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 voting 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 equilibrium 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 processes 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 candidate 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.

For decision making in groups, voting system has been used as a tool for centuries [

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 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 [

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. [

Meir et al. [

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 [

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.

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.

The formal model we use is based on the iterative model presented by Meir et al. [

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 [

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. [

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. [

Symbols and notations used in the analysis of Nauru Voting System are discussed here. Some of the notations and definitions are taken from [

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.

Voters’ preferences are the ranking of candidates according to their choice. The two types of voter preferences are described below.

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.

In declared preference, voters change their true preference depending on the situation and circumstances. Declared preferences are associated with a particular state.

Switching is the mechanism by which voters change their preference to obtain a desirable candidate or a favorable election outcome.

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.

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.

The following election rules are used a. Borda election rule b. Nauru scoring system. Copeland method and lexicographic rule are used to resolve ties.

In Borda rule, 0 point is assigned to the last election and 1 point to the penultimate election [

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
V_{7} |
---|---|---|---|---|---|---|---|

3 | A_{3} |
A_{3} |
A_{3} |
A_{3} |
B_{3} |
B_{3} |
B_{3} |

2 | B_{2} |
B_{2} |
B_{2} |
B_{2} |
D_{2} |
C_{2} |
D_{2} |

1 | D_{1} |
C_{1} |
D_{1} |
C_{1} |
C_{1} |
D_{1} |
C_{1} |

0 | C_{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.

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 [

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.

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 |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
---|---|---|---|---|

1 | A_{1} |
B_{1} |
C_{1} |
B_{1} |

2 | B_{1/2} |
A_{1/2} |
A_{1/2} |
A_{1/2} |

3 | C_{1/3} |
C_{1/3} |
B_{1/3} |
C_{1/3} |

Candidate B has maximum points so B is the winner of the election.

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 [

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
---|---|---|---|---|---|

1 | A_{1} |
B_{1} |
B_{1} |
A_{1} |
C_{1} |

2 | C_{1/2} |
A_{1/2} |
A_{1/2} |
B_{1/2} |
B_{1/2} |

3 | B_{1/3} |
C_{1/3} |
C_{1/3} |
C_{1/3} |
A_{1/3} |

A = 3.33 B = 3.33 C = 2.49

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.

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. [

Voters | TruePreference | DeclaredPreference |
---|---|---|

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 |

Let there are three candidates and four voters.

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
---|---|---|---|---|

1 | A_{1} |
B_{1} |
A_{1} |
B_{1} |

2 | B_{1/2} |
A_{1/2} |
B_{1/2} |
A_{1/2} |

3 | C_{1/3} |
C_{1/3} |
C_{1/3} |
C_{1/3} |

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.

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.

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 |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
---|---|---|---|---|

1 | A_{1x3} |
B_{1x2} |
C_{1x4} |
A_{1x2} |

2 | B_{1/2x3} |
C_{1/2x2} |
A_{1/2x4} |
B_{1/2x2} |

3 | C_{1/3x3} |
A_{1/3x2} |
B_{1/3x4} |
C_{1/3x2} |

In order to get the total score of candidates we multiply the Nauru points to the weights of the voters.

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.

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 |

True Preference result: In order to get the total score of candidate we multiply the Nauru points to the weights of the voters.

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
---|---|---|---|---|

1 | A_{1x2} |
B_{1x4} |
C_{1x3} |
A_{1x2} |

2 | B_{1/2x2} |
C_{1/2x4} |
A_{1/2x3} |
B_{1/2x2} |

3 | C_{1/3x2} |
A_{1/3x4} |
B_{1/3x3} |
C_{1/3x2} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
---|---|---|---|---|

1 | A_{1x2} |
B_{1x4} |
A_{1x3} |
A_{1x2} |

2 | B_{1/2x2} |
C_{1/2x4} |
C_{1/2x3} |
B_{1/2x2} |

3 | C_{1/3x2} |
A_{1/3x4} |
B_{1/3x3} |
C_{1/3x2} |

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.

Tie Breaking rules used are Copeland’s method and Lexicographical rule and voters start from their declared preferences.

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 |

In order to get the total score of candidate we multiply the Nauru points to the weights of the voters.

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | C_{1x2} |
B_{1x4} |
B_{1x2} |
C_{1x4} |
A_{1x4} |
A_{1x4} |

2 | B_{1/2x2} |
C_{1/2x4} |
C_{1/2x2} |
B_{1/2x4} |
B_{1/2x4} |
C_{1/2x4} |

3 | A_{1/3x2} |
A_{1/3x4} |
A_{1/3x2} |
A_{1/3x4} |
C_{1/3x4} |
B_{1/3x4} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | C_{1x2} |
B_{1x4} |
B_{1x2} |
C_{1x4} |
A_{1x4} |
C_{1x4} |

2 | B_{1/2x2} |
C_{1/2x4} |
C_{1/2x2} |
B_{1/2x4} |
B_{1/2x4} |
A_{1/2x4} |

3 | A_{1/3x2} |
A_{1/3x4} |
A_{1/3x2} |
A_{1/3x4} |
C_{1/3x4} |
B_{1/3x4} |

_{6}: ACB → CAB A = 9.96 B = 12.32 C = 14 → Winner

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}.

Let’s voters start from their declared state.

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 |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
C_{1} |
B_{1} |
A_{1} |
C_{1} |
B_{1} |

2 | B_{1/2} |
A_{1/2} |
C_{1/2} |
C_{1/2} |
B_{1/2} |
A_{1/2} |

3 | C_{1/3} |
B_{1/3} |
A_{1/3} |
B_{1/3} |
A_{1/3} |
C_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
C_{1} |
B_{1} |
A_{1} |
C_{1} |
B_{1} |

2 | C_{1/2} |
A_{1/2} |
C_{1/2} |
C_{1/2} |
B_{1/2} |
A_{1/2} |

3 | C_{1/3} |
B_{1/3} |
A_{1/3} |
B_{1/3} |
A_{1/3} |
C_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
C_{1} |
B_{1} |
A_{1} |
C_{1} |
B_{1} |

2 | C_{1/2} |
A_{1/2} |
C_{1/2} |
B_{1/2} |
B_{1/2} |
A_{1/2} |

3 | C_{1/3} |
B_{1/3} |
A_{1/3} |
C_{1/3} |
A_{1/3} |
C_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
C_{1} |
B_{1} |
A_{1} |
C_{1} |
B_{1} |

2 | B_{1/2} |
A_{1/2} |
C_{1/2} |
B_{1/2} |
B_{1/2} |
A_{1/2} |

3 | C_{1/3} |
B_{1/3} |
A_{1/3} |
C_{1/3} |
A_{1/3} |
C_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
C_{1} |
B_{1} |
A_{1} |
C_{1} |
B_{1} |

2 | B_{1/2} |
A_{1/2} |
C_{1/2} |
C_{1/2} |
B_{1/2} |
A_{1/2} |

3 | C_{1/3} |
B_{1/3} |
A_{1/3} |
B_{1/3} |
A_{1/3} |
C_{1/3} |

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.

Let’s voters start from their truthful state.

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 |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
B_{1} |
C_{1} |
A_{1} |
B_{1} |
C_{1} |

2 | B_{1/2} |
C_{1/2} |
A_{1/2} |
C_{1/2} |
A_{1/2} |
B_{1/2} |

3 | C_{1/3} |
A_{1/3} |
B_{1/3} |
B_{1/3} |
C_{1/3} |
A_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
C_{1} |
C_{1} |
A_{1} |
B_{1} |
C_{1} |

2 | B_{1/2} |
B_{1/2} |
A_{1/2} |
B_{1/2} |
A_{1/2} |
B_{1/2} |

3 | C_{1/3} |
A_{1/3} |
B_{1/3} |
C_{1/3} |
C_{1/3} |
A_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
B_{1} |
C_{1} |
A_{1} |
B_{1} |
C_{1} |

2 | B_{1/2} |
C_{1/2} |
A_{1/2} |
B_{1/2} |
A_{1/2} |
B_{1/2} |

3 | C_{1/3} |
A_{1/3} |
B_{1/3} |
C_{1/3} |
C_{1/3} |
A_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
B_{1} |
C_{1} |
A_{1} |
B_{1} |
C_{1} |

2 | B_{1/2} |
C_{1/2} |
A_{1/2} |
C_{1/2} |
A_{1/2} |
B_{1/2} |

3 | C_{1/3} |
A_{1/3} |
B_{1/3} |
B_{1/3} |
C_{1/3} |
A_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
V_{6} |
---|---|---|---|---|---|---|

1 | A_{1} |
C_{1} |
C_{1} |
A_{1} |
B_{1} |
C_{1} |

2 | B_{1/2} |
B_{1/2} |
A_{1/2} |
C_{1/2} |
A_{1/2} |
B_{1/2} |

3 | C_{1/3} |
A_{1/3} |
B_{1/3} |
B_{1/3} |
C_{1/3} |
A_{1/3} |

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.

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.

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 |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | B_{1} |
A_{1} |
C_{1} |

2 | A_{1/2} |
C_{1/2} |
B_{1/2} |

3 | C_{1/3} |
B_{1/3} |
A_{1/3} |

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

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | B_{1} |
A_{1} |
C_{1} |

2 | A_{1/2} |
B_{1/2} |
B_{1/2} |

3 | C_{1/3} |
C_{1/3} |
A_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | A_{1} |
A_{1} |
C_{1} |

2 | C_{1/2} |
B_{1/2} |
B_{1/2} |

3 | B_{1/3} |
C_{1/3} |
A_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | A_{1} |
A_{1} |
C_{1} |

2 | C_{1/2} |
C_{1/2} |
B_{1/2} |

3 | B_{1/3} |
B_{1/3} |
A_{1/3} |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | B_{1} |
A_{1} |
C_{1} |

2 | A_{1/2} |
C_{1/2} |
B_{1/2} |

3 | C_{1/3} |
B_{1/3} |
A_{1/3} |

Cycle is generated for the given set of moves.

Moves used are loser to winner, winner to new winner and winner to existing winner. Voters start from their declared preference list.

Voters | True |
Declared |
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 |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | B_{1x2} |
A_{1x2} |
C_{1x2} |

2 | A_{1/2x2} |
C_{1/2x2} |
B_{1/2x2} |

3 | C_{1/3x2} |
B_{1/3x2} |
A_{1/3x2} |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | A_{1x2} |
A_{1x2} |
C_{1x2} |

2 | C_{1/2x2} |
B_{1/2x2} |
B_{1/2x2} |

3 | B_{1/3x2} |
C_{1/3x2} |
A_{1/3x2} |

A = 3.66 B = 4 C = 3.32

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | B_{1x2} |
A_{1x2} |
C_{1x2} |

2 | A_{1/2x2} |
B_{1/2x2} |
B_{1/2x2} |

3 | C_{1/3x2} |
C_{1/3x2} |
A_{1/3x2} |

A = 4.66 B = 2.66 C = 3.66

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | A_{1x2} |
A_{1x2} |
C_{1x2} |

2 | C_{1/2x2} |
C_{1/2x2} |
B_{1/2x2} |

3 | B_{1/3x2} |
B_{1/3x2} |
A_{1/3x2} |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | B_{1x2} |
A_{1x2} |
C_{1x2} |

2 | A_{1/2x2} |
C_{1/2x2} |
B_{1/2x2} |

3 | C_{1/3x2} |
B_{1/3x2} |
A_{1/3x2} |

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.

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 |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
---|---|---|---|---|---|

1 | B_{1x1.9} |
E_{1x1.8} |
A_{1x1.9} |
A_{1x1.9} |
B_{1x1.8} |

2 | E_{1/2x1.9} |
D_{1/2x1.8} |
E_{1/2x1.9} |
B_{1/2x1.9} |
C_{1/2x1.8} |

3 | C_{1/3x1.9} |
B_{1/3x1.8} |
C_{1/3x1.9} |
C_{1/3x1.9} |
D_{1/3x1.8} |

4 | A_{1/4x1.9} |
A_{1/4x1.8} |
D_{1/4x1.9} |
D_{1/4x1.9} |
A_{1/4x1.8} |

5 | D_{1/5x1.9} |
C_{1/5x1.8} |
B_{1/5x1.9} |
E_{1/5x1.9} |
E_{1/5x1.8} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
---|---|---|---|---|---|

1 | B_{1x1.9} |
B_{1x1.8} |
A_{1x1.9} |
A_{1x1.9} |
B_{1x1.8} |

2 | E_{1/2x1.9} |
D_{1/2x1.8} |
E_{1/2x1.9} |
B_{1/2x1.9} |
C_{1/2x1.8} |

3 | C_{1/3x1.9} |
C_{1/3x1.8} |
C_{1/3x1.9} |
C_{1/3x1.9} |
D_{1/3x1.8} |

4 | A_{1/4x1.9} |
E_{1/4x1.8} |
D_{1/4x1.9} |
D_{1/4x1.9} |
A_{1/4x1.8} |

5 | D_{1/5x1.9} |
A_{1/5x1.8} |
B_{1/5x1.9} |
E_{1/5x1.9} |
E_{1/5x1.8} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
---|---|---|---|---|---|

1 | B_{1x1.9} |
B_{1x1.8} |
E_{1x1.9} |
A_{1x1.9} |
B_{1x1.8} |

2 | E_{1/2x1.9} |
D_{1/2x1.8} |
A_{1/2x1.9} |
B_{1/2x1.9} |
C_{1/2x1.8} |

3 | C_{1/3x1.9} |
C_{1/3x1.8} |
C_{1/3x1.9} |
C_{1/3x1.9} |
D_{1/3x1.8} |

4 | A_{1/4x1.9} |
E_{1/4x1.8} |
D_{1/4x1.9} |
D_{1/4x1.9} |
A_{1/4x1.8} |

5 | D_{1/5x1.9} |
A_{1/5x1.8} |
B_{1/5x1.9} |
E_{1/5x1.9} |
E_{1/5x1.8} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
---|---|---|---|---|---|

1 | B_{1x1.9} |
E_{1x1.8} |
E_{1x1.9} |
A_{1x1.9} |
B_{1x1.8} |

2 | E_{1/2x1.9} |
D_{1/2x1.8} |
A_{1/2x1.9} |
B_{1/2x1.9} |
C_{1/2x1.8} |

3 | C_{1/3x1.9} |
B_{1/3x1.8} |
C_{1/3x1.9} |
C_{1/3x1.9} |
D_{1/3x1.8} |

4 | A_{1/4x1.9} |
A_{1/4x1.8} |
D_{1/4x1.9} |
D_{1/4x1.9} |
A_{1/4x1.8} |

5 | D_{1/5x1.9} |
C_{1/5x1.8} |
B_{1/5x1.9} |
E_{1/5x1.9} |
E_{1/5x1.8} |

Choice | V_{1} |
V_{2} |
V_{3} |
V_{4} |
V_{5} |
---|---|---|---|---|---|

1 | B_{1x1.9} |
E_{1x1.8} |
A_{1x1.9} |
A_{1x1.9} |
B_{1x1.8} |

2 | E_{1/2x1.9} |
D_{1/2x1.8} |
E_{1/2x1.9} |
B_{1/2x1.9} |
C_{1/2x1.8} |

3 | C_{1/3x1.9} |
B_{1/3x1.8} |
C_{1/3x1.9} |
C_{1/3x1.9} |
D_{1/3x1.8} |

4 | A_{1/4x1.9} |
A_{1/4x1.8} |
D_{1/4x1.9} |
D_{1/4x1.9} |
A_{1/4x1.8} |

5 | D_{1/5x1.9} |
C_{1/5x1.8} |
B_{1/5x1.9} |
E_{1/5x1.9} |
E_{1/5x1.8} |

Moves = Loser-winner Winner-new Winner Winner –Existing Winner Winner – Loser

Voters start from their declared state

Voters | True |
Declared |
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 |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | B_{1x1.2} |
A_{1x1.2} |
C_{1x1.3} |

2 | A_{1/2x1.2} |
C_{1/2x1.2} |
B_{1/2x1.3} |

3 | C_{1/3x1.2} |
B_{1/3x1.2} |
A_{1/3x1.3} |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | B_{1x1.2} |
A_{1x1.2} |
C_{1x1.3} |

2 | A_{1/2x1.2} |
B_{1/2x1.2} |
B_{1/2x1.3} |

3 | C_{1/3x1.2} |
C_{1/3x1.2} |
A_{1/3x1.3} |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | A_{1x1.2} |
A_{1x1.2} |
C_{1x1.3} |

2 | C_{1/2x1.2} |
B_{1/2x1.2} |
B_{1/2x1.3} |

3 | B_{1/3x1.2} |
C_{1/3x1.2} |
A_{1/3x1.3} |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | A_{1x1.2} |
A_{1x1.2} |
C_{1x1.3} |

2 | C_{1/2x1.2} |
C_{1/2x1.2} |
B_{1/2x1.3} |

3 | B_{1/3x1.2} |
B_{1/3x1.2} |
A_{1/3x1.3} |

Choice | V_{1} |
V_{2} |
V_{3} |
---|---|---|---|

1 | B_{1x1.2} |
A_{1x1.2} |
C_{1x1.3} |

2 | A_{1/2x1.2} |
C_{1/2x1.2} |
B_{1/2x1.3} |

3 | C_{1/3x1.2} |
B_{1/3x1.2} |
A_{1/3x1.3} |

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.

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.