Ranked voting
Imagine an election where your vote isn't just for a single candidate, but a detailed expression of all your preferences, from first choice to last. This sophisticated approach is known as ranked voting, a system designed to capture the nuances of voter sentiment far beyond a simple 'X' on a ballot. It's a method with a surprisingly rich history and profound implications for how our leaders are chosen. Ranked voting systems allow voters to order candidates by preference, which can lead to more representative outcomes than traditional plurality voting. Different methods of tallying ranked ballots, like Instant-Runoff Voting and the Borda count, each possess unique mathematical properties and potential vulnerabilities. From ancient proposals to modern adoption, the evolution of ranked voting reflects a centuries-long quest to design fairer, more robust electoral processes.
AI Summary
Imagine an election where your vote isn't just for a single candidate, but a detailed expression of all your preferences, from first choice to last. This sophisticated approach is known as ranked voting, a system designed to capture the nuances of voter sentiment far beyond a simple 'X' on a ballot. It's a method with a surprisingly rich history and profound implications for how our leaders are chosen.
- Ranked voting systems allow voters to order candidates by preference, which can lead to more representative outcomes than traditional plurality voting.
- Different methods of tallying ranked ballots, like Instant-Runoff Voting and the Borda count, each possess unique mathematical properties and potential vulnerabilities.
- From ancient proposals to modern adoption, the evolution of ranked voting reflects a centuries-long quest to design fairer, more robust electoral processes.
What is Ranked Voting?
At its core, ranked voting is any electoral system that asks voters to do more than simply pick one candidate. Instead, you rank candidates in order of your preference — first choice, second choice, third choice, and so on. It's about giving a fuller picture of your will, not just a snapshot.
Diverse Approaches, Different Outcomes
While the basic idea is simple, how those ranked preferences are actually counted varies dramatically between systems. This means different ranked voting methods can have wildly different properties and results. Understanding these tabulation methods is key to understanding the systems themselves.
Contingent vs. Positional Systems
Broadly, ranked systems fall into two camps. Some, like Instant-Runoff Voting (IRV) and the Single Transferable Vote (STV), use your lower preferences as 'contingencies' or backups. They only come into play if your higher-ranked choices are eliminated or if their votes are surplus to requirement.
These 'contingent' systems are designed to ensure that expressing a lower preference won't accidentally harm your top choice. They aim to avoid wasted votes and find a candidate with broader support among the electorate.
Other systems, known as 'positional voting,' use ranks as weights. The Borda count is a classic example, where candidates receive points based on their position on each ballot. The further down a candidate is ranked, the fewer points they receive, reflecting a decreasing intensity of preference.
A Clarification on Terminology
In places like the United States and Australia, terms like 'ranked-choice voting' or 'preferential voting' are often used synonymously with Instant-Runoff Voting. However, these terms can sometimes refer to ranked systems more generally, so experts recommend being specific to avoid confusion.
Beyond the 'X' Mark
Ranked voting systems offer significantly more information than traditional 'X' voting systems, like first-past-the-post, which only record a single choice. However, they're distinct from 'rated voting' methods, where you might score candidates from 0 to 10, indicating how strongly you support them, not just the order.
This distinction in information captured is important. Certain theoretical results, such as Arrow's Impossibility Theorem, apply directly to ranked voting but not necessarily to rated voting, due to the different types of preferences they capture.
Flexibility in Ranking
The rules for how many candidates you must rank also vary. Some systems require you to rank a set number, while others, like optional preferential voting, give you full liberty to rank as few or as many candidates as you wish. Not all rankings on a ballot are necessarily used in every count for systems like STV or IRV.
A Glimpse into History
Early Seeds of Preference
The idea of ranked voting isn't new; its roots stretch back to the late 13th century. Ramon Llull, a Catalan philosopher, is credited with one of the earliest known proposals, developing a method similar to what we now call Copeland's or Condorcet's method in his 1299 treatise, Ars Electionis.
The Enlightenment Debate
Centuries later, the 18th century saw a resurgence of interest. Jean-Charles de Borda published a paper in 1781, advocating for his Borda count, which he called the 'order of merit.' This sparked a lively debate with the Marquis de Condorcet.
Condorcet, a brilliant mathematician and philosopher, criticized Borda's approach. He argued it didn't always reflect true group preferences, could be vulnerable to 'spoiler effects,' and might not elect the candidate preferred by a majority in head-to-head contests. This led him to develop his own set of methods.
19th Century Innovations
The 19th century continued this intellectual journey. Carl Andræ formulated a version of the Single Transferable Vote (STV) in Denmark, adopted in 1855. This early form transferred surplus votes from elected candidates but didn't yet include transfers from eliminated ones.
Interestingly, Condorcet himself considered Instant-Runoff Voting but dismissed it as 'pathological.' However, by 1864, Edward J. Nanson championed IRV, declaring it 'perfectly feasible and practicable' for elections of any size, challenging Condorcet's earlier reservations.
In 1871, MIT professor W. R. Ware held a groundbreaking STV demonstration with 150 students electing their favorite English authors. This proved that ranked transferable votes could indeed produce a clear winner, laying the groundwork for wider acceptance.
Multi-seat STV then gained traction for electing legislators in North America, Ireland, and other English-speaking regions. Single-winner versions, like Instant-Runoff Voting (then 'alternative voting') and contingent voting, became companions for mayoral and other single-winner contests.
Modern Theoretical Foundations
The mid-20th century saw a renewed academic focus, notably with Duncan Black's 1948 paper and Kenneth Arrow's seminal work on social choice theory. This branch of welfare economics explores how individual preferences can be aggregated into collective decisions, providing a rigorous mathematical framework for analyzing voting systems.
The Theory Behind Ranked Voting
The Quest for a Majority-Rule Winner
Many theoretical concepts dating back to Condorcet still shape our understanding of ranked voting. One pivotal idea is the 'Condorcet winner' — a candidate who would defeat every other candidate in a head-to-head contest. A system that always elects this candidate is called a Condorcet method.
However, sometimes a Condorcet winner simply doesn't exist, a situation known as a 'Condorcet cycle.' Imagine three candidates, A, B, and C, and three voters. One prefers A > C > B, another B > A > C, and the third C > B > A. In this 'rock-paper-scissors' scenario, no candidate can defeat all others in a two-way race.
Maximizing Social Well-being
Beyond simply electing a majority-preferred candidate, voting systems can also be evaluated on their ability to maximize overall 'social well-being.' This perspective aims to choose the candidate who best serves the collective good of society as a whole, a complex goal that different systems approach in various ways.
Spatial Voting Models
To visualize how voters make choices, political scientists use 'spatial voting models,' pioneered by Duncan Black and Anthony Downs. These models place voters and candidates in an ideological space, often a spectrum, assuming voters prefer candidates closer to their own positions.
A simple one-dimensional model might show candidate A on the left, C on the right, and B in the middle. Voters supporting A might rank A > B > C, while C supporters vote C > B > A. Those in the middle might be split on their second preference.
These models offer powerful insights, leading to theorems like Black's 'Median Voter Theorem.' This theorem suggests that in many spatial models, a Condorcet winner will exist, and it will be the candidate closest to the ideological median of the voters.
Inherent Limits and Strategic Play
Despite their strengths, all ranked voting systems face fundamental limitations. Arrow's Impossibility Theorem, a generalization of Condorcet's cycle issue, famously demonstrates that no ranked voting algorithm can entirely escape the 'spoiler effect' — where a third candidate can alter the outcome between two front-runners.
Building on this, Gibbard's Theorem reveals that no voting rule can guarantee a single, universally best strategy for voters that is independent of how others vote. This means 'strategic voting' – where you don't rank candidates truthfully to achieve a better outcome – is always a potential factor.
How Different Systems Work: Examples
The Borda Count
The Borda count is a classic positional system. If there are 'm' candidates, your first-ranked choice receives (m-1) points, your second (m-2) points, and so on, down to zero points for the last-ranked candidate. The candidate with the highest total points wins.
For instance, with 5 candidates, a first choice gets 4 points, a second gets 3, and so on. While simple, the Borda count doesn't guarantee a Condorcet winner and can be heavily influenced by the presence of 'spoiler' candidates who have no real chance of winning.
Score_{candidate} = \sum_{ballots} (m - rank_{ballot})Other Positional Systems
The Borda count is just one flavor of positional voting. Systems that award points using different formulas also fall into this category. For example, the Dowdall system uses a vector of (1, 1/2, 1/3, ..., 1/m), while the familiar 'first-past-the-post' plurality system can be thought of as (1, 0, ..., 0), only giving a point to the top choice.
Instant-Runoff Voting (IRV)
Often mistaken for all ranked-choice voting, Instant-Runoff Voting is a contingent method for single-winner elections. It works by eliminating the candidate with the fewest first-place votes in rounds. The votes from eliminated candidates are then redistributed to the voters' next preference until one candidate secures a majority of the remaining votes.
This process continues until a candidate achieves more than 50% of the active votes. IRV is designed to ensure the winner has majority support among those still in contention, but it does not necessarily guarantee a Condorcet winner.
Article
Ranked voting
Ranked voting is any voting system that uses voters' rankings of candidates to choose a single winner or multiple winners. More formally, a ranked vote system depends only on voters' order of preference of the candidates.
Ranked voting systems vary dramatically in how preferences are tabulated and counted, which gives them very different properties. In instant-runoff voting (IRV) and the single transferable vote system (STV), lower preferences are used as contingencies (back-up preferences) and are only applied when all higher-ranked preferences on a ballot have been eliminated or when the vote has been cast for a candidate who has been elected and surplus votes need to be transferred. Ranked votes of this type do not suffer the problem that a marked lower preference may be used against a voter's higher marked preference.
Some ranked vote systems use ranks as weights; these systems are called positional voting. In the Borda method, the 1st, 2nd, 3rd... candidates on each ballot receive 1, 2, 3... points, and the candidate with the fewest points is elected. Thus intensity of preference is assumed to be at ratios of 1 to 2, 2 to 3, etc.
In the United States and Australia, the terms ranked-choice voting and preferential voting, respectively, almost always refer to instant-runoff voting; however, because these terms have also been used to mean ranked systems in general, many social choice theorists recommend the use of the term instant-runoff voting in contexts where confusion might arise.
Ranked votes do not incorporate any information about intensity of preferences. Furthermore, common implementations do not account for equality of preference among two or more candidates.
Ranked voting systems of the instant-runoff voting type and the Borda count type are contrasted with rated voting methods, which allow voters to indicate how strongly they support different candidates (e.g. on a scale from 0 to 10). Ranked vote systems produce more information than X voting systems such as first-past-the-post voting. Rated voting systems produce more information than ordinal ballots; as a result, some common results like Arrow's theorem do not directly apply to them.
Some ranked voting systems require the voter rank a set number of candidates. Others, such as optional preferential voting, allow the voter full liberty as to how many candidates they rank. Under STV or IRV, not all rankings are used in any case.
History of ranked voting
Ranked voting
The earliest known proposals for a ranked voting system can be traced to the works of Ramon Llull in the late 13th century, who developed what would later be known as Copeland's method, which is similar to Condorcet's method. Copeland's method was devised by Ramon Llull in his 1299 treatise Ars Electionis, which was discussed by Nicholas of Cusa in the fifteenth century.
A second wave of analysis began when Jean-Charles de Borda published a paper in 1781, advocating for the Borda count, which he called the "order of merit". This methodology drew criticism from the Marquis de Condorcet, who developed his own methods after arguing Borda's approach did not accurately reflect group preferences, because it was vulnerable to spoiler effects and did not always elect the majority-preferred candidate.
Interest in ranked voting continued throughout the 19th century. Carl Andræ formulated a version of the single transferable vote election system, which was adopted by his country, Denmark, in 1855. This used contingent ranked votes to transfer votes of elected candidates, but it did not include transfers of votes from eliminated candidates.
In the 1700s, Condorcet considered instant-runoff voting but rejected it as pathological. In 1864, Edward J. Nanson, while noting Condorcet's concerns, said the IRV method is "a perfectly feasible and practicable one for elections on any scale".
MIT professor W. R. Ware held a demonstration STV election in 1871 by having 150 MIT students elect four English authors as their favorites. The field of candidates held the names of 35 authors in the first count and eventually concentrated a large proportion of the votes behind just four. Ware went on to surmise how a casual vacancy might be handled and found that countback would produce a winner. This proved that ranked transferable votes could be used to produce a single winner, despite the qualms of Condorcet and others.
Single transferable voting (multi-seat contests using contingent ranked transferable votes) later found common use in cities and to elect legislators in North America, Ireland and other parts of the English-speaking world, with single-winner versions, alternative voting (now known as instant-runoff voting) and contingent voting (also known as supplementary voting) being its companions for single-winner contests of mayors and others.
Theoretical exploration of electoral processes was revived by a 1948 paper from Duncan Black and Kenneth Arrow's investigations into social choice theory, a branch of welfare economics that extends rational choice to include community decision-making processes.
Adoption
Plurality voting is the most common voting system, and has been in widespread use since the earliest democracies. As plurality voting has exhibited weaknesses from its start, especially as soon as a third party joins the race, some social scientists began to recommend transferable votes (facilitated by contingent ranked ballots) to reduce the incidence of wasted votes and unrepresentative election results.
A form of the single transferable vote system was invented by Carl Andræ in Denmark, where it was used briefly before being abandoned in favour of open list list PR. STV still carried on in Denmark until 1953 for indirect election of the upper house.
At approximately the same time, the single transferable vote system was independently devised by British lawyer Thomas Hare, whose writings soon spread the method throughout the British Empire. Tasmania used STV (called the Hare system) in government elections in the 1890s. STV began permanent and wider adoption throughout Australia beginning in 1907 and the 1910s. The STV system, using contingent ranked votes, has been adopted in Ireland, South Africa, Malta, and approximately 40 cities in the United States and Canada. The single transferable vote system has also been used to elect legislators in Canada, South Africa, India and Nepal.
In the early 2020s, the use of contingent ranked votes saw a comeback in the United States. STV, for a time used only in Cambridge, Massachusetts, was adopted by Portland, Maine, and several other American cities beginning in 2022. Single-winner ranked voting (specifically instant-runoff voting) is used to elect politicians in the states of Maine and Alaska. In November 2016, the voters of Maine narrowly passed Question 5, approving ranked-choice voting (instant-runoff voting) for all elections. This was first put to use in 2018, marking the inaugural use of ranked votes in a statewide election in the United States (when Maine's statewide vote tally was used to determine the two at-large electoral college seats).
In November 2020, Alaska voters passed Measure 2, bringing ranked choice voting (instant-runoff voting) into use from 2022. After a series of electoral pathologies in Alaska's 2022 congressional special election, a poll found 54% of Alaskans supported a repeal of the system. This included a third of the voters who had supported Mary Peltola, the ultimate winner in the election. However, a referendum on the issue in 2024 saw a narrow majority in favour of retention of IRV.
Some local elections in New Zealand use the multi-winner single transferable vote system. STV is also used to elect local authorities in Scotland and Ireland. Nauru uses a rank-weighted positional method called the Dowdall system.
Equal-ranked ballots
Ranked voting
In voting with ranked ballots, a tied or equal-rank ballot is one where multiple candidates receive the same rank or rating. In instant runoff and first-preference plurality, such ballots are generally rejected; however, in social choice theory some election systems assume equal-ranked ballots are "split" evenly between all equal-ranked candidates (e.g. in a two-way tie, each candidate receives half a vote). Other election systems, the Borda count and the Condorcet method, can use different rules for handling equal-rank ballots. These rules produce different mathematical properties and behaviors, particularly under strategic voting.
Theoretical foundations of ranked voting
Majority-rule
Many concepts formulated by the Marquis de Condorcet in the 18th century continue to significantly impact the field. One of these concepts is the Condorcet winner, a candidate who would win against any other candidate in a two-way race. A voting system that always elects this candidate is called a Condorcet method; however, it is possible for an election to have no Condorcet winner, a situation called a Condorcet cycle. Suppose an election with 3 candidates A, B, and C has 3 voters. One votes A > C > B, one votes B > A > C, and one votes C > B > A. In this case, no Condorcet winner exists: A cannot be a Condorcet winner as two-thirds of voters prefer B over A. Similarly, B cannot be the winner as two-thirds prefer C over B, and C cannot win as two-thirds prefer A over C. This forms a rock-paper-scissors style cycle with no Condorcet winner.
Social well-being
Voting systems can also be judged on their ability to deliver results that maximize the overall well-being of society, i.e. to choose the best candidate for society as a whole.
Spatial voting models
Spatial voting models, initially proposed by Duncan Black and further developed by Anthony Downs, provide a theoretical framework for understanding electoral behavior. In these models, each voter and candidate is positioned within an ideological space that can span multiple dimensions. It is assumed that voters tend to favor candidates who closely align with their ideological position over those more distant. A political spectrum is an example of a one-dimensional spatial model.
A spatial model of voting
The accompanying diagram presents a simple one-dimensional spatial model, illustrating the voting methods discussed in subsequent sections of this article. It is assumed that supporters of candidate A cast their votes in the order of A > B > C, while candidate C's supporters vote in the sequence of C > B > A. Supporters of candidate B are equally divided between listing A or C as their second preference. From the data in the accompanying table, if there are 100 voters, the distribution of ballots will reflect the positioning of voters and candidates along the ideological spectrum.
Spatial models offer significant insights because they provide an intuitive visualization of voter preferences. These models give rise to an influential theorem—the median voter theorem—attributed to Duncan Black. This theorem stipulates that within a broad range of spatial models, including all one-dimensional models and all symmetric models across multiple dimensions, a Condorcet winner is guaranteed to exist. Moreover, this winner is the candidate closest to the median of the voter distribution. Empirical research has generally found that spatial voting models give a highly accurate explanation of most voting behavior.
Other theorems
Arrow's impossibility theorem is a generalization of Condorcet's result on the impossibility of majority rule. It demonstrates that every ranked voting algorithm is susceptible to the spoiler effect. Gibbard's theorem provides a closely related corollary, that no voting rule can have a single, always-best strategy that does not depend on other voters' ballots.
Examples
Borda count
The Borda count is a weighted-rank system that assigns scores to each candidate based on their position in each ballot. If m is the total number of candidates, the candidate ranked first on a ballot receives m − 1 points, the second receives m − 2, and so on, until the last-ranked candidate who receives zero. In the given example, candidate B emerges as the winner with 130 out of a total 300 points. While the Borda count is simple to administer, it does not meet the Condorcet criterion. Also, it is heavily affected by the entry of candidates who have no real chance of winning.
Other positional systems
Systems that award points in a similar way but possibly with a different formula are called positional systems. The score vector (m − 1, m − 2, ..., 0) is associated with the Borda count, (1, 1/2, 1/3, ..., 1/m) defines the Dowdall system and (1, 0, ..., 0) equates to first-past-the-post.
Instant-runoff voting
Instant-runoff voting, often conflated with ranked-choice voting in general, is a contingent ranked-vote voting method that recursively eliminates the plurality loser of an election until one candidate has the majority of the remaining votes. In the given example, candidate A is declared winner in the third round, having received a majority of votes through the accumulation of first-choice votes and redistributed votes from candidate B. This system embodies the voters' preferences between the final candidates, stopping when a candidate garners the preference of a majority of voters. Instant-runoff voting does not fulfill the Condorcet winner criterion.
Single transferable voting
Single transferable voting is a contingent ranked-vote election method that elects multiple members. It elects any candidates who achieve quota, and if necessary recursively eliminates the plurality loser at various stages of the vote count and transfers surplus votes of winners until enough are elected by quota or by still being in the running when the field of candidates is thinned to the number of remaining open seats. Because elected members are elected with the same or about the same number of votes, each party popular enough for representation receives a number of seats appropriate to the vote tallies of its candidates. The transfers reduce waste to about one quota - which in a five-seat district is about 17 percent of valid votes; in districts with more members than five, the waste is smaller. All but one quota of votes approximately are used to actually elect someone in the district so the percentage of effective votes is dependably about 80 to 90 percent of valid votes in the district.
Defeat-dropping Condorcet
The defeat-dropping Condorcet methods all look for a Condorcet winner, i.e. a candidate who is not defeated by any other candidate in a one-on-one majority vote. If there is no Condorcet winner, they repeatedly drop (set the margin to zero) for the one-on-one matchups that are closest to being tied, until there is a Condorcet winner. How "closest to being tied" is defined depends on the specific rule. For the Minimax Condorcet method, the elections with the smallest margin of victory are dropped, whereas in ranked pairs only elections that create a cycle are eligible to be dropped (with defeats being dropped based on the margin of victory).
How-To-Vote cards
Ranked voting
In Australian Commonwealth and state elections, voters are handed How-To-Vote cards (HTVC) which recommend how best to support a chosen candidate. In South Australia, the Electoral Commission publishes a combined HTVC for all candidates which saves time and money. This also benefits minor parties and independents who are unable to staff all polling booths.