What is the fundamental definition of a Condorcet winner in the context of elections?

A Condorcet winner is a candidate who would win a head-to-head election against every other candidate in the race. This means that if you were to compare the Condorcet winner directly against any single opponent, more than half of the electorate would prefer the Condorcet winner.

What are the alternative names or descriptions used for a Condorcet winner?

A Condorcet winner is also referred to as a majority winner, a majority-preferred candidate, a beats-all winner, or a tournament winner. These names highlight its ability to defeat all other candidates in pairwise comparisons.

How does the Condorcet winner criterion relate to the principle of majority rule?

The Condorcet winner criterion extends the principle of majority rule to elections involving multiple candidates. It ensures that if a candidate is preferred by a majority over every other individual candidate, that candidate will win the election.

Does a Condorcet winner always exist in every election?

No, a Condorcet winner does not always exist. It is possible for a situation to arise, known as Condorcet's voting paradox, where there is a cycle of preferences among candidates, meaning no single candidate can defeat all others in pairwise matchups.

What is Condorcet's voting paradox, and what does it describe?

Condorcet's voting paradox describes a situation where voter preferences create a cycle, similar to a 'rock, paper, scissors' dynamic. For example, Candidate A might be preferred over Candidate B, Candidate B over Candidate C, and Candidate C over Candidate A, meaning no Condorcet winner emerges.

What is the Smith set in relation to Condorcet criteria?

The Smith set is a concept that generalizes Condorcet criteria. It is defined as the smallest set of candidates where every candidate within the set is preferred, either directly or indirectly, over every candidate outside the set. A Smith set will always exist, even if a Condorcet winner does not.

Under what specific conditions is a majority-rule winner guaranteed to exist?

A majority-rule winner is guaranteed to exist if voters are positioned along a single-dimensional political spectrum (like left-right) and consistently vote for candidates closer to their own position. In such a scenario, the candidate most representative of the electorate's median position wins.

What is the name of the theorem that describes the guaranteed winner in a single-dimensional political spectrum scenario?

This outcome, where the candidate closest to the median voter's position wins in a single-dimensional political spectrum, is known as the median voter theorem.

Why might real-life political electorates not perfectly align with the median voter theorem's assumptions?

Real-life political electorates are inherently multidimensional, meaning voters' preferences cannot be accurately represented by a single spectrum like left-right, or even just two dimensions. This complexity can lead to situations where the median voter theorem's conditions for a guaranteed winner are not met.

What have studies estimated about the frequency of cyclical preferences (Condorcet paradoxes) in actual elections?

Studies suggest that cyclical preferences, which prevent a Condorcet winner from emerging, are relatively rare in real-world elections. Estimates of their prevalence typically range from 1% to 10% of races.

Which voting systems are known to guarantee the election of a Condorcet winner when one exists?

Voting systems that guarantee the election of a Condorcet winner, when one exists, include methods like Ranked Pairs, Schulze's method, and the Tideman alternative method. These are often referred to as Condorcet methods.

Who is credited with the early detailed study of Condorcet methods, and when did this occur?

The Spanish philosopher and theologian Ramon Llull is credited with the first detailed study of Condorcet methods in the 13th century. His work, however, was lost for about 500 years.

Who rediscovered and advanced the study of Condorcet methods during the Age of Enlightenment?

Nicolas de Caritat, Marquis de Condorcet, a mathematician and political philosopher, rediscovered and significantly advanced the study of these methods during the Age of Enlightenment, leading to their association with his name.

Describe the hypothetical scenario presented in the article involving government funds and voting options.

The article presents a scenario where a government receives a windfall of funds and must decide between three options: spending it, cutting taxes, or paying off debt. Voters participate in pairwise comparisons to determine preferences.

In the hypothetical government funds example, which option was identified as the 'beats-all' winner?

In the hypothetical example, the option to 'Pay debt' was identified as the beats-all winner because it was preferred over both 'Cut taxes' and 'Spend more' in pairwise comparisons, indicating it would defeat any other single option.

How are majority-rule winners generally determined from voter rankings?

Majority-rule winners can be determined from voter rankings by conducting pairwise comparisons between all candidates. The winner is the candidate who receives more than 50% of the votes in each of these head-to-head matchups.

How do Condorcet methods demonstrate resistance to spoiler effects?

Condorcet methods are highly resistant to spoiler effects because the only way to prevent a Condorcet winner from being elected is to directly defeat them in a pairwise contest. This implies that spoiler effects, where a minor candidate draws votes away from a major candidate, are less likely to alter the outcome if a true Condorcet winner exists.

What is the participation criterion in the context of voting systems?

The participation criterion is a standard for voting systems that requires a candidate's election to be unaffected by the withdrawal of voters who do not support them. In simpler terms, it means that if a candidate wins, they should still win if some voters who didn't vote for them abstain.

What do studies suggest about the empirical rarity of participation failures for modern Condorcet methods like ranked pairs?

Studies analyzing election datasets suggest that participation failures are empirically rare for modern Condorcet methods, such as ranked pairs. One survey found no instances of participation failures for methods within the ranked pairs-minimax family.

How does the Condorcet criterion relate to the majority criterion?

The Condorcet criterion implies the majority criterion. This is because a candidate who is ranked first by a majority of voters is, by definition, preferred over every other candidate by that same majority, thus satisfying the conditions of the majority criterion.

What is the mutual majority criterion, and do Condorcet methods satisfy it?

The mutual majority criterion states that if there is a group of candidates who are all mutually preferred over any candidate outside the group, then one of the candidates from that group should win. Condorcet methods do pass this criterion, especially in cases where a Condorcet winner exists, as such a winner is part of the smallest mutual majority set.

What does the Smith criterion guarantee, and how does it relate to Condorcet systems?

The Smith criterion guarantees a stronger form of majority rule. It ensures that if no single majority-rule winner exists, the winner must come from the 'top cycle' or Smith set, which includes candidates who can beat every other candidate, directly or indirectly. Most Condorcet systems satisfy this criterion.

Why does plurality voting fail the Condorcet criterion?

Plurality voting fails the Condorcet criterion primarily due to vote-splitting effects. In this system, voters only cast a single vote for their top choice, and the full spectrum of their preferences is not recorded or considered, which can lead to a candidate winning without being preferred by a majority in pairwise contests.

Describe the hypothetical election scenario used to illustrate the failure of plurality voting.

In the example, 30% of voters prefer A over B over C, 30% prefer C over A over B, and 40% prefer B over A over C. Plurality voting would elect B (40%), but A is the Condorcet winner, beating B 60% to 40% and C 70% to 30%.

What real-life election is cited as a potential example of plurality voting failure due to a spoiler candidate?

The 2000 United States presidential election in Florida is cited as a potential real-life example. It's suggested that many voters preferred Al Gore over George Bush, but Bush won due to Ralph Nader acting as a spoiler candidate.

How does Instant-Runoff Voting (IRV) work?

In Instant-Runoff Voting (IRV), voters rank candidates. The candidate with the fewest first-place votes is eliminated, and their votes are redistributed based on the voters' next preferences. This process continues until one candidate has a majority.

Why does Instant-Runoff Voting (IRV) not comply with the Condorcet criterion?

IRV can fail the Condorcet criterion because it is possible for it to elect a candidate who would lose in a head-to-head contest against another candidate. This happens when the Condorcet winner is eliminated early due to having fewer first-place votes, even if they are preferred by a majority overall.

Describe the hypothetical IRV election scenario where the Condorcet winner is not elected.

In an example with 35 votes for A>B>C, 34 for C>B>A, and 31 for B>C>A, candidate B is the Condorcet winner. However, under IRV, B is eliminated first (fewest first-place votes), and C wins with the transferred votes. This illustrates IRV's failure to elect the Condorcet winner.

What real-life election is mentioned as an instance where IRV failed the Condorcet criteria?

The 2009 mayoral election in Burlington, Vermont, is cited as a real-life example where instant-runoff voting (IRV) failed to comply with the Condorcet criteria.

How does the Borda count system determine a winner?

The Borda count system assigns points to candidates based on their position in a voter's ranked preference list. The candidate who receives the most total points across all ballots is declared the winner.

Why does the Borda count system fail the Condorcet criterion?

The Borda count fails the Condorcet criterion because it can elect a candidate who is not preferred by a majority in pairwise comparisons. It prioritizes overall ranking points rather than direct head-to-head majority support, potentially leading to a winner who loses to another candidate in a one-on-one race.

Describe the hypothetical election scenario used to illustrate the failure of the Borda count.

In an example with five voters, three candidates (A, B, C), and preferences A>B>C (3 voters) and B>C>A (2 voters), B wins the Borda count with 7 points. However, A is the Condorcet winner, preferred by 3 out of 5 voters over B.

How does the highest medians (or Bucklin) system determine a winner?

The highest medians system, also known as Bucklin voting, involves voters rating candidates on a scale (e.g., excellent, good, fair, poor). The winner is the candidate with the best median rating across all voters.

Why does the highest medians system fail the Condorcet criterion?

The highest medians system can fail the Condorcet criterion because it focuses on the median rating rather than pairwise majority preferences. A candidate might achieve a better median rating through widespread 'good' ratings, while another candidate, preferred by a majority in direct contests, might receive a lower median rating.

Describe the hypothetical election scenario used to illustrate the failure of the highest medians system.

In a scenario with 35 voters rating A 'excellent', B 'fair', C 'poor'; 34 voters rating C 'excellent', B 'fair', A 'poor'; and 31 voters rating B 'excellent', C 'good', A 'poor'. B is the Condorcet winner, but C wins the highest medians system with a 'good' median rating compared to B's 'fair' median.

How does Approval Voting work?

In Approval Voting, voters can approve of, or vote for, as many candidates as they wish on a ballot. The candidate who receives the most approvals wins.

Why does Approval Voting fail the Condorcet criterion?

Approval Voting can fail the Condorcet criterion because voters might approve of multiple candidates, potentially leading to a situation where a candidate with broad approval but not necessarily majority preference in pairwise contests wins. For instance, if voters approve their top two choices, a candidate might win with 100% approval even if another candidate is the Condorcet winner.

Describe the hypothetical scenario illustrating Approval Voting's failure to meet the Condorcet criterion.

If 70% of voters prefer A>B>C and 30% prefer C>B>A, and all voters approve their top two choices, Candidate B would win with 100% approval. However, A is the Condorcet winner, beating B 70% to 30%.

How does Score Voting work?

Score Voting, also known as rating voting, requires voters to assign a score to each candidate from a predefined scale (e.g., 0 to 5). The candidate with the highest total score across all voters is the winner.

Why does Score Voting fail the Condorcet criterion?

Score Voting fails the Condorcet criterion because it can elect a candidate who is not the Condorcet winner. This occurs when a candidate's supporters are more enthusiastic (give higher scores) than the supporters of the Condorcet winner, leading to a higher total score despite lower pairwise preference.

Provide an example of how Score Voting might fail to elect the Condorcet winner.

In an example with 45 voters giving A a score of 5/5, B 1/5, C 0/5; 40 voters giving C 5/5, B 1/5, A 0/5; and 15 voters giving B 5/5, C 4/5, A 2/5. C wins with an average score of 2.6, but B is the Condorcet winner, preferred by a majority in head-to-head contests. This also shows that adding a runoff to score voting doesn't always guarantee Condorcet compliance.

What is the relationship between the Condorcet criterion and the majority criterion?

The Condorcet criterion is a stronger condition than the majority criterion. If a voting system satisfies the Condorcet criterion, it automatically satisfies the majority criterion, because a candidate who beats everyone head-to-head must also be preferred by a majority over any single opponent.

What are some specific Condorcet methods listed in the article?

Specific Condorcet methods mentioned include Copeland's method, Dodgson's method, Kemeny method, Minimax Condorcet method, Nanson's method, Ranked Pairs, and Schulze method.

What are some voting systems that do *not* satisfy the Condorcet criterion, according to the article?

The article lists plurality voting, instant-runoff voting (IRV), Borda count, Approval Voting, Coombs' rule, Bucklin voting, and Score Voting as systems that do not satisfy the Condorcet criterion.

What is the significance of the 'Condorcet methods' category on Wikipedia?

The 'Category:Condorcet methods' on Wikipedia serves as a collection point for more detailed information on various voting systems that adhere to the Condorcet criterion, providing a broader overview of these specific electoral approaches.

What is the 'Smith criterion' in the context of electoral systems?

The Smith criterion is a voting system criterion that ensures that if there is a group of candidates who are all mutually preferred over any candidate outside the group (the Smith set), then the winner must be one of these candidates. It's considered a stronger form of majority rule than the Condorcet criterion alone.

What is the 'participation criterion' for voting systems?

The participation criterion is a desirable property for voting systems, ensuring that a candidate's victory is not jeopardized by the abstention of voters who do not support them. It means that if a candidate wins, they should still win if some non-supporters abstain from voting.

What is the 'independence of clones' criterion, and how does it relate to spoiler effects?

The independence of clones criterion is a voting system property that states that if a candidate is replaced by a clone (a candidate with identical preferences), the outcome should not change. This is related to spoiler effects, as clones can sometimes act as spoilers.

What is the 'mutual majority criterion'?

The mutual majority criterion states that if a set of candidates are all mutually preferred over any candidate outside that set, then one of the candidates from that set should win the election. Condorcet methods generally satisfy this criterion.

What are the main categories of electoral systems discussed in the sidebar navigation?

The sidebar navigation categorizes electoral systems into Single-winner systems, Proportional representation systems (further divided by systems, allocation, and quotas), Mixed systems, Semi-proportional systems, and discusses various Criteria and Other related topics.

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What is a Condorcet Winner?

The Majority's Choice

A Condorcet winner is a candidate who would prevail in a head-to-head election against every other candidate. In essence, they are preferred by more than half of the electorate when compared pairwise.^[1]^[2] This principle extends the concept of majority rule to multi-candidate elections, ensuring the winner possesses broad support.^[3] The term originates from the 18th-century French mathematician and political philosopher, Nicolas de Condorcet.^[11]

Pronunciation and Terminology

The name "Condorcet" is pronounced approximately /kənˈdɔːrsɛ/ (kun-DOR-set) in English. A candidate satisfying this criterion is also referred to as a majority winner, a beats-all winner, or a tournament winner, reflecting their dominance in pairwise comparisons.^[3]^[4]

The Ideal of Consensus

Electoral systems that consistently elect a Condorcet winner, when one exists, are said to satisfy the Condorcet winner criterion. This criterion is highly valued as it aligns with the democratic ideal of electing a candidate with the broadest consensus support.

Condorcet's Voting Paradox

The Cycle of Preferences

A significant challenge in social choice theory is the possibility of Condorcet's voting paradox, also known as the majority impossibility or conflicting majorities paradox. This occurs when pairwise comparisons result in a cycle, where candidate A beats B, B beats C, but C beats A. Such cycles mean no single candidate can defeat all others head-to-head, thus no Condorcet winner exists.^[6] This phenomenon is analogous to intransitive dice in probability.

Real-World Prevalence

While theoretically possible, empirical studies suggest that such cycles are relatively infrequent in actual elections, with estimates of their occurrence ranging from 1% to 10% of races.^[10] This suggests that in many real-world scenarios, a Condorcet winner does emerge.

The Smith Set

When a Condorcet cycle occurs, the Smith set (or top cycle) provides a generalization. It is the smallest set of candidates where every candidate within the set beats every candidate outside the set, either directly or indirectly through a chain of pairwise victories. The Smith set always exists and represents the set of "undefeated" candidates in a cycle.

The Median Voter Theorem

A Simplified Political Landscape

In a simplified, one-dimensional political spectrum (e.g., left-right), where voters consistently prefer candidates closer to their own position, the median voter theorem guarantees the existence of a Condorcet winner. This winner is the candidate whose position aligns most closely with the median voter in the electorate.^[7]

Beyond One Dimension

However, real-world political landscapes are inherently multidimensional. Reducing them to a single dimension, or even two, often fails to accurately capture voter preferences. In these multidimensional spaces, the median voter theorem's guarantee of a Condorcet winner does not necessarily hold, increasing the potential for cycles.^[8]^[9]

Voting Methods & Condorcet

Methods Satisfying the Criterion

Several voting systems are designed to ensure that if a Condorcet winner exists, they will be elected. These methods are often considered robust and fair:

Black's method
Kemeny-Young method
Dodgson's method
Minimax Condorcet
Nanson's method
Ranked Pairs
Schulze method
Tideman's alternative method

These are often grouped under the umbrella of "Condorcet methods" or "tournament solutions."^[12]

Methods Failing the Criterion

Conversely, many common voting systems do not guarantee the election of a Condorcet winner, even when one is present. This failure can lead to outcomes that may not reflect the broadest consensus:

Plurality voting (First-past-the-post)
Instant-runoff voting (IRV)
Borda count
Approval Voting
Coombs' rule
Bucklin voting / Median voting
Score voting

These systems may be susceptible to issues like vote-splitting or strategic voting that can obscure the true majority preference.^[13]

Historical Roots

Ancient Origins

The foundational ideas behind Condorcet methods trace back to the 13th century with the work of Ramon Llull, a philosopher and theologian who explored methods for church governance. His manuscript, Ars Electionis, was lost for centuries, delaying the formal study of these concepts.^[11]

The Enlightenment and Condorcet

The formal development of social choice theory and the Condorcet criterion gained significant momentum during the Age of Enlightenment. Nicolas de Caritat, Marquis de Condorcet, a prominent mathematician and political philosopher, extensively studied and advanced these ideas, laying the groundwork for modern understanding of voting systems and collective decision-making.^[11]

Illustrative Example

Windfall Funds Allocation

Consider a scenario where a government receives unexpected funds and must decide how to allocate them: spend more, cut taxes, or pay off debt. Voters express preferences through pairwise comparisons:

	vs. Spend more	vs. Cut taxes
Pay debt	403–305	496–212	2–0
Cut taxes	522–186	1–1
Spend more	0–2

In this example, "Pay debt" wins against "Spend more" (403 vs. 305) and against "Cut taxes" (496 vs. 212). Therefore, "Pay debt" is the Condorcet winner, securing a majority against all other options.^[1]

Key Properties

Stability and Spoiler Resistance

Condorcet methods are highly resistant to spoiler effects. A spoiler candidate is one who, by running in an election, causes a less preferred candidate to win due to vote-splitting. Condorcet methods generally avoid this, as the winner must beat all other candidates head-to-head, implying spoilers can only emerge if there isn't a clear majority winner.^[12]

Participation Criterion

While most Condorcet methods are robust, they can theoretically fail the participation criterion (where adding or removing a non-voting participant should not change the outcome) in specific constructed examples. However, empirical studies suggest this is rare for modern Condorcet methods like Ranked Pairs.^[12]

Majoritarian Alignment

The Condorcet criterion implies other majoritarian criteria, such as the majority criterion (a candidate receiving >50% of first-place votes should win) and the mutual majority criterion (if a group of candidates is mutually preferred by a majority over any candidate outside the group, one of them should win). Systems that satisfy the Condorcet criterion generally align well with these fundamental principles of majority rule.

Method Compliance

Systems That Pass

Many voting systems satisfy the Condorcet winner criterion, ensuring that if a majority winner exists, they are elected. These include:

Black
Kemeny-Young
Dodgson's method
Minimax Condorcet
Nanson's method
Ranked Pairs
Schulze method
Total Vote Runoff
Tideman's alternative method

Systems That Fail

Conversely, several widely used systems do not guarantee the election of a Condorcet winner:

Plurality voting
Instant-runoff voting (IRV)
Borda count
Approval Voting
Coombs' rule
Bucklin voting / Median voting
Score voting

These failures can arise from phenomena like vote-splitting or strategic manipulation.

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References

A full list of references for this article are available at the Condorcet winner criterion Wikipedia page

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The Unassailable Victor

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What is a Condorcet Winner? ℹ️

👑 The Majority's Choice

🗣️ Pronunciation and Terminology

⚖️ The Ideal of Consensus

Condorcet's Voting Paradox 🔄

🎲 The Cycle of Preferences

📊 Real-World Prevalence

🔗 The Smith Set

The Median Voter Theorem 📊

📏 A Simplified Political Landscape

🌐 Beyond One Dimension

Voting Methods & Condorcet ✅

👍 Methods Satisfying the Criterion

❌ Methods Failing the Criterion

Historical Roots 📜

⏳ Ancient Origins

💡 The Enlightenment and Condorcet

Illustrative Example 📝

💰 Windfall Funds Allocation

Key Properties ⚙️

🛡️ Stability and Spoiler Resistance

🚶 Participation Criterion

🤝 Majoritarian Alignment

Method Compliance 📚

✔️ Systems That Pass

❌ Systems That Fail

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Dive in with Flashcard Learning!

What is a Condorcet Winner?

The Majority's Choice

Pronunciation and Terminology

The Ideal of Consensus

Condorcet's Voting Paradox

The Cycle of Preferences

Real-World Prevalence

The Smith Set

The Median Voter Theorem

A Simplified Political Landscape

Beyond One Dimension

Voting Methods & Condorcet

Methods Satisfying the Criterion

Methods Failing the Criterion

Historical Roots

Ancient Origins

The Enlightenment and Condorcet

Illustrative Example

Windfall Funds Allocation

Key Properties

Stability and Spoiler Resistance

Participation Criterion

Majoritarian Alignment

Method Compliance

Systems That Pass

Systems That Fail

Teacher's Corner

True or False?

Test Your Knowledge!

Gamer's Corner

Discover other topics to study!

References

Feedback & Support

Disclaimer

Important Notice