, 2014) authors explain how electors don’t re-

ally make a personal ranking of candidates, as tradi-

tional methods input assume, and that this is the rea-

son behind the inadequacy of traditional voting mod-

els. Forcing electors to rank candidates leads to inco-

herence, impossibility and incompatibility. 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. Analyzing data from the 2002 election, the au-

thors describe the limits of the system First Pass the

Post (FFP), which allows voters to express just one

preference. Voters are induced to strategic voting:

voting the candidate who is most likely to win against

those deemed worst rather than voting the preferred

candidate. In the 2002 French presidential election,

Jospin, the left’s leading candidate, was eliminated

Clustered Majority Judgement

513

in the ﬁrst round, despite being one of the favorites

according to polls. In the second round Chirac of

the moderate right beat Le Pen, representative of the

far right. The vast majority of Chirac’s votes were

against Le Pen rather than him. They were, therefore,

strategic votes. As a result of the 2002 experience,

in the 2007 elections the number of registered vot-

ers increased sharply, from 41.2 million in 2002 to

44.5 million in 2007 and it was much discussed in the

media whether there could have been strategic votes

aimed at to avoid sensational defeats of the favorite

candidates in the ﬁrst round. According to a poll

conducted on the election day, 30% of French vot-

ers voted strategically in 2007. The minor left candi-

dates, in fact, obtained 27% in 2002 and 11% in 2007,

the minor right 16%. in 2002 and 3% in 2002. This

is a perfect example of Arrow’s paradox: the winner

depends on the presence or absence of candidates, in-

cluding those who have absolutely no chance of win-

ning. Strategic nominations are also encouraged.

2.2 Formal Aspects

To introduce social choice theory formally, consider

a simple decision problem: a collective choice be-

tween two alternatives. The ﬁrst involves imposing

some ‘procedural’ requirements on the relationship

between individual votes and social decisions and

showing that majority rule is the only aggregation rule

satisfying them. May (1952) (May, 1952) (Caroprese

and Zumpano, 2020a) introduced four such require-

ments 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 < v1, v2, ..., vn >, if

the votes for the two alternatives are reversed, the

social decision is reversed too (to treat all alterna-

tives equally), i.e.

f (−v

, −v

, ..., −v

) = − f (v

, v

, ..., v

) (2)

• Positive responsiveness: For any admissible pro-

ﬁle < 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 social

decision does not change in the opposite direc-

tion; 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.

The May theorem (Theorem: ”An aggregation rule

satisﬁes universal domain, anonymity, neutrality, and

positive responsiveness if and only if it is majority

rule”) provides an argument for the majority rule

based on four plausible procedural desires and the

theorem helps us characterize other aggregation rules

in terms of which desiderata they violate.

But that’s with regards to binary choice. Now, we

consider a set N = [1, 2, ..., n] of individuals (n ≥ 2).

Let X = [x, y, z, ...] be a set of social alternatives, for

example possible policy platforms, election candi-

dates, or other(List, 2022). Each individual i ∈ N has

a preference ordering R

over these alternatives that

rapresents a complete and transitive binary relation

on X. For any x, y ∈ X, xR

y means that individual i

weakly prefers x to y. We write xP

y if xR

y and not

x (‘individual i strictly prefers x to y’), and xI

if xR

y and yR

x (‘individual i is indifferent between

x and y’). But we must specify that at the heart of

social choice theory is the analysis of preference ag-

gregation (Caroprese and Zumpano, 2020b), under-

stood as the aggregation of several individuals’ prefer-

ence rankings of two or more social alternatives into a

single, collective preference ranking (or choice) over

these alternatives (Garcia-Bermejo, 2011). In case of

many successful alternatives, we need a more sophis-

ticated model to deal with preferences’ aggregation

(Klaus Nehring a, 2011). 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 subset of size k winning candi-

dates. A voting rule can perform its role on different

types of ordered preferences, even though the most

common refers to a pre-ﬁxed linear order on the alter-

natives. 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 re-

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

(Fabre, 2018). Indeed:

• Representation: for each partition of voters

∈ V (with

≥

k

(3)

at least one successful candidate is elected from

that partition;

DATA 2022 - 11th International Conference on Data Science, Technology and Applications

514

• Proportionality: for each partition of voters

∈ V (with

≥

k

(4)

number of elected candidate is proportional to the

partition’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.

In general, letting α = (α

, α

, ..., α

) be a candidate

A’s set of n grades (written from highest to lowest,

≥ α

i+1

for all i), there is a majority of (at least)

n − k + 1 for n A’s grade to beat most α

and at least

n−k+1

, for all 1 ≤ k ≤

(n+1)

. We call this the (n-k+1)

- majority for [α

, α

n−k+1

As already mentioned any possible ranking tuple that

we choose to describe must follow ordering relations.

So the ranking should respect domination: namely,

evaluate one candidate above another when that can-

didate’s grades dominate the other’s.

The described majority judgement is a single winner

system, found comparing recursively median grade

between candidates: ﬁrst, grades are ordered in

columns from the highest to the lowest according

to the order relation, then the middle column (lower

middle if number of grades are even) with the highest

grade between candidates’row is selected. If there’s

a tie, algorithm keeps on discarding grades equal in

value to the shared median, until one of the tied can-

didate is found to have the highest median. Before

describing how it’s possible to generalize this single

winner system to a multi winner strategy, thanks to the

use of clusters, we focus our attention on how these

works, analyzing in particular K-medoids.

Figure 1: Example with 5 grades, between the dashed lines

it’s reported the median grade. Highest occurrences in

”Good” determines the winner.

3 CLUSTERING APPROACH

3.1 How Clusters Work

There’s no precise deﬁnition of clustering, mostly due

to the huge variety in different clustering algorithms.

We can state that they share the ability to divide data

into groups with some common features. According

to some general traits, we can distinguish types of

clustering:

1. Connectivity Models: data points in a sam-

ple space exhibits similarity according to the dis-

tance between them. Two approaches are equally

valid: bottom-up where each observation constitutes

a group and then pairs of clusters are merged; top-

down, where observation are included in one cluster

and then it’s segregated; but in both approaches is not

included the possibility of modifying a cluster once

created;

2. Distribution Models: once created a cluster,

model check probabilities on observations following a

particular distribution. Good performances are not al-

ways guaranteed since these models are prone to over-

ﬁt data if no constraint on complexity is made;

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

Clustering is the process of grouping a set of objects

in order to have each similar object to each other in

one cluster, that are dissimilar to objects in other clus-

ters.

For our goal, namely selecting winners from a group

of candidates, K-medoids clustering are used, because

Clustered Majority Judgement

515

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.

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.

The K-medoid we use is part of the python sklearn

library (Pedregosa et al., 2011), which is oriented to

machine learning. This library supports partition-

ing around medoids (PAM) (Leonard Kaufman, 2015)

proposed by Kaufman and Rousseeuw (1990), that is

known to be most powerful. 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;

• 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

Multi winner majority judgement exploits clustering

approach to apply to each group majority judgement

(Andrea Loreggia, 2020). Given k the number of

candidates to be elected, algorithm seeks the optimal

number of cluster to create.

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

tant additional requirement: once selected a number

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.

3.4 Algorithm

In order to explain how the algorithm deals with po-

larization problem, most relevant steps are described

in pseudocode and in annotated strides:

1. set the number of winners as maximum number of

clusters;

DATA 2022 - 11th International Conference on Data Science, Technology and Applications

516

2. cluster are created decreasing the previous maxi-

mum number 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 (line 8 in

pseudocode);

3. the function winners calculates the median for ev-

ery created cluster (line 15 of pseudocode);

4. check that winners from cluster are different be-

tween each other (line 29 in pseudocode); in case

it’s not true (condition=”ko” on pseudocode) al-

gorithm 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

3.5 Case Studies

In this section, we describe two interesting compar-

isons of majority judgement (MJ) and clustered ma-

jority judgement (CMJ).

The ﬁrst one is a simulation of the use of MJ for decid-

ing the name of a new security software to be release

for the bank employees (we point out here that is has

been a real life case study but the bank management

asked to hide the references for privacy reasons). The

second scenario we tested MJ is the case of a local

bank BCC Naples that used MJ in order to test if their

credit policy could be encoded in an automatic sys-

tem.

3.5.1 Case Study 1: President of the Republic

Election

In order to test our algorithm, we asked an hetero-

geneous group of voters to express judgements on a

pre-deﬁned list of possible candidates as President of

the Republic before the elections took place. This

list has been created according to the rumours circu-

lated on that period, creating a bias effect on our re-

sults, as it was excluded a possible rielection of Sergio

Mattarella.

In spite of it, we focus on how the algorithm has

worked in order to balance polarization, returning a

subset of winners with size chosen a priori, that we

may interpret as best solutions for majority of people

who took part into the venture.

Input parameters of Clustered Majority Judgement

test are Excellent, Very Good, Good, Acceptable,

Poor, To Reject, No Opinion and the number of win-

ners is set a priori equal to 3. 125 voters took part into

this election and the algorithm form three clusters, ex-

actly like the number of winners.

Testing our algorithm on the described election has

shown how difference preference has a leverage on

judgement aggregation: for example, voters in Clus-

ter 1 are more bound to express ”Good” judgement

for candidates considered neutral in terms of politi-

cal ideas, than the cluster 3 in which voters have a

tendency in judging neutral ones as ”Fair” or ”Poor”.

Cluster 2 has intermediate traits and no particular ten-

dency is emphasized. We can compare CMJ results

Table 2: CMJ results.

Cluster Cluster size Winner

Cluster 1 65 Mario Draghi

Cluster 2 35 Paolo Gentiloni

Cluster 3 25 Anna Finocchiaro

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

plied to voters.

Ranking MJ Candidate

1 Mario Draghi

2 Paolo Gentiloni

3 Emma Bonino

with single-winner MJ ranking, comparing the tables

tab3 and tab2. The comparison shows different re-

sults for the third candidate, highlighting how clus-

Clustered Majority Judgement

517

tering inﬂuences outcome, giving more weight to mi-

norities’ judgement.

3.5.2 Case Study 2: Deciding the Name of

Security Software

Another interesting comparison case is our second

case study, which concerns the application of MJ

and CMJ to a business case where a well-known in-

ternational credit institution shared the need to have

an custom branded proprietary cyber security sys-

tem. The same judgments paradigma of the case

study 1 were used: Excellent, Very Good, Good, Ac-

ceptable, Poor, To Reject, No Opinion. Every voter

will have to respond with these judgments to these

alternatives proposed by managers: 3PM, ARTOOL,

CARLO/CARLOW, KY3P, MEDUSA, PANORAMA,

TRIMTOOL.

The results of MJ for this election appear to lead to a

tie; let’s consider the judgments attributed to the al-

ternative MEDUSA and ARTOOL:

• MEDUSA: {Excellent, Very Good, Acceptable,

Acceptable, Poor, To Reject}

• ARTOOL: {Excellent, Very Good, Acceptable,

Acceptable, Poor, No Opinion}

In this case, that is if two alternatives have the same

majority grade, a ballot eliminates the common ma-

jority grade. If the latter is also common, the proce-

dure is repeated until these bonds are broken, apply-

ing this algorithm, we get the following results:

• MEDUSA: {To Reject}

• ARTOOL: {No Opinion}

MEDUSA has the highest rating, so it wins. This it-

erative proceedings is capable to break the tie, but it

leads to different results, in the case of an even num-

ber of voters, if the highest median rating is consid-

ered as majority grade. This discrepancy is due to the

deﬁnition of median and therefore of majority grade:

in case of an even number of voters, there are two

median judgments and therefore it is difﬁcult to deter-

mine exactly which one to consider as majority grade.

Furthermore, considering the lowest median rating,

there is an absolute majority of ratings at least equal

to the majority grade.

Also in this case we want to see if there are substan-

tial differences in the result using the CMJ. Unlike the

ﬁrst case study, we point out number of voters here is

very small.

By setting 2 as the number of winners, clustered re-

sults is: MEDUSA as the ﬁrst winner and ARTOOL

as the second. As we can see, the results of the CMJ

coincide with that of the traditional MJ. A naive ex-

planation could be linked to the small number of vot-

ers, who show a marked polarization for this vote.

3.6 Case Study 3: Working Hours per

Week

The last case study is a good paradigm for deciding

how to manage working hours in the ofﬁce, given a

ﬁxed number of working hours to be done (18 hours).

In this case, we asked 160 students of University Fed-

erico II of Naples to choose the best combination of

working hours, in presence (P) or with online lectures

(O). We used again the grades Excellent, Very Good,

Good, Acceptable, Poor, To Reject, No Opinion and

the ﬁve options are:

1. 6 hours (P) - 6 hours (O) - 6 hours (P or O)

2. 10 hours (P) - 4 hours (O) - 4 hours (P or O)

3. 8 hours (P) - 6 hours (O) - 4 hours (P or O)

4. 7 hours (P) - 9 hours (O) - 2 hours (P or O)

5. 5 hours (P) - 5 hours (O) - 8 hours (P or O)

The results of MJ method, with the traditional com-

pute of medians takes back as winner the option 4 (7

hours (P) - 9 hours (O) - 2 hours (P or O)) that has the

highest number of ”Good” votes.

Instead the compute of winner with CMJ method

takes back a different situation: we ﬁxed 2 as num-

ber of winners (and number of clusters) and the ﬁrst

one is the option 3 (8 hours (P) - 6 hours (O) - 4 hours

(P or O)) and the second one is the option 4, the same

winner of MJ method.

As we can see, probably because the number of voters

is quite high, the results are not the same like in case

study 2. With CMJ, we take into account the wide

spectrum of preferences, with special regards for the

most polarising ones, which are the most inﬂuent in

creating different clusters.

Especially for this reason, we may prefer CMJ to

MJ for this case-study’s lookalike situations, where a

shared solution should be taken, considering the dif-

ferent impact it can have on the heterogenous groups

(clusters) the judgement is made by.

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, Ma-

jority 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 clustering ap-

proach has been reported in order to choose the ﬁttest

one for our generalization of Majority Judgement as a

DATA 2022 - 11th International Conference on Data Science, Technology and Applications

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multi-winner strategy. After that, three different case

studies are reported, with a particular attention to the

comparison between MJ and CMJ results.

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 winners, 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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