The Mathematics of Ranked-Choice Single-Winner Voting Systems. Can Different Systems of Voting Affect the Results?

Scientific Essay , 2021 , 23 Pages , Grade: 7 (IB)

Mathematics - Statistics

The aim of this paper is to investigate to what extent different systems of voting can affect the results and the distribution of votes in single-winner elections. Moreover, during elections, there are always politicians who hold extreme beliefs and are either loved or hated by people and those who hold tempered beliefs but are tolerable by society. Consequently, the research also examines which of the two mentioned characteristics a candidate should have to increase the chances of winning by applying basic statistical measures such as mean or standard deviation.

The subject is relevant because it allows us to get a mathematical insight into single-winner electoral systems and can show whether a particular system of voting is only a tool, or directly contributes to the results. The paper analyses 4 different methods of voting that are or were used in the world to elect a President or other representatives for single-member posts: Supplementary Vote (SV), Instant-runoff voting (IRV), Bucklin Voting, and Coombs’ method.

It is worth adding that these systems are ranked-choice systems, that is voters rank their candidates from the most favorable one to the one that in their opinion is completely not suitable to accede to a particular office. Ranked-choice systems were chosen because this paper does not investigate only who wins the election but also how the situation of candidates in other positions changes.

I.Introduction

It is not an exaggeration to say that mathematics is omnipresent in the surrounding world. As far as the eye can see, there are plenty of signs of mathematical activity. Consequently, there is no surprise that this science can also be found in politics and, more specifically, it has a lot in common with various types of elections.When I am writing this essay, exactly one month has passed since the last presidential elections in my home country, Poland, which for many weeks engaged the attention of citizens and caused extreme emotions. The elections were marginally won by Andrzej Duda, who became the President of Poland for the second term. I started wondering whether it is possible to change the results of a single-winner election by applying a different voting system.

That is why the aim of this paper is to investigate to what extent different systems of voting can affect the results and the distribution of votes in single-winner elections. Moreover, during elections, there are always politicians who hold extreme beliefs and are either loved or hated by people and those who hold tempered beliefs but are tolerable by society. Consequently, the research also examines which of the two mentioned characteristics a candidate should have to increase the chances of winning by applying basic statistical measures such as mean or standard deviation. The subject is relevant because it allows us to get a mathematical insight into single-winner electoral systems and can show whether a particular system of voting is only a tool, or directly contributes to the results. Moreover, I am deeply passionate about politics and I am curious whether it is doable to manipulate the results of the election by applying a particular voting system. I have decided to analyze 4 different methods of voting that are or were used in the world to elect a President or other representatives for single-member posts: Supplementary Vote (SV), Instant-runoff voting (IRV), Bucklin Voting, and Coombs' method. It is worth adding that these systems are ranked-choice systems, that is voters rank their candidates from the most favorable one to the one that in their opinion is completely not suitable to accede to a particular office. Ranked-choice systems were chosen because this paper does not investigate only who wins the election but also how the situation of candidates in other positions changes.

2. Procedure

The investigation started by conducting an anonymous poll which had to gather data needed for further analysis. Voluntary sampling was used because of its simplicity and low time consumption. The participants had to be over 18 years old as in Poland this is a requirement to be legally allowed to vote. The majority of data was gathered using an anonymous, online poll where the responses were obtained from people aged from 18 to 56 years old. The rest of the data was obtained from older people using the same, but paper-based survey. It was done to avoid a non-response bias, at the same time improving the representativeness of the poll. Moreover, the reliability was increased by the fact that there was a similar representation of each age group so it was not possible that one age group could distort the results as all groups were of a similar size. Fifty four participants ranked the candidates according to their preferences. The 6 candidates were the most known politicians who applied for the position of the President of Poland in 2020: Andrzej Duda, Rafai Trzaskowski, Szymon Hoiownia, Krzysztof Bosak, Wiadysiaw Kosiniak-Kamysz, and Robert Biedron who were marked with letters A, B, C, D, E, and F respectively to make the research more clear and transparent. These candidates were chosen because the poll was conducted shortly after the presidential elections, so the majority of participants have already formed an impression about each of the mentioned candidates and therefore, they did not have any problems to rank them. The main plan for this investigation is to use the data collected in the poll to explain how particular systems of voting work and investigate how the results differ among these systems. The pictures below show an online ballot the participants had to fill out (left-most) and a sample ballot from the Instant run-off election^[1] (right-most).

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Picture 1: A picture showing an online ballot used to collect data.

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Picture 2: A picture showing a sample ballot used in the ranked-choice elections in Australia.

The results of the survey are summarized in the tables below. The results of each candidate are presented in a separate table - only ballots where a particular candidate (marked as a letter) was the 1st choice, are included in respective tables. It was done to make the process of analysis easier and better-organized. As a result, 6 tables were created. At this stage, colors do not have any significance.

Candidate A

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Table 1: Table showing the distribution of votes when the voters' first choice was candidate A.

Candidate B

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Table 2: Table showing the distribution of votes when the voters’ first choice was candidate B.

Candidate C

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Table 3: Table showing the distribution of votes when the voters’ first choice was candidate C.

Candidate D

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Table 4: Table showing the distribution of votes when the voters’ first choice was candidate D.

Candidate E

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Table 5: Table showing the distribution of votes when the voters’ first choice was candidate E.

Candidate F

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Table 6: Table showing the distribution of votes when the voters’ first choice was candidate F.

3. Analysis of voting systems

- First past the post (Plurality voting)

Although this system does not belong to ranked voting systems and is not mathematically complicated, it is an excellent introduction to the systems described in the later part of this paper. The first past the post (FPTP) system, defined also as a plurality voting, functions as a way to elect a president in over 20 countries, for example in Honduras, Iceland, or Singapore. FPTP is a simple and logical method based on two main rules by which every citizen casts only one vote and the candidate who obtained the most votes wins.

Investigation

Firstly, in the tables from section 2 the candidate with most first-choice votes can be found. This candidate is the winner of these elections (Candidate B-16 votes). The rest of the candidates have been ranked according to this criterion.

Final results

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Table 1: Table showing how the results of the election would look like if the FPTP method had been used.

Although FPTP is a simple and time-saving method it has serious disadvantages. The main drawback is that the candidate can win presidential elections by having the plurality (the highest number of votes), not the majority (over 50%), of votes. The best example of it is the presidential election won by Fidel Ramos with only 24 % of the popular vote in the Philippines in 1992. It was enough that he had the plurality, although the rest of entitled to vote citizens voted for the other 6 candidates. The next issue concerns the point of view of extremists who are usually the minorities ignored by the FPTP method. Therefore, FPTP is suitable for elections where only 2 candidates compete with each other.

- Supplementary vote

The first preferential voting system called Supplementary Vote (SV) is applied during the London mayoral election. The voters are given a ballot with two columns. In the first column, they must mark their favorite candidate whereas in the second one they can choose their second preference. The election is won by a candidate with over 50% of votes (in this case, the majority is 28 or more votes). If none of the candidates receives the majority, the top two candidates compete with each other in the second run. The rest of them are eliminated, however, the second choice is still important. If the second preferred candidate is one of the top two candidates, then this vote can be regarded as an extra one for a particular candidate.

Investigation

Consider the first preferences to check whether a certain candidate obtained the majority of the first-choice votes. Table 1 from section 3 shows that there is no winner in the first round, but there are two top candidates: candidate B (16 votes) and C (13 votes). The remaining candidates are eliminated, however, the second preferences of these candidates' voters are taken into account. The second-choice votes for the top two candidates are colored in green in tables from section 2.

Candidate B: 16 first-place votes +12 second-place votes = 28 votes, the majority

Candidate C: 13 first-place votes+18 second-place votes = 31 votes, the winner with a greater majority of votes

Important note: In this part of the paper the reader can come across on words „the majority” and „the greater majority” as mentioned above which may be confusing and illogical at first sight. Therefore, I would like to rectify that „the majority” always means 28 votes or more, no matter how many next-preference votes are added. The majority is equal to 28 and is constant because there were 54 voters. If there are at least 2 candidates who obtained 28 votes or more, the elections are won by the candidate with the „greater majority” of votes, that is the candidate who obtained the most votes. For example, in the Supplementary Vote system described above, there are top two candidates and each of them has the majority, that is at least 28 votes. However, candidate C has more votes than candidate B and therefore, has the greater majority of votes. This rule applies to every voting system described in this paper.

Final results

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Table 2: Table showing how the results of the election would look like if the Supplementary vote method had been used.

The asset of this system is that it is simple to understand. Moreover, the voter has more power since both their first and second choices might count whereas the majority criterion prevents from obtaining the confusing results, as in the FPTP method. On the other hand, SV may involve tactical voting because the second preferences may also be important. Furthermore, there are a lot of „wasted votes” as the first choices of candidates who are eliminated in the first round no longer count, only second choices may. Additionally, if a voter is not sure whether there will be a winner after the first round, they may try to second-guess which two candidates will be in the second round and allocate their votes in such a way to make their prediction true. In such situation there is a possibility that a voter will inadvertently beat their favorite candidate. Table 2 indicates that in this method a candidate should have a similar number of first and second-choice votes to win rather than trying to get as many first-choice votes as possible.

- Instant-runoff voting (IRV or Plurality with elimination)

The following system is used to elect the President of Ireland and forms an extended version of FPTP. Firstly, the voters have to rank the candidates according to their preferences using numbers, starting with using "1" next to their favorite candidate on the ballot. If over half of the voters ranked the same candidate in the first place, then this candidate wins. If not, the candidate who obtained the fewest first-place votes is eliminated. However, the votes of those who voted for this candidate are still valid and move to the one who was their next preference. This procedure is repeated until one of the candidates has the majority and wins the election. If two or more candidates obtain the majority, the candidate with the greater majority wins. Here is a simple explanation of how it works in practice.

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Table 3: Table explaining the mechanism of the IRV method.

It is worth adding that if a candidate takes the last place, votes for them are lost since there is no preference they could be added to.

Investigation

Determine the candidate with the fewest number of first-place votes. Table 1 indicates that this is candidate E (3 first-place votes), hence, they are eliminated. Their votes are highlighted in yellow in tables in section 2.

Candidate A: 7 first-place votes

Candidate B: 16 first-place votes

Candidate C: 13 first-place votes

Candidate D: 6 first-place votes

Candidate E: 3 first-place votes, 1st eliminated

Candidate F: 9 first-place votes

Move the votes from Candidate E to the candidate that was the next preference.

Candidate A: 7+8 = 15 votes , 2nd eliminated

Candidate B: 16+7 = 23 votes

Candidate C: 13+5 = 18 votes

Candidate D: 6+13 = 19 votes

Candidate E: 3, 1st eliminated

Candidate F: 9+18 = 27, a half of the number of votes, but not the majority.

[...]

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^[1] Source: www.aec.gov.au

• Grade: 7 (IB)
• Author: Maciej Nodzyński (Author)
• Publication Year: 2021
• Pages: 23
• Catalog Number: V1059528
• ISBN (eBook): 9783346473493
• ISBN (Book): 9783346473509
• Language: English
• Tags: voting system, election