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Analysis of Iterative Process for Nauru Voting System
Neelam Gohar
1
,*
, Sidra Niaz
1
, Mamoona Naveed Asghar
2
and Salma Noor
1
1
Shaheed Benazir Bhutto Women University Peshawar Pakistan
2
Athlone Institute of Technology Athlone, Ireland
Corresponding Author: Neelam Gohar. Email: neelam.gohar@sbbwu.edu.pk
Received: 23 November 2020; Accepted: 29 January 2021
Abstract: Game theory is a popular area of artificial intelligence in which the
voter acknowledges his own desires and favors the person he wants to be his
representative. In multi-agent systems, social choice functions help aggregate
agents’different preferences over alternatives into a single choice. Since all vot-
ing rules are susceptible to manipulation, the analysis of elections is complicated
by the possibility of voter manipulation attempts. One approach to understanding
elections is to treat them as an iterative process and see if we can reach an equili-
brium point. Meir et al. proposed an iterative process to reach a stable outcome,
i.e., Nash Equilibrium. This technique, explored in previous work, converges to a
Nash equilibrium for plurality voting, along with a tie-breaking rule that chooses a
winner according to a linear order of preferences over candidates. Almost all the
scoring rules have been studied in previous work, we identified the iterative pro-
cesses of the Nauru voting system. We analyzed the Nauru voting system with
Copelands and lexicographic rule for tiebreaking. Nauru is the modified version
of Borda counting. Like Borda counting, Nauru voting system scores each candi-
date with different points. In the iterative behavior analysis of the Nauru voting
system, when two or more winning candidates have the same score, a tie occurs.
To break the tie, we use the Copeland method, which is a pairwise comparison to
rank the candidates. If there is still a tie, we break it using the traditional linear
ordering method, the lexicographic rule. We have observed cycles for different
manipulative moves.
Keywords: Multi agent system (MAS); iterative voting; manipulation; game
theory; group decision making
1 Introduction
For decision making in groups, voting system has been used as a tool for centuries [1–3]. Voting and
aggregation mechanisms have also been used for other tasks such as aggregating search results from the
Web [4] and collaborative filtering [5].
Social choice is concerned with the analysis and design of methods for aggregating the preferences of
multiple agents and is considered a fundamental tool for the study of multi-agent systems. A multi-agent
system requires a mechanism to aggregate opinions for a joint decision. One such mechanism is a voting
This work is licensed under a Creative Commons Attribution 4.0 International License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.
Intelligent Automation & Soft Computing
DOI:10.32604/iasc.2021.015461
Article
ech
T
PressScience
system in which each voter ranks several options (such as people, common plans, rooms for a meeting, etc.).
The result is in the form of an option using a voting rule after aggregating the voters’preferences over the
available options. Sometimes voters misstate their preference order to make the outcome more favorable to
them; such type of voting is called strategic or tactical voting.
In social choice theory, the important result of Gibbard-Satterthwaite Theorem [6,7] states that any non-
dictatorial “reasonable”voting rule is susceptible to manipulation, so developing voting models for truthful
voters is pointless. Effective manipulation is related to the knowledge that agents have about the true
preferences of other voters. Much work is based on complete information.
One way to understand the electoral process is to look at the dynamics of manipulation to reach an
equilibrium state where voters are satisfied with their manipulations and no voter wants to switch
preferences. The most stable solution is Nash Equilibrium. It can be used as a tool for analyzing the
election process. Such an approach has been defined in terms of a formal model of iterative voting by
Meir et al. [8], their work can be considered as a decision process where voters can change their vote at
any time (just like in committees or online facilities). A similar process can be seen on various websites
to coordinate dates of an event such as www.doodle.com; after casting an initial vote, each participant is
allowed to change their vote. Players change their choices one at a time; the iterative voting process is
better suited for a small number of agents or when a choice is very close.
Meir et al. [8] showed that under the plurality rule with deterministic tie-breaking, voters must converge
to a Nash equilibrium when choosing the best available answers in a given state. However, convergence for
better answers is not guaranteed under the setting with weighted voters.
Deterministic tie-breaking rule plays an important role in the convergence of equilibria, under positional
scoring rules, convergence is not guaranteed unless we restrict the choice of tie-breaking rule. In the current
paper, we investigate the non-convergence of the Nauru voting system with deterministic tie-breaking rule in
the context of iterative voting for the Nauru voting system.
The paper studied the multiprocessor machine scheduling problems in the context of coordination
mechanisms. The properties of the pure strategy Nash Equilibrium were considered for the selfish
scheduling games. The system converges to a pure strategy Nash Equilibrium in linear time, the
convergence rate is computed by iteratively playing the jobs their best responses starting from an
arbitrary state. This is important for the price of anarchy and upper and lower bounds on the price of
anarchy have also been derived [9].
Voters have incomplete information about the preference of other voters, they only know the score of all
candidates to change their vote. Iterative processes that converge to a steady state or generate cycles are often
needed to analyze the dynamics. Our contribution is the game theoretic analysis of the Nauru voting system
with various manipulative moves in weighted and unweighted settings.
1.1 Contribution
We study iterative voting for the Nauru voting rule in a simple form for a given set of moves, and our
results are for the set of manipulations where the process leads to a cycle. We find that even in this simple
model, the dynamics for the Nauru voting rule do not converge. This is the first time this iterative process for
Nauru has been analyzed, and we take into account that voters have incomplete information about the
preferences of other voters. Our results show that voters can assume declared or truthful preferences and
the process does not converge for a given set of manipulations.
242 IASC, 2021, vol.28, no.1
2 Related Work
The formal model we use is based on the iterative model presented by Meir et al. [8]. The notion of
iterative voting and the existence of equilibria for manipulation dynamics emerge from previous literature.
Voting systems and convergence to equilibria have been studied in computer science [8,10–13]. In group
decision making, an iterative process to reach a stable outcome was operated for agents [14], but the
method used was “value like money”among agents, thus not relevant to a voting system. Airiau et al. [15]
analyzed the possibility of equilibrium in such an iterative process by using voting rules such as plurality.
Achieving equilibrium with an iterative process has been considered by previous researchers, but mainly
when deciding on an allocation of public goods. A summary of most of the work can be found in the work of
Laffont [16], however the preferences chosen are single peaked preferences. Feddersen et al. [17] also
restrict their search for equilibria to single peaked preferences. A more different approach was taken by
Messner and Polborn [18], analyzing equilibria through coalitional manipulation.
Most of the above work assumed that voters have some knowledge about the preferences of the other
players, and this is one of the main limitations of their work.
In terms of knowledge of other players’preferences, Chopra et al. [19] showed the results by restricting a
player’s knowledge of other players’preferences, they analyzed the iterative process for plurality voting rule.
A model was proposed by Myerson and Weber [20], it is assumed that voters have some knowledge about the
winning candidate (e.g., based on pre-election polls), a Nash equilibrium for scoring rules was found, but not
every election leads to an equilibrium. The limitations of the iterative method and the importance of the
equality rule were considered by Lev et al. [21] and it was proved that the use of a constrained equality
rule, such as linear order, does not by itself guarantee convergence. The work of Hwang, Ching Lai et al.
[22]. has given an overview of group decision making under multiple criteria, existing methods,
properties and applicability in the complexity of group decision making. In [23], the authors presented a
multi-agent system for resource allocation and process scheduling in which agents communicate and
cooperate with each other to produce an optimal schedule.
Group decisions were explored in terms of voting properties, which provides well formalized techniques
for preference aggregation. Voting also provides some efficient algorithms for resolving disagreements among
participants. There are many works on game-theoretic analysis of voting that consider voters as strategic agents
and reach Nash Equilibrium as a solution concept for preference aggregation scenarios. The iterative voting
model introduced by Meir et al. [8] is used for non-convergence of Nauru voting system. Another
framework adopted by researchers is adding truth biasness to games proposed by Thompson et al. [24].
Truth biasness means that voters are truthful because they cannot influence the election outcome.
3 Preliminaries
Symbols and notations used in the analysis of Nauru Voting System are discussed here. Some of the
notations and definitions are taken from [8,10,12].
3.1 Nauru Scoring System
Nauru is a position scoring rule. Voters give different points to candidates, such as (A1, B1/2, C1/3, Z.1/n)
A1 means that the voter gives 1 point to the first candidate, B1/2 means that the voter gives 0.50 points to the
second candidate, and so on.
3.2 Preferences
Voters’preferences are the ranking of candidates according to their choice. The two types of voter
preferences are described below.
IASC, 2021, vol.28, no.1 243
3.3 True/Fixed Preference
True preference is the actual ranking of candidates according to their true choice. True preferences are
fixed, voters do not change them and want their favorite candidate to win the election.
3.4 Declared/Stated Preference
In declared preference, voters change their true preference depending on the situation and circumstances.
Declared preferences are associated with a particular state.
3.5 Switching
Switching is the mechanism by which voters change their preference to obtain a desirable candidate or a
favorable election outcome.
3.6 Notation for Ranking
Voters rank candidates by using notation such as“”. A B C means that A is the first choice, B is the
second choice and so on.
3.7 Transition
Transition is the mechanism by which the voter changes his vote and the transition is from one state to
another state. We allow one voter at a time to change his preference list.
3.8 Manipulative Move
Manipulative move is the rational move where the voter changes his preference in support of a preferred
candidate to win the election. There are the different moves defined in [10,12].
The following election rules are used a. Borda election rule b. Nauru scoring system. Copeland method
and lexicographic rule are used to resolve ties.
3.9 Borda Rule
In Borda rule, 0 point is assigned to the last election and 1 point to the penultimate election [25]. Borda
ranks each candidate in whole numbers, starting with the least preferred candidate. The points of each
candidate are added and the candidate with the highest score is the winner of the election [26].
3.10 Example 1
Table 1: Borda Rules
Choice V
1
V
2
V
3
V
4
V
5
V
6
V
7
3A
3
A
3
A
3
A
3
B
3
B
3
B
3
2B
2
B
2
B
2
B
2
D
2
C
2
D
2
1D
1
C
1
D
1
C
1
C
1
D
1
C
1
0C
0
D
0
C
0
D
0
A
0
A
0
A
0
A = 12, B = 17, C = 6, D = 7
B is the Winner according to the Borda rule.
244 IASC, 2021, vol.28, no.1
3.11 Nauru Scoring System
Nauru, the modified form of the Borda rule, awards 1 point for the first choice, 1/2 point for the second
choice, 1/3 for the third choice, 1/ 4 for the fourth choice, 1/5 for the fifth choice, etc. In the Nauru scoring
system, the score is derived by assigning points to the various positions in a predetermined ranking order.
Thus, the points of each candidate are added together and the candidate with the highest score is the
winner [27]. The Borda voting rule ranks candidates in whole numbers and assigns zero to the least
preferred candidate, but in Nauru every candidate is ranked in a decimal point system except the first,
hence it is sometimes called the decimal point system. The following example shows how the Nauru
electoral system works.
3.12 Example 2
No of Candidates = 03 No of Voters = 04 Candidates = A, B and C Voter’s=V
1,
V
2
,V
3
,V
4
Voting rule is Nauru point system and voters start from their declared Preferences.
Candidate B has maximum points so B is the winner of the election.
3.13 Tie Breaking Rules
During the election, when two candidates receive the same points, a tie occurs between the candidates.
To break the tie, we use two methods, the Copeland method and the lexicographic rule. The Copeland
method is a pairwise comparison where one candidate is compared to every other candidate. In [28], we
use the Copeland method as the tie-breaking rule because it performs a comparison between candidates.
A candidate beats every other candidate in the comparison and selects the best one [29]. Copeland
method can select a more desirable winner than just following the linear order.
3.14 Example 3
In above example we have three candidates A, B and C and tie occur between candidates A and B, so we
use the pair wise comparison rule (Copeland’s method) to break the tie between the candidates. In three
candidate case the comparison is A.B, A.C, B.C.
Table 2: Nauru point System
Voters True Preference Declared Preference
V
1
B>A>C A>B>C
V
2
A>B>C B>A>C
V
3
A>C>B C>A>B
V
4
A>B>C B>A>C
Table 2a: State = 0
Choice V
1
V
2
V
3
V
4
1A
1
B
1
C
1
B
1
2B
1/2
A
1/2
A
1/2
A
1/2
3C
1/3
C
1/3
B
1/3
C
1/3
State = 0 A = 2.5 B = 2.83 C = 1.99
IASC, 2021, vol.28, no.1 245
We compare A.B in the above table V
1
prefers candidate A over B, so candidate A gets 1 point V
2
prefers candidate B over A, so candidate B gets 1 point.
A.B A = 2 B = 3 Result: B = 1
If tie occurs between A and B, then each candidate receives 0.5.
A.C A = 1 + 1 + 1 + 1 = 4 C = 1 Result: A = 1
B.C B = 1 + 1 + 1 = 3 C = 1 + 1 = 2 Result: B = 1
A=1 B=1+1=2 C=0
If the tie is not broken by the Copeland method, then we break the tie by the lexicographic rule. The
lexicographic method is the traditional tie-breaking rule and selects candidates according to linear order.
[30] In lexicographic rules, the voting systems are not dictatorial. These classical methods are most
commonly used as tie-breaking rules in plurality voting systems. [10,26]
3.15 Example 4
Let there are three candidates and four voters.
If there is a tie between candidates A and B, we use the Copeland method; if there is another tie between
candidates A and B, we use the lexicographic rule.
Table 3: Copeland’s Method
Choice V
1
V
2
V
3
V
4
V
5
1A
1
B
1
B
1
A
1
C
1
2C
1/2
A
1/2
A
1/2
B
1/2
B
1/2
3B
1/3
C
1/3
C
1/3
C
1/3
A
1/3
A = 3.33 B = 3.33 C = 2.49
Table 4: Lexicographical method
Voters True
Preference
Declared
Preference
V
1
A>B>C A>B>C
V
2
B>A>C B>A>C
V
3
A>C>B A>B>C
V
4
B>A>C B>A>C
Table 4a: State = 0
Choice V
1
V
2
V
3
V
4
1A
1
B
1
A
1
B
1
2B
1/2
A
1/2
B
1/2
A
1/2
3C
1/3
C
1/3
C
1/3
C
1/3
State = 0 A = 3 B = 3 C = 1.32
246 IASC, 2021, vol.28, no.1
3.16 Weighted Voting System
In a weighted voting system, weights are assigned to the voters. Each participant/voter has different
weights. Weighted system means that some preferences have more weight than the preferences of the
other voters. Voters have equal influence on the outcome of an election. Voter weights can represent a
group of like-minded voters who have the same preference. The weights remain the same throughout the
election. In the Nauru electoral system, an individual voter can hardly have a significant impact on the
election outcome, so we assign a weight to voters that is in real positive numbers so that individual
weighted voters can change the election outcome more effectively and efficiently than unweighted voters.
For weighted Nauru voters, we multiply the Nauru numbers by the weight of the voters, which are
converted to floating point numbers. The following example shows how the Nauru voting system works
in a weighted setting.
3.17 Example 5
In order to get the total score of candidates we multiply the Nauru points to the weights of the voters.
3.18 Problem Statement
In this research we have analysed the dynamics of the Nauru electoral system, allowing voters to choose
their favourite candidate in each state and make him their representative. Nauru is the modified version of
Borda and is more accurate. In Borda, voters give 0 point for last preference and 1 point for penultimate
preference but in Nauru voting system, voters give 1 point for first choice and ½ point for second choice
and so on. In this paper, we have used Copeland’s method as a tie rule along with the traditional linear
comparison method, i.e., lexicographic method.
In the election, voters are allowed to change their preferences and try to make their favourite candidate
the winner of the election. The manipulation dynamics of the election for the Nauru electoral system has been
analysed for the first time where the voters have incomplete information, they only know about the current
scores of the candidates. We look for the scenarios where convergence is not guaranteed.
Table 5: Nauru Weighted Point System
Voters True
Preference
Declared
Preference
Weights
V
1
B>A>C A>B>C 3
V
2
C>B>A B>C>A 2
V
3
A>C>B C>A>B 4
V
4
B>A>C A>B>C 2
State = 0 A=8/Winner B = 5.82 C = 6.65
Table 5a: State = 0
Choice V
1
V
2
V
3
V
4
1A
1x3
B
1x2
C
1x4
A
1x2
2B
1/2x3
C
1/2x2
A
1/2x4
B
1/2x2
3C
1/3x3
A
1/3x2
B
1/3x4
C
1/3x2
IASC, 2021, vol.28, no.1 247
3.19 Nauru Weighted Voting System
Example 6 Voters start from their truthful state.
State = 0
True Preference result: In order to get the total score of candidate we multiply the Nauru points to the
weights of the voters.
In this example candidate B is the winner in State S
0
but in state S
1
only voter V
3
can change his
preference from CAB /ACB as a result the outcome of election is changed.
3.20 Example 7
Tie Breaking rules used are Copeland’s method and Lexicographical rule and voters start from their
declared preferences.
Table 6: Nauru Weighted Point System
Voters True Preference True Preference Weights
V
1
A>B>C A>B>C 2
V
2
B>C>A B>C>A 4
V
3
C>A>B C>A>B 3
Table 6a: State = 0
Choice V
1
V
2
V
3
V
4
1A
1x2
B
1x4
C
1x3
A
1x2
2B
1/2x2
C
1/2x4
A
1/2x3
B
1/2x2
3C
1/3x2
A
1/3x4
B
1/3x3
C
1/3x2
State = 0 A = 6.82 B = 6.991 C = 6.32
Table 6b: State =1 V
3
: CAB /ACB
Choice V
1
V
2
V
3
V
4
1A
1x2
B
1x4
A
1x3
A
1x2
2B
1/2x2
C
1/2x4
C
1/2x3
B
1/2x2
3C
1/3x2
A
1/3x4
B
1/3x3
C
1/3x2
State = 1 ACB A = 8.32 B = 6.99 C = 4.82
Table 7: Tie Breaking rules for Nauru weighted point system
Voters True Preference Declared Preference Weights
V
1
B>C>A C>B>A 2
V
2
C>B>A B>C>A 4
V
3
C>B>A B>C>A 2
V
4
B>C>A C>B>A 4
V
5
B>A>C A>B>C 4
V
6
C>A>B A>C>B 4
248 IASC, 2021, vol.28, no.1
State = 0
In order to get the total score of candidate we multiply the Nauru points to the weights of the voters.
According to the lexicographical rule B is the winner of the election.
In this example candidate B is the winner at State S
0
but in state S
1
voter V
6
changes his preference
from ACB /CAB, this transition effects the overall outcome of the election and candidate C becomes
winner of state S
1
.
4 Cycles of Iterative Voting for Nauru Voting System
4.1 Example 8
Let’s voters start from their declared state.
Table 7a: State = 0
Choice V
1
V
2
V
3
V
4
V
5
V
6
1C
1x2
B
1x4
B
1x2
C
1x4
A
1x4
A
1x4
2B
1/2x2
C
1/2x4
C
1/2x2
B
1/2x4
B
1/2x4
C
1/2x4
3A
1/3x2
A
1/3x4
A
1/3x2
A
1/3x4
C
1/3x4
B
1/3x4
State = 0 A = 11.9 B = 12.32 C = 12.32
Table 7b: State = 1
Choice V
1
V
2
V
3
V
4
V
5
V
6
1C
1x2
B
1x4
B
1x2
C
1x4
A
1x4
C
1x4
2B
1/2x2
C
1/2x4
C
1/2x2
B
1/2x4
B
1/2x4
A
1/2x4
3A
1/3x2
A
1/3x4
A
1/3x2
A
1/3x4
C
1/3x4
B
1/3x4
State = 1 V
6
: ACB /CAB A = 9.96 B = 12.32 C = 14 /Winner
Table 8: Cycles of Iterative Voting
Voters True Preference Declared Preference
V
1
C>B>A A>B>C
V
2
C>A>B C>A>B
V
3
B>C>A B>C>A
V
4
A>B>C A>C>B
V
5
C>B>A C>B>A
V
6
B>A>C B>A>C
IASC, 2021, vol.28, no.1 249
The above example shows the cycle for the Nauru voting system when Type 1 and Type 3 moves are
considered and when voters start from their stated preferences.
Table 8a: State = 0
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
C
1
B
1
A
1
C
1
B
1
2B
1/2
A
1/2
C
1/2
C
1/2
B
1/2
A
1/2
3C
1/3
B
1/3
A
1/3
B
1/3
A
1/3
C
1/3
State = 0 A = 3.66 B = 3.66 C = 3.66
Table 8b: State = 1 V1: ABC /ACB
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
C
1
B
1
A
1
C
1
B
1
2C
1/2
A
1/2
C
1/2
C
1/2
B
1/2
A
1/2
3C
1/3
B
1/3
A
1/3
B
1/3
A
1/3
C
1/3
State = 1 A = 3.66 B = 3.49 C = 3.83
Table 8c: State = 2 V4: ACB /ABC
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
C
1
B
1
A
1
C
1
B
1
2C
1/2
A
1/2
C
1/2
B
1/2
B
1/2
A
1/2
3C
1/3
B
1/3
A
1/3
C
1/3
A
1/3
C
1/3
State = 2 A = 3.66 B = 3.66 C = 3.66
Table 8d: State = 3 V1: ACB /ABC
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
C
1
B
1
A
1
C
1
B
1
2B
1/2
A
1/2
C
1/2
B
1/2
B
1/2
A
1/2
3C
1/3
B
1/3
A
1/3
C
1/3
A
1/3
C
1/3
State = 3 A = 3.66 B = 3.83 C = 3.49
Table 8e: State = 4 V4: ABC /ACB
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
C
1
B
1
A
1
C
1
B
1
2B
1/2
A
1/2
C
1/2
C
1/2
B
1/2
A
1/2
3C
1/3
B
1/3
A
1/3
B
1/3
A
1/3
C
1/3
State = 4 A = 3.66 B = 3.66 C = 3.66
250 IASC, 2021, vol.28, no.1
4.2 Example 9
Let’s voters start from their truthful state.
Table 9: Cycle for Nauru Voting System
Voters True Preference True Preference
V
1
A>B>C A>B>C
V
2
B>C>A B>C>A
V
3
C>A>B C>A>B
V
4
A>C>B A>C>B
V
5
B>A>C B>A>C
V
6
C>B>A C>B>A
Table 9a: State = 0
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
B
1
C
1
A
1
B
1
C
1
2B
1/2
C
1/2
A
1/2
C
1/2
A
1/2
B
1/2
3C
1/3
A
1/3
B
1/3
B
1/3
C
1/3
A
1/3
State = 0 A = 3.66 B = 3.66 C = 3.66
Table 9b: State = 1 V2: BCA /CBA
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
C
1
C
1
A
1
B
1
C
1
2B
1/2
B
1/2
A
1/2
B
1/2
A
1/2
B
1/2
3C
1/3
A
1/3
B
1/3
C
1/3
C
1/3
A
1/3
State = 1 A = 3.66 B = 3.16 C = 4.16
Table 9c: State = 2 V4: ACB /ABC
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
B
1
C
1
A
1
B
1
C
1
2B
1/2
C
1/2
A
1/2
B
1/2
A
1/2
B
1/2
3C
1/3
A
1/3
B
1/3
C
1/3
C
1/3
A
1/3
State = 2 A = 3.66 B = 3.33 C = 3.99
Table 9d: State =3 V2: CBA /BCA
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
B
1
C
1
A
1
B
1
C
1
2B
1/2
C
1/2
A
1/2
C
1/2
A
1/2
B
1/2
3C
1/3
A
1/3
B
1/3
B
1/3
C
1/3
A
1/3
State = 3 A = 3.66 B = 3.83 C = 3.49
IASC, 2021, vol.28, no.1 251
The above example shows the non-convergence result for the Nauru voting system when type 1, type 3,
and type 4b moves are considered and when voters assume their truthful state.
Example 8 and 9 show that regardless of whether voters assume their truthful or declared state, the
iterative voting process for Nauru voting does not converge when these three manipulative moves are
used in the unweighted setting. A mechanism is needed to break the cycle and achieve equilibrium.
5 Dynamics for Nauru Weighted Voting System
5.1 Example 10
The moves used in this example are loser-winner, winner-new winner, and winner-existing winner.
Voters start with their stated preferences and use the moves above. The following tables show the
transition from one state to another by executing a manipulative move in the weighted setting.
State = 0
Declared Preference result: In order to get the total score of candidate we multiply the Nauru points to
the weights of the voters. A = 1.83 B = 1.83 C = 1.83
Table 9e: State = 4 V4: ABC /ACB
Choice V
1
V
2
V
3
V
4
V
5
V
6
1A
1
C
1
C
1
A
1
B
1
C
1
2B
1/2
B
1/2
A
1/2
C
1/2
A
1/2
B
1/2
3C
1/3
A
1/3
B
1/3
B
1/3
C
1/3
A
1/3
State = 4 A = 3.66 B = 3.66 C = 3.66
Table 10: Nauru Weighted Voting System
Voters True Preference Declared Preference
V
1
A>B>C B>A>C
V
2
C>A>B A>C>B
V
3
B>C>A C>B>A
Table 10a: State = 0
Choice V
1
V
2
V
3
1B
1
A
1
C
1
2A
1/2
C
1/2
B
1/2
3C
1/3
B
1/3
A
1/3
Table 10b: State = 1 V2: ACB /ABC
Choice V
1
V
2
V
3
1B
1
A
1
C
1
2A
1/2
B
1/2
B
1/2
3C
1/3
C
1/3
A
1/3
State = 1 A = 1.83 B = 2 C = 1.66
252 IASC, 2021, vol.28, no.1
Cycle is generated for the given set of moves.
5.2 Example 11
Moves used are loser to winner, winner to new winner and winner to existing winner. Voters start from
their declared preference list.
Table 10c: State = 2 V1: BAC /ACB
Choice V
1
V
2
V
3
1A
1
A
1
C
1
2C
1/2
B
1/2
B
1/2
3B
1/3
C
1/3
A
1/3
State = 2 A = 2.33 B = 1.33 C = 1.83
Table 10d: State = 3 V2: ABC /ACB
Choice V
1
V
2
V
3
1A
1
A
1
C
1
2C
1/2
C
1/2
B
1/2
3B
1/3
B
1/3
A
1/3
State = 3 A = 1.83 B = 0.99 C = 2
Table 10e: State = 4 V1: ACB /BAC
Choice V
1
V
2
V
3
1B
1
A
1
C
1
2A
1/2
C
1/2
B
1/2
3C
1/3
B
1/3
A
1/3
State = 4 A = 1 + 0.5 + 0.33 = 1.83 B = 1 + 0.5 + 0.33 = 1.83 C = 1 + 0.5 + 0.33 = 1.83
Table 11: Nauru weighted voting system
Voters True
Preference
Declared
Preference
Weights
V
1
A>B>C B>A>C 2
V
2
C>A>B A>C>B 2
V
3
B>C>A C>B>A 2
Table 11a: State = 0
Choice V
1
V
2
V
3
1B
1x2
A
1x2
C
1x2
2A
1/2x2
C
1/2x2
B
1/2x2
3C
1/3x2
B
1/3x2
A
1/3x2
State = 0 A = 3.66 B = 3.66 C = 3.66
IASC, 2021, vol.28, no.1 253
5.3 Example 12
Weighted Nauru point system is used and the weights are real numbers. The moves used in this example
are loser-winner, winner-new winner, winner-existing winner, and winner-loser.
Voters start with their stated preferences.
Table 11b: State = 1 V2: ACB /ABC
Choice V
1
V
2
V
3
1A
1x2
A
1x2
C
1x2
2C
1/2x2
B
1/2x2
B
1/2x2
3B
1/3x2
C
1/3x2
A
1/3x2
A = 3.66 B = 4 C = 3.32
Table 11c: State = 2 V1: BAC /ACB
Choice V
1
V
2
V
3
1B
1x2
A
1x2
C
1x2
2A
1/2x2
B
1/2x2
B
1/2x2
3C
1/3x2
C
1/3x2
A
1/3x2
A = 4.66 B = 2.66 C = 3.66
Table 11d: State = 3 V2: ABC /ACB
Choice V
1
V
2
V
3
1A
1x2
A
1x2
C
1x2
2C
1/2x2
C
1/2x2
B
1/2x2
3B
1/3x2
B
1/3x2
A
1/3x2
State = 3 A = 4.66 B = 2.32 C = 4
Table 11e: State = 4 V1: ACB /BAC
Choice V
1
V
2
V
3
1B
1x2
A
1x2
C
1x2
2A
1/2x2
C
1/2x2
B
1/2x2
3C
1/3x2
B
1/3x2
A
1/3x2
State = 4 A = 2 + 1 + 0.66 = 3.66 B = 2 + 1 + 0.66 = 3.66 C = 2 + 1 + 0.66 = 3.66
Table 12: Weighted Nauru point system
Votes True Preference Declared Preference Weighs
V
1
A>B>C>E>D B>E>C>A>D 1.9
V
2
D>B>A>C>E E>D>B>A>C 1.8
V
3
C>A>E>D>B A>E>C>D>B 1.9
V
4
A>D>B>C>E A>B>C>D>E 1.9
V
5
D>C>B>A>E B>C>D>A>E 1.8
254 IASC, 2021, vol.28, no.1
Table 12a: State = 0
Choice V
1
V
2
V
3
V
4
V
5
1B
1x1.9
E
1x1.8
A
1x1.9
A
1x1.9
B
1x1.8
2E
1/2x1.9
D
1/2x1.8
E
1/2x1.9
B
1/2x1.9
C
1/2x1.8
3C
1/3x1.9
B
1/3x1.8
C
1/3x1.9
C
1/3x1.9
D
1/3x1.8
4A
1/4x1.9
A
1/4x1.8
D
1/4x1.9
D
1/4x1.9
A
1/4x1.8
5D
1/5x1.9
C
1/5x1.8
B
1/5x1.9
E
1/5x1.9
E
1/5x1.8
State = 0 A = 5.175 B = 5.6 C = 3.15 D = 2.83 E = 4.44
Table 12b: State = 1 V2: EDBAC /BDCEA
Choice V
1
V
2
V
3
V
4
V
5
1B
1x1.9
B
1x1.8
A
1x1.9
A
1x1.9
B
1x1.8
2E
1/2x1.9
D
1/2x1.8
E
1/2x1.9
B
1/2x1.9
C
1/2x1.8
3C
1/3x1.9
C
1/3x1.8
C
1/3x1.9
C
1/3x1.9
D
1/3x1.8
4A
1/4x1.9
E
1/4x1.8
D
1/4x1.9
D
1/4x1.9
A
1/4x1.8
5D
1/5x1.9
A
1/5x1.8
B
1/5x1.9
E
1/5x1.9
E
1/5x1.8
State = 1 A = 5.085 B = 6.83 C = 3.39 D = 2.83 E = 3.11
Table 12c: State = 2 V3: AECDB /EACDB
Choice V
1
V
2
V
3
V
4
V
5
1B
1x1.9
B
1x1.8
E
1x1.9
A
1x1.9
B
1x1.8
2E
1/2x1.9
D
1/2x1.8
A
1/2x1.9
B
1/2x1.9
C
1/2x1.8
3C
1/3x1.9
C
1/3x1.8
C
1/3x1.9
C
1/3x1.9
D
1/3x1.8
4A
1/4x1.9
E
1/4x1.8
D
1/4x1.9
D
1/4x1.9
A
1/4x1.8
5D
1/5x1.9
A
1/5x1.8
B
1/5x1.9
E
1/5x1.9
E
1/5x1.8
State = 2 A = 4.135 B = 6.83 C = 3.394 D = 2.83 E = .04
Table 12d: State = 3 V2: BDCEA /EDBCA
Choice V
1
V
2
V
3
V
4
V
5
1B
1x1.9
E
1x1.8
E
1x1.9
A
1x1.9
B
1x1.8
2E
1/2x1.9
D
1/2x1.8
A
1/2x1.9
B
1/2x1.9
C
1/2x1.8
3C
1/3x1.9
B
1/3x1.8
C
1/3x1.9
C
1/3x1.9
D
1/3x1.8
4A
1/4x1.9
A
1/4x1.8
D
1/4x1.9
D
1/4x1.9
A
1/4x1.8
5D
1/5x1.9
C
1/5x1.8
B
1/5x1.9
E
1/5x1.9
E
1/5x1.8
State = 3 A = 4.22 B = 5.63 C = 2.82 D = 2.83 E = 5.39
IASC, 2021, vol.28, no.1 255
5.4 Example 13
Moves = Loser-winner Winner-new Winner Winner –Existing Winner Winner –Loser
Voters start from their declared state
Table 12e: State = 4 V3: EACDB /AECDB
Choice V
1
V
2
V
3
V
4
V
5
1B
1x1.9
E
1x1.8
A
1x1.9
A
1x1.9
B
1x1.8
2E
1/2x1.9
D
1/2x1.8
E
1/2x1.9
B
1/2x1.9
C
1/2x1.8
3C
1/3x1.9
B
1/3x1.8
C
1/3x1.9
C
1/3x1.9
D
1/3x1.8
4A
1/4x1.9
A
1/4x1.8
D
1/4x1.9
D
1/4x1.9
A
1/4x1.8
5D
1/5x1.9
C
1/5x1.8
B
1/5x1.9
E
1/5x1.9
E
1/5x1.8
State = 4 A = 5.175 B = 5.63 C = 3.15 D = 2.83 E = 4.44
Table 13: Nauru Weighted Voting System
Voters True
Preference
Declared
Preference
Weights
V
1
A>B>C B>A>C 1.2
V
2
C>A>B A>C>B 1.2
V
3
B>C>A C>B>A 1.3
Table 13a: State = 0
Choice V
1
V
2
V
3
1B
1x1.2
A
1x1.2
C
1x1.3
2A
1/2x1.2
C
1/2x1.2
B
1/2x1.3
3C
1/3x1.2
B
1/3x1.2
A
1/3x1.3
State = 0 A = 2.23 B = 2.25 C = 2.3
Table 13b: State = 1 V2: ACB /ABC
Choice V
1
V
2
V
3
1B
1x1.2
A
1x1.2
C
1x1.3
2A
1/2x1.2
B
1/2x1.2
B
1/2x1.3
3C
1/3x1.2
C
1/3x1.2
A
1/3x1.3
State = 1 A = 23 B = 2.45 C = 2.1
Table 13c: State = 2 V1: BAC /ACB
Choice V
1
V
2
V
3
1A
1x1.2
A
1x1.2
C
1x1.3
2C
1/2x1.2
B
1/2x1.2
B
1/2x1.3
3B
1/3x1.2
C
1/3x1.2
A
1/3x1.3
State = 2 A = 2.83 B = 1.65 C = 2.3
256 IASC, 2021, vol.28, no.1
Examples 10, 11, 12, and 13 show that the Nauru voting system generates cycles in weighted settings for
various subsets of manipulative moves, regardless of whether we consider real or integer weights. Voters
assume their stated preference and have incomplete information about the preferences of other voters, in
case of ties, equality rules are used and voters choose better and best answers. An interesting direction is
how to break the cycle to converge the process of manipulation.
6 Conclusions
The challenge in the electoral system is the issue of manipulation, with some manipulations being
NP-hard. As an alternative game-theoretic approach to evaluating the process of preference aggregation,
Nash equilibria can be considered as a solution concept for aggregating the preferences of strategic voters.
For the iterative process for Nauru with tie breaking such as lexicographic or Copeland method, cycles
are observed for different sets of moves. The iterative process for Nauru was analysed for the first time and
we consider that voters have incomplete information about candidate preferences. Our results show that
voters may assume a declared or truthful preference, but the process does not converge for a given set of
moves. Further analysis can be done for the Nauru voting rule and can be constrained like convergence in
the case of best-response dynamics. Randomised tie-breaking can also be applied to Nauru. Other profiles
that contain cycles or convergence are of interest for the study of dynamics. In the context of voting
systems, it can also be considered as a method for consensus building and can be used in some
applications such as Doodle for event scheduling.
This analysis of dynamics helps us to compare and select the relevant voting model based on the
properties of Nash Equilibrium or non-convergence. Another direction to explore is the dependence of
convergence or non-convergence on various attributes such as tie-breaking rule or weighted settings.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the
present study.
Table 13d: State = 3 V2: ABC /ACB
Choice V
1
V
2
V
3
1A
1x1.2
A
1x1.2
C
1x1.3
2C
1/2x1.2
C
1/2x1.2
B
1/2x1.3
3B
1/3x1.2
B
1/3x1.2
A
1/3x1.3
State = 3 A = 2.83 B = 1.45 C = 2.5
Table 13e: State = 4 V1: ACB /BAC
Choice V
1
V
2
V
3
1B
1x1.2
A
1x1.2
C
1x1.3
2A
1/2x1.2
C
1/2x1.2
B
1/2x1.3
3C
1/3x1.2
B
1/3x1.2
A
1/3x1.3
State = 4 A = 2.23 B = 2.25 C = 2.3
IASC, 2021, vol.28, no.1 257
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