“*Election processes are fundamentally mathematical, so some algorithm – and every algorithm is mathematical and is used to add up the votes. There are many methods of counting votes, which, among other things, depend on how much and in what way information is collected from voters. The more information is taken as input, the better mathematics can calculate the output, that is, declare the winner who more objectively and completely represents the true will of the voters.*“, said Dr. Ismar Volić.

However, the more one tries to make this election fairer, closer to the objective will of the voters, the more complicated and therefore more expensive the electoral system becomes.

The way most of the world, including America and BiH, votes in elections where one winner is chosen (presidents, mayors, mayors, representatives from single-mandate districts, etc.) is a **relative majority**, so the voter chooses one candidate and the one with the most votes wins.

“*This is a simple voting system that we are all used to, but it is also the worst, precisely because of its simplicity. He does not take in enough information to always know how to declare the real winner, someone who really represents the ‘will of the people’. Therefore, the input is deficient and thus the output cannot be expected to be adequate.*“, remarked Volić.

## What is May’s theorem in the context of electoral systems and what are its key claims?

May’s Theorem (or May’s Theorem, as you prefer) of simple majority is an important result in social choice theory related to voting and collective decision-making. The theorem was developed by the mathematician Kenneth May in 1952, and it describes the conditions under which the simple majority voting system (known as the majority system) is unique and optimal.

“May’s* theorem confirms what intuitively makes a lot of sense, namely that when we have an election with only two candidates, then the best method for choosing one of them is an absolute majority. So the voter chooses one of the two names and the candidate with the most votes wins. The crucial hypothesis is that this is a race with only two candidates, which means that the one with more votes also necessarily wins an absolute majority, i.e. more than 50% of the votes. As soon as we have three or more candidates, the most votes or a relative majority does not mean an absolute majority, and this is where the breakdown of this system begins.*“, Volić explained.

This theorem is significant because it emphasizes the simplicity and fairness of the majority voting system when it comes to binary decisions (choosing between two options). However, in more complex situations with *more **than two options*, the majority system can have disadvantages, such as vote splitting and voting paradox. This is the reason why different election systems are used in practice, such as the proportional system, preferential voting and others, in order to cover more complex election scenarios.

“*It should be mentioned that the word ‘best’ has a precisely defined meaning in Mayo’s theorem, which of course must be the case if we want a rigorous and universal result. May’s theorem belongs to the field called the theory of social choice, which aims to formally axiomatize the problem of aggregation of individual preferences into group preferences. So the fundamental problem of this theory is how a group can reach a decision that is the best or fairest for as many of its members as possible. That theory, like any other, starts with certain axioms or postulates, which are built upon, new concepts are defined and theorems about them are proved. In the case of social choice theory, it is the axioms that mathematically precisely define the meaning of the words ‘best’ and ‘fairest’ decision-making systems. May’s theorem states that, when these axioms are taken as a starting point, then absolute majority is the best method for two-candidate elections.*“, Ismar Volić explained further.

## “Dispersion of votes” and “spoiler” candidate in the election process

Dispersion of votes and the spoiler effect are two phenomena that often occur in electoral processes and can significantly influence the outcome of the election.

*Vote ***splitting** occurs when voters’ votes are split between two or more similar candidates or options, resulting in a reduced chance for any of these candidates to win. This most often happens in systems where a simple majority is used, better known as “first-past-the-post”.

**How does the dispersion of votes occur?**

If voters who support the positions of candidates A and B split their votes between the two, this may result in neither of them receiving a sufficient majority. For example, candidate A gets 30% of the vote, candidate B 30%, and candidate C 40%. Candidate C wins because of the dispersion of votes between candidates A and B.

Multiple candidates of similar ideology may apply. When several candidates who share similar political views or belong to the same party participate in elections, the votes that would otherwise go to one candidate are divided between them. Also, the voter base itself may be fragmented – voters who support similar policies may be dispersed among different candidates, thus reducing the total number of votes any candidate receives. This also arises as a consequence **of the lack of strategic voting** : voters may vote for their preferred candidate regardless of his chances of winning, instead of strategically voting for the candidate who has a better chance of winning.

“*Dispersion and spoiler are the most common side effects of the relative majority. The first can happen when there are several similar candidates who split the votes and thereby enable another, often extreme or polarizing candidate, to win. In the 2016 elections for the presidential candidate ahead of the Republican Party in America, Donald Trump won with 44% of the vote because the three other main candidates divided the votes and individually won them less than him. However, polls tell us that Republican voters would be happier with any of those three candidates instead of Trump. But the system of relative majority is not able to gather that kind of deeper and more complete preferences from the voters.*” Professor Volić gave an example of this phenomenon.

“*Another important example for me is because my current representative in the US House of People, Jake Auchincloss. He is in 2020. won with 22% of the vote while three fantastic, progressive women made up the dispersion where they each received about 20% of the vote. Polls tell us that those of us who voted for one of the three, that is, 60% of the voters, would rather have any of them as a representative than Auchincloss, but the system did not allow us to somehow express that opinion.*” Volić added.

**The spoiler effect** occurs when a candidate who has no real chance of winning takes votes from a candidate who might otherwise win, thereby enabling a third candidate to win, often one who is ideologically opposed to the majority of voters who voted for the spoiler candidate. Votes are scattered between the main candidate and the spoiler, allowing the candidate with the fewest votes to win.

“*Spoiler happens when a minor candidate who has no chance of winning takes enough votes from another candidate and thus makes it impossible for him to win. A standard example is the US presidential election in 2000. It all came down to the result in Florida where George W. Bush won 537 more votes than Al Gore. However, the candidate of the Green Party, Ralph Nader, won about 90,000 votes, and polls showed that almost everyone who voted for him would rather have Gore than Bush as president. But unfortunately, the relative majority does not allow this information to be included in the calculation.*“, Volić explained.

If candidate B has no real chance of winning, but takes enough votes from candidate A, candidate C can win. For example, candidate A would get 60% of the vote without candidate B, and candidate C 40%. With candidate B in the race, candidate A gets 40%, candidate B 15%, and candidate C 45%. Candidate C wins because of the spoiler effect.

A spoiler candidate can further polarize the electorate and reduce the chances of rallying votes around one main candidate.

“*At the moment, there was a lot of talk about Robert F. Kennedy as an independent candidate who could easily be a spoiler in the upcoming November elections for the President of America. A system in which such a thing is even a possibility should absolutely be eliminated and replaced with something better.*“, Volić added.

Dispersion of votes and the spoiler effect are related, and look similar, the key difference lies in their causes and consequences. Dispersion of votes occurs when similar candidates share votes among themselves, while the spoiler effect occurs when a candidate with no real chance of winning influences the outcome of the election by taking votes from the main candidate. These phenomena emphasize the importance of carefully designing electoral systems to minimize negative effects on the fairness and representativeness of elections.

“*In any race with three or more candidates in which a decision is made by a relative majority, there is a danger of dispersion or spoilers. And if someone wins with less than 50% of the vote, the question can be raised as to whether there was a dispersion or spoiler that allowed them to do so. If so, then that candidate is not the right choice of the people. Of course, by then it’s too late to do anything. This is equally valid in America and in BiH*“, remarked Dr. Ismar Volić.

## What is preferential voting and how does it differ from the traditional voting system?

In preferential voting, voters rank candidates in order of preference, from most favorite to least favorite. Then there are rounds of vote counting, where candidates with the lowest number of votes are eliminated.

“*Preferential voting means that the voter gives more information in the form of candidate ranking. He indicates his first choice on the ballot, as is the case with the relative majority, but also says who is his second choice, third choice, etc. With this information, mathematics can do much more. There are various methods to draw a winner from such preferential ballots.*“, Volić said.

Preferential voting better reflects the will of the electorate, but it also has its drawbacks. It can be more complicated for voters, especially those who are not familiar with the system. Voters must rank the candidates, which requires more thought and time than simply selecting one candidate. Counting preferential votes is more complex and requires more time and resources compared to a simple majority system. Multiple rounds of counting are required, and automatic counting software can be expensive and require staff training. The mere existence of multiple voting rounds makes administration and the election process more expensive.

The results of preferential voting can be unpredictable, and the outcome can depend on how the voters ranked the candidates. This can lead to situations where the winner is not necessarily the one with the most first preferences. Minority opinions may also be marginalized.

## Hare preferential voting system

The Hare voting system, also known as **the Single Transferable Vote** (**STV**), is a form of proportional voting used for elections in multi-member districts. The aim of this system is to ensure proportional representation, where the number of mandates won reflects the support that candidates or parties have among voters.

The Hare voting system is named after Thomas Hare, a British jurist and electoral system reformer, who was one of the pioneers of this voting approach. Hare developed this method in the 19th century with the goal of creating a fairer and more proportional electoral system that would better reflect the actual preferences of voters.

“*The Hare system is one of the methods of announcing the winner of the preferential vote. If no one gets more than 50% of the first votes, then the candidate with the least first votes is eliminated and his votes are transferred to the candidates who are next on each ballot. If then someone has more than 50% of the votes, that is the winner, and if not, then the process is repeated. This is how iteratively we arrive at a candidate who will necessarily be the choice of the absolute majority. Effectively, this method allows the voter, in case his first choice does not have enough support, to declare who his next choice is, and that candidate now gets his support. If that choice is also eliminated, then the vote is switched to the third choice, and so on. In any case, the voter’s opinion continues to influence the outcome beyond and deeper than his first choice.* “, Volić explained.

## How can preferential voting help in reducing the dispersion of votes, the influence of “spoiler” candidates, but also in reducing polarization?

“*Preferential voting, and we include the Hare method and several other popular systems, completely eliminate dispersion and spoiler precisely because we know from the preferential ballots who is the second, third, and further voter’s choice. If, for example, preferential voting had been used in the Bush-Gore-Nader race, Nader would have dropped out and his votes would have gone to Gore, and Gore would have emerged above Bush with an absolute majority. A similar thing would happen with Trump, because one of the other candidates would take the lead after a few steps of elimination and redistribution of votes.*“, Volić explained.

In addition, preferential voting supports independent candidates and those from smaller parties because there is a greater chance of victory or at least a good result if all other candidates do not have to be defeated from the start. It is enough for enough voters to place someone high on their preference list and they can achieve a good result.

“*That’s why preferential voting often elects reasonable centrists and eliminates extreme or polarizing candidates. Women, for example, are elected much more often when preferential voting is used. Voters also do not have to fear ‘wasted’ votes and are free to vote for minor or independent candidates because, if those candidates are eliminated, the votes are carried forward regardless and continue to contribute to the process.*“, Volić added.

## What is the monotony of the electoral system?

No, this does not mean that the elections are boring, it just speaks to the stability of the system, in the sense that the risk of unpredictable results is reduced. Monotonicity is a property of an electoral system that refers to how the outcome of an election changes when voters’ preferences or votes change in a way that should be intuitive or expected.

There is monotonicity towards a candidate – if voters prefer a certain candidate more (for example, give him more votes) or give him a higher rank on their ballots, the electoral system should ensure that the candidate’s chances of winning or his position in the election results improve, and does not worsen.

We can also talk about monotonicity according to a group of candidates – if the number of votes or preferences changes in favor of a group of candidates (for example, by giving more votes to an entire party or block of candidates), the electoral system should result in a better or equally good outcome for that group, not a worse one.

**Monotonicity is an important characteristic because it ensures that voters can safely express their preferences without fear that their changes in voting will lead to paradoxical or undesirable results.**

Systems such as Instant-Runoff Voting (IRV) or Single Transferable Vote (STV) are often studied to determine whether they are monotonic or whether there is the possibility of tactical voting that could lead to undesirable results.

## Is there any voting system that we could call “almost perfect” at least from a mathematical point of view?

There is no perfect voting system that meets all criteria of fairness and efficiency, but certain systems tend to approach the ideal better than others. One way of assessment is to use **Arrow’s impossibility theorem**, which asserts that no electoral system can simultaneously satisfy all reasonable fairness criteria (such as universal domain, transitivity, independence from irrelevant alternatives, monotonicity, and non-dictatorial properties).

According to Volić, one of the central results in the theory of elections, which was proved by Kenneth Arrow in 1952., and for which he received the Nobel Prize in Economics in 1972, says that there is not and cannot be an ideal preferential voting system.

*If voting is approached axiomatically, as with May’s theorem, and the desirable criteria that a good preferential system should satisfy are listed, Arrow proves that there is no method that will satisfy them all. It is one of the so-called impossibility theorems that pervade social choice theory.*“, Volić added.

“*In the last seventy years, therefore, research work in this area has been focused on the question of which preferential method (Hare, Borda, Condorcet, and many others) is the least non-ideal, that is, which least satisfies the given axioms. Through the inclusion of empirical studies made possible recently by the proliferation of data science techniques, we are now slowly learning that Hare’s method is most likely the most desirable. It will most rarely lead to the strange, undesirable or paradoxical results that Arrow warns against.*“, he explained.

At the same time, Hare’s method is growing in popularity. In America, it is used for all elections in two states (Maine and Nebraska) and in many cities (New York, Minneapolis, San Francisco, etc.). There are currently over 90 initiatives across America to replace the relative majority with the Hare method. Wherever Hare is used, people are satisfied and see its value. It empowers voters because it represents them better and enables more credible representativeness.

## What are the main mathematical approaches used for the analysis and optimization of electoral processes

“*As I already mentioned, data science is currently entering election theory by the big door. It is currently one of the most exciting interdisciplinary applications of mathematics and statistics to the social sciences. Masses of data are accessible to us – population censuses, results of previous elections, demographic statistics – and data science is finally ready to analyze and process them effectively. Now, for the first time, we are ready to test a political-mathematical theory such as electoral systems on clouds of data from previous elections, to model it on real populations numbering in the millions, to generate test elections with machine learning trained on real data, quickly include or exclude various variables to test correlations, to incorporate existing electoral laws and electoral legislation into the models, etc. Even if we don’t have any practical application of all this, this would be incredibly exciting for us in the academic world, another proof that interdisciplinarity is the real future research activity.*“, Volić said.

## What is included in “mathematical” elective engineering? How is gerrymandering used as a tool in electoral engineering?

As Ismar Volić points out, “electoral systems are only the beginning of the connection between mathematics and democracy”. There are other things that can affect the results of electoral processes.

– Democratic processes, such as the allocation of mandates, proportionality in representation, the size of legislative bodies and many others, are fundamentally mathematical and as such should be questioned and criticized. There are various mathematical options for each of these processes, and it is up to us to educate ourselves and demand that those that are more coherent and representative be used. One of these processes is districtization, or determining the boundaries of voting districts. If boundaries are carefully crafted to intentionally include or exclude certain populations in specific districts, it can affect election outcomes. Determining district boundaries for political purposes is called * gerrymandering*. In America, this is one of the biggest threats to democracy because gerrymandering has made most districts uncompetitive. In them, it is predetermined who will win and thus millions of voters feel neglected because they effectively have no vote.

*Gerrymandering* is a phenomenon named after Elbridge Gerry, the early 19th century governor of Massachusetts, who in 1812 approved a snake-shaped electoral district to benefit his party politically. The name “gerrymandering” is a combination of his name and the word “salamandering”, due to the similarity of the electoral district to a snake. The main goal *of gerrymandering* is to manipulate the composition of electoral districts in order to secure an advantage in the electoral process, often at the expense of fair representation of voters or political rivals.

“*In BiH, there is also gerrymandering at the local level, and we can freely say that the districtization that was established by Dayton is actually one big gerrymander that still hasn’t been fixed.*“, Volić said.

Mathematicians have become involved in the study of gerrymandering in the last ten years, and now we have powerful tools to recognize when a district is the intended product of malfeasance and when its “strange” boundaries are derived from geography or laws that must be respected and are not politically motivated.

“*Districtization is another field where theoretical mathematics, statistics and data science meet in a completely new way. Even politicians, activists, and courts see the value of a mathematical approach to this problem, and we are increasingly being invited to participate in discussions and be expert witnesses in gerrymandering litigation.*”

Mathematics and electoral processes, and thus democracy, are deeply connected. Through mathematical models and analyses, we can recognize the imperfections of our electoral systems and work on their improvement. In this way, mathematics not only helps to understand democratic processes, but also to improve them, ensuring that democracy works in the best possible way for all citizens. **Quantitative literacy** and science literacy are therefore essential elements that voters should have in order to understand electoral processes.

**“ In any case, this is an incredibly exciting time for scientific work at the intersection of mathematics and politics with many open questions and opportunities for academic work that has the potential to improve society.“, Dr. Ismar Volić concluded.**

*Note: cover illustration created using generative AI tools.*