Section 7.3: Condorcet Criterion

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Learning Objectives

• Decide if an election violates the Condorcet criterion

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The plurality method has been used successfully for centuries in stable democracies. It has been used in the United States since 1789 and even the United Kingdom sin medieval times. The method has had a proven track record of peaceful transitions of power. 

However, despite its simplicity, the plurality method has serious flaws that can produce arguably unfair outcomes.

• Winner Without Majority Support: A candidate can win with only 30-40% of votes, meaning a majority opposes them.
• Vote Splitting (Spoiler Effect): Similar candidates can "split" votes between them, causing both to lose to a less popular candidate.
• Strategic Voting (Insincere Voting): Voters may not vote for their true favorite because they think that candidate can't win.
• Lack of Information About Preferences: Plurality only captures first choices, ignoring all other preferences.
• Condorcet Criterion Violation: Plurality can fail to elect the Condorcet winner (the candidate who would beat all others head-to-head).

The last flaw is one that we will further focus on. 

When we evaluate voting methods, we need ways to measure whether they produce "fair" outcomes. Fairness criteria are mathematical standards that describe desirable properties we want in a voting system.

Think of fairness criteria like quality standards for any measurement tool:

• A thermometer should give consistent readings
• A scale should measure the same object the same way each time
• A voting method should satisfy certain logical fairness principles

The big question is if a voting method is "fair," what exactly does that mean? This leads us to the following:

 

Fairness Criteria

A fairness criterion is a precise statement of a specific behavior that we expect to happen if an election is to be considered fair. (Note that “criterion” is the singular form, and “criteria” is the plural form.)

 

The Four Main Fairness Criteria

We'll examine the four most important and intuitive fairness criteria:

• Condorcet Criterion
• Majority Criterion
• Monotonicity Criterion
• Independence of Irrelevant Alternatives Criterion (IIA)

Fairness criteria are tests we use to evaluate whether a voting method produces fair outcomes.

The Process for Fairness

1. Apply a voting method: (Plurality, Borda Count, Plurality with Elimination, or Pairwise Comparison) to determine a winner
2. Test the fairness criteria: Check whether the winner also satisfies each of the four fairness criteria:
   • Condorcet Criterion
   • Majority Criterion
   • Monotonicity Criterion
   • Independence of Irrelevant Alternatives Criterion (IIA)
3. Evaluate the result:
   1. No violation: If the voting method's winner satisfies the criterion
   2. Violation: If the criterion identifies a different candidate who "should" have won

The Goal: Find a voting method that satisfies all four fairness criteria for every possible election scenario.

 

Condorcet Criterion (Fairness Criterion #1)

If a candidate would defeat every other candidate in a head-to-head (one-on-one) comparison, called the Condorcet Criterion, that candidate should win the election. Such a candidate is called a Condorcet winner.

 

Satisfying Fairness Criteria

• When we can show that a voting method is guaranteed to fulfill a given fairness criterion in any possible election, we say that the method satisfies the criterion and there is no violation.
• If a voting method does not satisfy a given fairness criterion, even if it fails in only one election, then we say that the method violates the given criterion.

 

Example #7.3.1 🤔

Consider the following preference table with Candidates A, B, & C:

Number of Voters	5	4	3
First Choice	A	B	C
Second Choice	B	A	A
Third Choice	C	C	B

1. Use the plurality method to determine a winner.
2. Does the election violate the Condorcet criterion?

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✅ Solution:

• Looking at the First Choice row, Candidate A has 5 first place votes, Candidate B has 4 first place votes, and Candidate C has 3 first place votes. Thus, Candidate A is the winner by the plurality method.
• We need to check the one-on-one scores for each possible pair of candidates. The match ups are A vs. B, A vs. C, and B vs. C.

Use a table for each match-up to record your results.

\(\begin{array}{r|l} \text{A}&\text{B} \\ \hline 5& \\  &4 \\ 3& \\ \hline \large \textbf{8} & \large \textbf{4} \\ \end{array}\)

So, for A vs. B,

• Who ranks higher in the first column?: A does, so 5 votes for A.
• Who ranks higher in the second column?: B does, so 4 votes for B.
• Who ranks higher in the third column?: A does, so 3 votes for A.
• Result: A beats B, 8-4

 

\(\begin{array}{r|l} \text{A}&\text{C} \\ \hline 5& \\  4& \\ &3 \\ \hline \large \textbf{9} & \large \textbf{3} \\ \end{array}\)

So, for A vs. C, 

• Who ranks higher in the first column?: A does, so 5 votes for A.
• Who ranks higher in the second column?: A does, so 4 votes for A.
• Who ranks higher in the third column?: C does, so 3 votes for C.
• Result: A beats C, 9-3

 

\(\begin{array}{r|l} \text{B}&\text{C} \\ \hline 5& \\  4& \\ &3 \\ \hline \large \textbf{9} & \large \textbf{3} \\ \end{array}\)

So, for B vs. C, 

• Who ranks higher in the first column?: B does, so 5 votes for B.
• Who ranks higher in the second column?: B does, so 4 votes for B.
• Who ranks higher in the third column?: C does, so 3 votes for C.
• Result: B beats C, 9-3

 

It would be simpler to summarize everything all at once in a table with all of the head-to-head match-ups from above.

\(\begin{array}{r|l} \text{A}&\text{B} \\ \hline 5& \\ &4 \\ 3& \\ \hline \large \textbf{8} & \large \textbf{4} \\ \end{array} \)           \(\begin{array}{r|l} \text{A}&\text{C} \\ \hline 5& \\ 4& \\ &3 \\ \hline \large \textbf{9} & \large \textbf{3} \\ \end{array} \)           \(\begin{array}{r|l} \text{B}&\text{C} \\ \hline 5& \\ 4& \\ &3 \\ \hline \large \textbf{9} & \large \textbf{3} \\ \end{array} \)

A wins B           A wins C          B wins C

 

Here are the head-to-head records:

    A: 2 wins (beat B and C)
    B: 1 win (beat C only)
    C: 0 wins

Thus, Candidate A beats everyone head-to-head. Candidate A is the Condorcet winner.

Since, Candidate A wins by the plurality method and is also the Condorcet winner, there is not a violation of the Condorcet criterion. (NO)

Note: Technically, when using the Condorcet criterion, since Candidate A beat Candidate B and then Candidate A beat Candidate C, we technically do not need to show the last head-to-head match-up, since Candidate A already won the election.

Example #7.3.2 🤔

Consider the following preference table with Candidates X, Y, & Z:

Number of Voters	40	35	25
First Choice	X	Z	Y
Second Choice	Z	Y	Z
Third Choice	Y	X	X

1. Use the plurality method to determine a winner.
2. Does the election violate the Condorcet criterion?

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✅ Solution:

• Looking at the First Choice row, Candidate X has 40 first place votes, Candidate Y has 35 first place votes, and Candidate Z has 25 first place votes. Thus, Candidate X is the winner by the plurality method.
• We need to check the one-on-one scores for each possible pair of candidates. The match ups are X vs. Y, X vs. Z, and Y vs. Z.

Use a table for each match-up to record your results.

\(\begin{array}{r|l} \text{X}&\text{Y} \\ \hline 40& \\ &35 \\ &25 \\ \hline \large \textbf{40} & \large \textbf{60} \\ \end{array}\)           \(\begin{array}{r|l} \text{X}&\text{Z} \\ \hline 40& \\ &35 \\ &25 \\ \hline \large \textbf{40} & \large \textbf{60} \\ \end{array}\)           \(\begin{array}{r|l} \text{Y}&\text{Z} \\ \hline &40 \\ &35 \\ 25& \\ \hline \large \textbf{25} & \large \textbf{75} \\ \end{array}\)

  Y wins X               Z wins X               Z wins Y

 

Here are the head-to-head records:

    X: 0 wins
    Y: 1 win (beat X only)
    Z: 2 wins (beat X and Y)

Thus, Candidate Z beats everyone head-to-head. Candidate Z is the Condorcet winner.

Since, Candidate X wins by the plurality method, yet Candidate Z is the Condorcet winner, there is a violation of the Condorcet criterion. (YES)

Example #7.3.3 🤔

A survey of 23 customers was conducted asking them to rank which protein they prefer at a local Chipotle restaurant. The results are listed below:

Number of Voters	8	7	5	3
First Choice	Steak	Carnitas	Chicken	Steak
Second Choice	Chicken	Barbacoa	Carnitas	Barbacoa
Third Choice	Barbacoa	Chicken	Steak	Chicken
Fourth Choice	Carnitas	Steak	Barbacoa	Carnitas

1. Use the plurality method to determine which protein is the winner.
2. Does the election violate the Condorcet criterion?

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✅ Solution:

• Looking at the First Choice row, Steak has 8 + 3 = 11 first place votes, Carnitas has 7 first place votes, Chicken has 5 first place votes, and Barbacoa has 0 first place votes. Thus, Steak is the winner by the plurality method.
• We need to check the one-on-one scores for each possible pair of candidates until a winner if possible exists. We do not necessarily need to show ALL pairings. If we find a winner first, you may not need to show all head-to-head matchups. Notice here we will start witch Steak first, since they were the winner using the plurality method. This strategy sometimes reduces the number of head-to-head matchups when determining if there was a violation.

    Summarizing the results from the preference table in another table with some to all of the head-to-head match-ups from above, we have

\(\begin{array}{c|c} \text{ Steak }&\text{Carnitas} \\ \hline 8& \\ &7 \\ &5 \\ 3& \\ \hline \large \textbf{11} & \large \textbf{12} \\ \end{array}\)           \(\begin{array}{c|c} \text{Carnitas}&\text{Chicken} \\ \hline &8 \\ 7& \\ &5 \\ &3 \\ \hline \large \textbf{7} & \large \textbf{16} \\ \end{array}\)           \(\begin{array}{c|c} \text{Chicken}&\text{Barbacoa} \\ \hline 8& \\ &7 \\ 5& \\ &3 \\ \hline \large \textbf{13} & \large \textbf{10} \\ \end{array}\)           \(\begin{array}{c|c} \text{Chicken}&\text{ Steak } \\ \hline &8 \\ 7& \\ 5& \\ &3 \\ \hline \large \textbf{12} & \large \textbf{11} \\ \end{array} \)

 Carnitas wins Steak            Chicken wins Carnitas            Chicken wins Barbacoa            Chicken wins Steak

• In the first match-up, Carnitas vs. Steak, Carnitas wins. So, in the next match-up, use Carnitas again against another protein.
• In the second match-up, Carnitas vs. Chicken, Chicken wins. So, in the next match-up, use Chicken again against another protein.
• In the third match-up, Chicken vs. Barbacoa, Chicken wins again. So, in the next match-up, use Chicken again against the final protein.
• In the fourth match-up, Chicken vs. Steak, Chicken wins again. We do not need to show any more match-ups, since we have a winner. We don't need all of the head-to-head records. If possible, once a winner has been obtained, there is no need to show the rest of the other match-ups.

Thus, Chicken beats every other protein head-to-head. Chicken is the Condorcet winner.

Since, Steak wins by the plurality method, yet Chicken is the Condorcet winner, there is a violation of the Condorcet criterion. (YES)

Example #7.3.4 🤔

City council members need to choose a new location for the city's community center. Four sites are being considered:

Number of Voters	35	30	20	15
First Choice	Downtown	Northside	Eastside	Westside
Second Choice	Northside	Eastside	Westside	Eastside
Third Choice	Eastside	Westside	Northside	Northside
Fourth Choice	Westside	Downtown	Downtown	Downtown

1. Use the plurality method to determine which new location is the winner.
2. Does the election violate the Condorcet criterion?

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✅ Solution:

• Looking at the First Choice row, Downtown has 35 first place votes, Northside has 30 first place votes, Eastside has 20 first place votes, and Westside has 15 first place votes. Thus, Downtown is the winner by the plurality method.
• We need to check the one-on-one scores for each possible pair of candidates until a winner if possible exists. We do not necessarily need to show ALL pairings. If we find a winner first, you may not need to show all head-to-head matchups. Notice here we will start witch Downtown first, since they were the winner using the plurality method. This strategy sometimes reduces the number of head-to-head matchups when determining if there was a violation.

    Summarizing the results from the preference table in another table with some to all of the head-to-head match-ups from above, we have

\(\begin{array}{c|c} \text{Downtown}&\text{Northside} \\ \hline 35& \\ &30 \\ &20 \\ &15 \\ \hline \large \textbf{35} & \large \textbf{65} \\ \end{array}\)           \(\begin{array}{c|c} \text{Northside}&\text{Eastside} \\ \hline 35& \\ 30& \\ &20 \\ &15 \\ \hline \large \textbf{65} & \large \textbf{35} \\ \end{array}\)           \(\begin{array}{c|c} \text{Northside}&\text{Westside} \\ \hline 35& \\ 30& \\ &20 \\ &15 \\ \hline \large \textbf{65} & \large \textbf{35} \\ \end{array}\)

Northside wins Downtown           Northside wins Eastside           Northside wins Westside

• In the first match-up, Downtown vs. Northside, Northside wins. So, in the next match-up, use Northside again against another location.
• In the second match-up, Northside vs. Eastside, Northside wins again. So, in the next match-up, use Northside again against the final location.
• In the third match-up, Northside vs. Westside, Northside wins again. We do not need to show any more match-ups, since we have a winner. We don't need all of the head-to-head records. If possible, once a winner has been obtained, there is no need to show the rest of the other match-ups.

Thus, Northside beats every other site head-to-head. Northside is the Condorcet winner.

Since, Downtown wins by the plurality method, yet Northside is the Condorcet winner, there is a violation of the Condorcet criterion. (YES)

Example #7.3.5 🤔

Consider the following preference table with Candidates A, B, & C:

Number of Voters	7	5	2	2
First Choice	C	B	A	C
Second Choice	B	C	B	A
Third Choice	A	A	C	B

1. Use the plurality method to determine which new location is the winner.
2. Does the election violate the Condorcet criterion?

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✅ Solution:

• Looking at the First Choice row, Candidate A has 2 first place votes, Candidate B has 5 first place votes, and Candidate C has 7 + 2 = 9 first place votes. Thus, C is the winner by the plurality method.
• We need to check the one-on-one scores for each possible pair of candidates until a winner if possible exists. We do not necessarily need to show ALL pairings. If we find a winner first, you may not need to show all head-to-head matchups. Notice here we will start witch Candidate C first, since they were the winner using the plurality method. This strategy sometimes reduces the number of head-to-head matchups when determining if there was a violation.

    Summarizing the results from the preference table in another table with some to all of the head-to-head match-ups from above, we have

\(\begin{array}{l|c} \text{C}&\text{A} \\ \hline 7& \\ 5& \\ &2 \\ 1& \\ \hline \large \textbf{2} & \large \textbf{13} \\ \end{array}\)           \(\begin{array}{l|r} \text{C}&\text{B} \\ \hline 7& \\ &5 \\ &2 \\ 1& \\ \hline \large \textbf{8} & \large \textbf{7} \\ \end{array}\)

 C wins A            C wins B

• In the first match-up, C vs. A, C wins. So, in the next match-up, use C again another candidate.
• In the second match-up, C vs. B, C wins again. We do not need to show A vs. B, since we have a winner. We don't need all of the head-to-head records. If possible, once a winner has been obtained, there is no need to show the rest of the other match-ups.

Thus, Candidate C beats everyone head-to-head. Candidate C is the winner by the Condorcet criterion.

Since, Candidate C wins by the plurality method and is also the Condorcet winner, there is not a violation of the Condorcet criterion. (NO)

Example #7.3.6 🤔

A survey of 13 students was conducted to determine how they ranked the Astronomy, Biology, and Chemistry labs from most to least preferred.

Number of Voters	6	5	2
First Choice	Astronomy	Biology	Chemistry
Second Choice	Chemistry	Astronomy	Biology
Third Choice	Biology	Chemistry	Astronomy

1. Use the plurality method to determine which new location is the winner.
2. Does the election violate the Condorcet criterion?

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✅ Solution:

• Looking at the First Choice row, Astronomy has 6 first place votes, Biology has 5 first place votes, and Chemistry has 2 first place votes. Thus, Astronomy is the winner by the plurality method.
• We need to check the one-on-one scores for each possible pair of candidates until a winner if possible exists. We do not necessarily need to show ALL pairings. If we find a winner first, you may not need to show all head-to-head matchups. Notice here we will start witch Astronomy first, since they were the winner using the plurality method. This strategy sometimes reduces the number of head-to-head matchups when determining if there was a violation.

    Summarizing the results from the preference table in another table with some to all of the head-to-head match-ups from above, we have

\(\begin{array}{c|c} \text{Astronomy}&\text{Biology} \\ \hline 6& \\ &5 \\ &2 \\ \hline \large \textbf{6} & \large \textbf{7} \\ \end{array}\)           \(\begin{array}{c|c} \text{Astronomy}&\text{Chemistry} \\ \hline 6& \\ 5& \\ &2 \\ \hline \large \textbf{11} & \large \textbf{2} \\ \end{array}\)           \(\begin{array}{c|c} \text{Biology}&\text{Chemistry} \\ \hline &6 \\ 5& \\ &2 \\ \hline \large \textbf{5} & \large \textbf{8} \\ \end{array}\)

Biology wins Astronomy           Astronomy wins Chemistry           Chemistry wins Biology

• In the first match-up, Astronomy vs. Biology, Biology wins. So, in the next match-up, use Biology and again the other subject.
• In the second match-up, Biology vs. Chemistry, Chemistry wins. So, in the next match-up, use Chemistry again against the Astronomy.
• In the third and final match-up, Astronomy vs. Chemistry, Astronomy wins.

Thus, there is no winner by the Condorcet criterion, because no one particular subject won out the other two.

Since, Astronomy wins by the plurality method, yet there is no Condorcet winner, there is a violation of the Condorcet criterion. (YES)

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Even though a criterion may be violated by a voting method in one particular contest scenario, it does not mean that the voting method always violates the criterion. It simply demonstrates that the voting method has the potential to violate the criterion in certain election scenarios.

With the examples above, we saw a case where the Condorcet criterion satisfied the plurality method as well as a case where the Condorcet criterion violated the plurality method. So, this means that the plurality method is flawed and is not considered a fair method. In the next two sections, we will look at three more major voting methods along with three additional fairness criterion, in hopes to find a voting method that is flawless.

A voting method is considered completely fair or flawless if and only if it satisfies all reasonable fairness criteria under all possible election scenarios.

Requirements for a "fair" method:

• ✓ Never violates the Condorcet criterion
• ✓ Never violates the Majority criterion
• ✓ Never violates the Monotonicity criterion
• ✓ Remains independent of irrelevant alternatives
• ✓ Never exhibits vulnerability to strategic manipulation
• ✓ Treats all voters and candidates symmetrically
• ✓ Produces consistent, predictable outcomes

In other words: A fair method must have zero structural flaws; no scenarios where it produces arguably unjust outcomes.

So, here is our initial table that with regards to the four fairness criteria:

Fairness Criterion \(\Large\longrightarrow\) Major Voting Method \(\Large\downarrow\)	Condorcet Criterion	Majority Criterion	Monotonicity Criterion	Independence of Irrelevant Alternatives Criterion
Plurality	Violation Possible	?	?	?
Borda Count	?	?	?	?
Plurality-with-Elimination	?	?	?	?
Pairwise Comparison	?	?	?	?

The table summarizes the four major voting methods. When a ✔ is present this means that the fairness criteria are always satisfied by the voting methods. If there is at least one case for a violation using a fairness criterion with a voting method, that means that particular voting method is not always fair and marked "Violation Possible". From the five examples above, there were at least three cases where there was a violation when using the plurality method, thus we place "Violation Possible" instead of a ✔.

For the next six sections, we will introduce the rest of the major voting methods as well as the rest of the fairness criterions and fill in the rest of the table above with either a ✔ or a "Violation Possible" in each cell. We are looking for a major voting method that has ✔'s across all four fairness criterion's and unfortunately, we have determined that the plurality method will not satisfy at least one of them.

 

Historical Note: Marquis de Condorcet

The Condorcet criterion is named after Marie Jean Antoine Nicolas de Caritat, Marquis de Condorcet (1743–1794), a French philosopher, mathematician, and political thinker from the Enlightenment era.

Condorcet is best known today for his work on voting theory. He formulated what is now called the Condorcet method of voting, based on pairwise (head-to-head) comparisons between candidates as we learned in this section. This was one of the earliest demonstrations that democratic decision-making can have logical problems.

Condorcet was also a close associate of leading Enlightenment figures such as Voltaire and d’Alembert and was actively involved in the French Revolution, particularly in debates over democracy and education. These experiences shaped his strong advocacy for democratic government, human rights, and freedom of expression. Unusually for the eighteenth century, he supported equal political and civil rights for women, openly opposed slavery, and believed in universal public education grounded in reason and scientific knowledge.

Section 7.3: Condorcet Criterion [In-Class Exercises]

1. Consider the following preference table with Candidates X, Y, & Z:

   Number of Voters	5	4	2	1
   First Choice	X	Y	X	Z
   Second Choice	Y	X	Z	Y
   Third Choice	Z	Z	Y	X

   1. Use the plurality method to determine the winner.
   2. Does the election violate the Condorcet criterion?

2. Consider the following preference table with Candidates A, B, & C:

   Number of Voters	7	6	6	5	3
   First Choice	C	B	A	B	A
   Second Choice	A	A	B	C	C
   Third Choice	B	C	C	A	B

   1. Use the plurality method to determine the winner.
   2. Does the election violate the Condorcet criterion?

3. A group of 41 people is choosing where to eat lunch during their meeting. The four fast food delivery options are: Board & Brew, Naugles, T.K. Burgers, or In-N-Out Burgers. Each voter ranks the restaurants from most preferred to least preferred.

   Number of Voters	11	9	8	6	4	3
   First Choice	In-N-Out	T.K. Burgers	Naugles	Board & Brew	T.K.Burgers	In-N-Out
   Second Choice	Naugles	Naugles	T.K. Burgers	T.K. Burgers	In-N-Out	T.K. Burgers
   Third Choice	T.K. Burgers	Board & Brew	In-N-Out	In-N-Out	Naugles	Board & Brew
   Fourth Choice	Board & Brew	In-N-Out	Board & Brew	Naugles	Board & Brew	Naugles

   1. Use the plurality method to determine which restaurant is the winner.
   2. Does the election violate the Condorcet criterion?

Answers

1. 1. Candidate X (7 votes)
   2. Candidate X wins by the plurality method and is also the Condorcet winner. Thus, there is no violation of the Condorcet criterion. (NO)
2. 1. Candidate B (11 votes).
   2. Candidate B wins by the plurality method, yet there is no Condorcet winner. Thus, the method violates the Condorcet criterion. (YES)
3. 1. In-N-Out (14 votes)
   2. In-N-Out wins by the plurality method, yet T.K. Burgers is the Condorcet winner. Thus, the method violates the Condorcet criterion. (YES)

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