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Fairness in Voting Methods

Compare and contrast fairness of voting using majority criterion.

Learning Objectives

After completing this section, you should be able to:

  1. Compare and contrast fairness of voting using majority criterion.
  2. Compare and contrast fairness of voting using head-to-head criterion.
  3. Compare and contrast fairness of voting using monotonicity criterion.
  4. Compare and contrast fairness of voting using irrelevant alternatives criterion.
  5. Apply Arrow’s Impossibility Theorem when evaluating voting fairness.

The Majority Criterion

One of the most fundamental concepts in voting is the idea that most voters should be in favor of a candidate for a candidate to win, and that a candidate should not win without majority support. This concept is known as the majority criterion.

With respect to the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, and the Borda count method—we will explore two important questions:

  1. Which of these voting systems satisfy the majority criterion and which do not?
  2. Is it always “fair” for a voting system to satisfy the majority criterion?

Keep in mind that this criterion only applies when one of the candidates has a majority. So, the examples we will analyze will be based on scenarios in which a single candidate has more than 50 percent of the vote.

Roommates Choose Fast Food

Try it.

It’s final exams week and seven college students are hungry. They must get food, but from which drive thru? Their preferences are listed in the table below. The majority have listed McDonald’s as their top choice. Let’s calculate what the results of the election will be using various voting methods.

VotersABCDEFG
(M) McDonald’s1111555
(B) Burger King2222222
(T) Taco Bell4434331
(O) Pollo Tropical3543413
(I) Pizza Hut5355144
  1. Which restaurant is the winner using the plurality voting method?
  2. Which restaurant is the winner using the ranked-choice voting method?
  3. Does the majority criterion apply? If so, for which of voting method(s), if any, did the majority criterion fail?
Solution

  1. For plurality voting, we only need to count the first-place votes for each candidate. In this case, McDonald’s has four first place votes, which is a majority and wins the election automatically.
  2. For ranked-choice voting, McDonald’s also wins because it has a majority at the end of Round 1.
  3. The majority criterion does apply because one candidate had a majority of the first-place votes. The majority criterion did not fail by either method, because in each case the majority candidate won.

From , it appears that the plurality and ranked-choice voting methods satisfy the majority criterion. In general, the majority candidate always wins in a plurality election because the candidate that has more than half of the votes has more votes than any other candidate. The same is true for ranked-choice voting; and there will never be a need for a second round when there is a majority candidate. Let’s examine how some of the other voting methods stand up to the majority criterion.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Head-to-Head Criterion

Another fairness criterion you must consider as you select a voting method for Imaginaria is the Condorcet criterion, also known as the head-to-head criterion. An election method satisfies the Condorcet criterion provided that the Condorcet candidate wins the election whenever a Condorcet candidate exists. A Condorcet method is any voting method that satisfies the Condorcet criterion.

Recall from Three Key Questions that not every election has a Condorcet candidate; the Condorcet criterion will not apply to every election. Also recall that a Condorcet candidate cannot lose an election by pairwise comparison. So, the pairwise comparison voting method is said to satisfy the Condorcet criterion.

Spending Tax Refund

Try it.

A survey asked a random sample of 100 people in the United States to rank their priorities for spending their tax refund. The options were (V) go on vacation, (S) put into savings, (D) pay off debt, or (T) other. The pairwise comparison matrix for the results is in . Determine whether the Condorcet criterion applies.

Solution

The Condorcet criterion only applies when there is a Condorcet candidate. “Pay off debt” (D) is a Condorcet candidate because D wins every matchup. Yes, the Condorcet criterion applies to this election.

As we have seen, the plurality method, ranked-choice voting, and the Borda count method each fail the Condorcet criterion in some circumstances. Of the four main ranked voting methods we have discussed, only the pairwise comparison method satisfies the Condorcet criterion every time. A summary of each voting method as it relates to the Condorcet criterion is found in the following table.

Voting MethodCondorcet Criterion
PluralityViolates
Ranked-choiceViolates
Pairwise comparisonSatisfies
Borda countViolates

Condensed — the full section is in OpenStax Contemporary Mathematics.

Monotonicity Criterion

The citizens of Imaginaria might be surprised to learn that it is possible for a voter to cause a candidate to lose by ranking that candidate higher on their ballot. Is that fair? Most voters would say, “Absolutely not!!” This is an example of a violation of the fairness criterion called the monotonicity criterion, which is satisfied when no candidate is harmed by up-ranking nor helped by down-ranking, provided all other votes remain the same.

Consider a scenario in which voters are permitted a first round that is not binding, and then they may change their vote before the second round. Such a first round can be called a “straw poll.” Now, let’s suppose that a particular candidate won the straw poll. After that, several voters are convinced to increase their support, or up-rank, that winning candidate and no voters decrease that support. It is reasonable to expect that the winner of the first round will also win the second. Similarly, if some of the voters decide to decrease their support, or down-rank, a losing candidate, it is reasonable to expect that candidate will still lose in the second round.

You might be wondering why it’s called the monotonicity criterion. In mathematics, the term monotonicity refers to the quality of always increasing or always decreasing. For example, a person’s age is monotonic because it always increases, whereas a person’s weight is not monotonic because it can increase or decrease. If the only changes to the votes for a particular candidate after a straw poll are in one direction, this change is considered monotonic.

If you are going to make an informed decision about which voting method to use in Imaginaria, you need to know which of the four main ranked voting methods we have discussed—plurality, ranked-choice, pairwise comparison, and the Borda count method—satisfy the monotonicity criterion.

The last few examples illustrate that the plurality method, pairwise comparison voting, and the Borda count method each satisfy the monotonicity criterion. Of the four main ranked voting methods we have discussed, only the ranked-choice method violates the monotonicity criterion. A summary of each voting method as it relates to the Condorcet criterion is found in the table below.

Voting MethodMonotonicity Criterion
PluralitySatisfies
Ranked-choiceViolates
Pairwise comparisonSatisfies
Borda countSatisfies

Condensed — the full section is in OpenStax Contemporary Mathematics.

Irrelevant Alternatives Criterion

We have covered a lot about voting fairness, but there is one more fairness criterion that you and the other Imaginarians should know. Consider this well-known anecdote that is sometimes attributed to the American philosopher Sidney Morgenbesser:

A man is told by his waiter that the dessert options this evening are blueberry pie or apple pie. The man orders the apple pie. The waiter returns and tells him that there is also a third option, cherry pie. The man says, “In that case, I would like the blueberry pie.” (Gaming the Vote: Why Elections Aren’t Fair (and What We Can Do About It), William Pound stone, p. 50, ISBN 0-8090-4893-0)

This story illustrates the concept of the Irrelevant Alternatives Criterion, also known as the Independence of Irrelevant Alternatives Criterion (IIA), which means that the introduction or removal of a third candidate should not change or reverse the rankings of the original two candidates relative to one another. In particular, if a losing candidate is removed from the race or if a new candidate is added, the winner of the race should not change.

Apple, Blueberry, or Cherry?

Try it.

Suppose that 30 students in a class are going to vote on whether to have apple, blueberry, or cherry pie. Use the summary of ranked ballots in below to answer each question.

Number of Ballots14124
(A) Apple Pie133
(B) Blueberry Pie212
(C) Cherry Pie321
  1. Determine the winner of the election by plurality.
  2. Which candidate would win a plurality election if cherry pie were removed from the ballot?
  3. Does this election violate the IIA?
Solution

  1. The number of first place votes for each candidate is: A with 14, B with 12, and C with 4. Apple pie has the most first-place votes and wins the election.
  2. If cherry pie is removed from the ballot, then the four voters in the third column now rank blueberry pie as their first choice. So the four votes for C now belong to B. This means that blueberry pie has 16 votes compared to the 14 votes for apple pie. Blueberry pie now wins the plurality election.
  3. Yes, the election violates the IIA because the removal of a losing candidate from the ballot changed the winner of the election.

We have seen that all four of the main voting systems we are working with fail the Irrelevant Alternatives Criterion (IIA). A summary of each voting method as it relates to the IIA criterion is found in the table below.

Voting MethodIrrelevant Alternatives Criterion
PluralityViolates
Ranked-choiceViolates
Pairwise comparisonViolates
Borda countViolates

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

  • There are several common measures of voting fairness, including the majority criterion, the head-to head criterion, the monotonicity criterion, and the irrelevant alternatives criterion.
  • According to Arrow’s Impossibility Theorem, each voting method in which the only information is the order of preference of the voters will violate one of the fairness criteria.

Practice (8)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. It’s final exams week and seven college students are hungry. They must get food, but from which drive thru? Their preferences are listed in the table below. The majority have listed McDonald’s as their top choice. Let’s calculate what the results of the election will be using various voting methods.

    VotersABCDEFG
    (M) McDonald’s1111555
    (B) Burger King2222222
    (T) Taco Bell4434331
    (O) Pollo Tropical3543413
    (I) Pizza Hut5355144
    1. Which restaurant is the winner using the plurality voting method?
    2. Which restaurant is the winner using the ranked-choice voting method?
    3. Does the majority criterion apply? If so, for which of voting method(s), if any, did the majority criterion fail?
    Révèle la réponse

    1. For plurality voting, we only need to count the first-place votes for each candidate. In this case, McDonald’s has four first place votes, which is a majority and wins the election automatically.
    2. For ranked-choice voting, McDonald’s also wins because it has a majority at the end of Round 1.
    3. The majority criterion does apply because one candidate had a majority of the first-place votes. The majority criterion did not fail by either method, because in each case the majority candidate won.

  2. Those seven college students are hungry again! Their preferences haven’t changed, as shown below. Let’s calculate if the results change when we use different voting methods.

    VOTERSABCDEFG
    (M) McDonald’s1111555
    (B) Burger King2222222
    (T) Taco Bell4434331
    (O) Pollo Tropical3543413
    (I) Pizza Hut5355144

    1. Which restaurant is the winner using the pairwise comparison voting method?
    2. Which restaurant is the winner using the Borda count voting method?
    3. Does the majority criterion apply? If so, for which voting method(s), if any, did the majority criterion fail?
    Révèle la réponse
    1. For pairwise comparison, notice that McDonald’s is a Condorcet candidate because it wins every pairwise comparison. So, McDonald’s is the winner.
    2. For the Borda count, we must calculate the Borda score for each candidate: McDonald’s is 16, Burger King is 21, Taco Bell is 13, Pollo Tropical is 12, Pizza Hut is 8. The winner is Burger King!
    3. Yes, the majority criterion applies because McDonald’s has the majority of first place votes. The majority criterion only fails using the Borda method.

  3. A survey asked a random sample of 100 people in the United States to rank their priorities for spending their tax refund. The options were (V) go on vacation, (S) put into savings, (D) pay off debt, or (T) other. The pairwise comparison matrix for the results is in . Determine whether the Condorcet criterion applies.

    Révèle la réponse

    The Condorcet criterion only applies when there is a Condorcet candidate. “Pay off debt” (D) is a Condorcet candidate because D wins every matchup. Yes, the Condorcet criterion applies to this election.

  4. Let’s return to the survey about tax refund spending from . We know that the Condorcet criterion applies because Option D, “Pay off debt,” is a Condorcet candidate, which wins every pairwise match up.

    Use the information in the ballot summary from the table below to find the winner and determine whether the Condorcet criterion is satisfied in this election when each of the following voting methods are used.

    Votes3332314
    On a vacation (V)1332
    Put into savings (S)3121
    Pay off debt (D)2214
    Other (T)4443
    1. Plurality
    2. Ranked-choice voting
    3. Borda count
    Révèle la réponse
    1. V wins 33 first place votes; S, 36; D, 31; and T, 0. So candidate S, “Put into savings,” has a plurality and wins. Since the Condorcet candidate D didn’t win, the Condorcet criterion is violated.
    2. Use the steps outlined in Ranked-Choice Voting for determining the winner of an election by ranked-choice voting, the application of the Hare method in which instant runoffs are used.

      Step 1: The number of votes needed to achieve a majority is 51.

      Step 2: As illustrated in part 1, no candidate has a majority of first-place votes; so the candidate with the fewest votes, T, must be eliminated.

      Step 3: Reallocate votes to the remaining candidates for the second round:

      Votes3332314
      On a vacation (V)1332
      Put into savings (S)3121
      Pay off debt (D)2213

      Step 4: Repeat the process from Step 2. Count the first-place votes for each candidate: V has 33 votes, S has 36 votes, D has 31. votes. Eliminate candidate D, “Pay off debt,” for the third round.

      Step 5: Repeat the process from Step 3. Reallocate votes to the remaining candidates for the third round:

      Votes3332314
      On a vacation (V)1222
      Put into savings (S)2111

      Step 6: Repeat the process from Step 2 one last time. Count the first-place votes for each candidate: V has 33, S has 67. Candidate S, “Put into savings,” has a majority and wins. Since the Condorcet candidate D candidate D, “Pay off debt,” didn’t win, the Condorcet criterion is violated.

    3. Calculate the Borda score for each candidate.

      V: \(33(4-1)+32(4-3)+31(4-3)+4(4-2)=170\)

      S: \(33(4-3)+32(4-1)+31(4-2)+4(4-1)=203\)

      D: \(33(4-2)+32(4-2)+31(4-1)+4(4-4)=223\)

      T: \(33(4-4)+32(4-4)+31(4-4)+4(4-3)=4\)

    Candidate D, “Pay off debt,” has the highest Borda score and wins. Since D was the Condorcet candidate, this election satisfies the Condorcet criterion.

  5. The local animal shelter is having a vote-by-donation charity event. For a $10 donation, an individual can complete a ranked ballot indicating their favorite large dog breed: standard poodle, golden retriever, Labrador retriever, or bulldog. Use the summary of ballots below to answer each question.

    Votes42536124
    (S) Standard Poodle1321
    (G) Golden Retriever3144
    (L) Labrador Retriever4212
    (B) Bulldog2433
    1. Determine the winner of the election by plurality.
    2. Suppose that the 53 voters in the second column increased their ranking of the winner by 1. Determine the winner by plurality with the new rankings.
    3. Does this election violate the monotonicity criterion?
    4. Do you think the result of part 3 is also true for plurality voting and the monotonicity criterion in general? Why or why not?
    Révèle la réponse

    1. The number of votes for each candidate are: S 66, G 53, L 61, and B 0. The winner is the standard poodle.
    2. If the 53 voters in the second column rank S as 2 and L as 3, then the number of votes for each candidate are: S with 66, G with 53, L with 61, and B with 0. The winner is still the standard poodle.
    3. This election does not violate the monotonicity criterion because the winner was not hurt by up-ranking.
    4. In general, increasing the ranking for a winner of a plurality election will either leave them with the same or more first place votes while leaving the other candidates with the same or fewer first place votes. So a plurality election will never violate the monotonicity criterion.

  6. Earlier, we discovered that the summary of ranked ballots shown in the table below results in the pairwise comparison matrix in . Use this information to answer the questions.

    Number of Ballots9590110115
    Option A4411
    Option B2222
    Option C3134
    Option D1343

    1. Determine the winner of the election by the pairwise comparison method.
    2. Suppose that the 95 voters in the first column increased their ranking of the winner by 1. Determine the winner by the pairwise comparison method with the new rankings.
    3. Does this election violate the monotonicity criterion?
    4. Do you think the result of part 3 is true for the pairwise comparison method and the monotonicity criterion in general? Why or why not?
    Révèle la réponse

    1. By the pairwise comparison method, Option A wins with three points.
    2. If the 95 voters in the first column of increased their ranking of the winner by 1, then C would fall into fourth place and A would move up to third place on those ballots. This would only affect the matchup between A and C, and the result would be that A would gain 95 votes while C would lose 95 votes. This means A would have 320 votes and C would have 90. Since A already was ahead of C, this just puts A further ahead and causes no change to the election results.
    3. Since the winner A is not hurt by an up-rank, and the loser C is not helped by a down-rank, this election is fair by the monotonicity criterion.
    4. Yes, the monotonicity criterion would be satisfied by the pairwise comparison method, because an up-rank of the winner can never decrease the number of pairwise wins. Similarly, a down-rank can never increase the number of pairwise wins.

  7. Suppose that 30 students in a class are going to vote on whether to have apple, blueberry, or cherry pie. Use the summary of ranked ballots in below to answer each question.

    Number of Ballots14124
    (A) Apple Pie133
    (B) Blueberry Pie212
    (C) Cherry Pie321
    1. Determine the winner of the election by plurality.
    2. Which candidate would win a plurality election if cherry pie were removed from the ballot?
    3. Does this election violate the IIA?
    Révèle la réponse

    1. The number of first place votes for each candidate is: A with 14, B with 12, and C with 4. Apple pie has the most first-place votes and wins the election.
    2. If cherry pie is removed from the ballot, then the four voters in the third column now rank blueberry pie as their first choice. So the four votes for C now belong to B. This means that blueberry pie has 16 votes compared to the 14 votes for apple pie. Blueberry pie now wins the plurality election.
    3. Yes, the election violates the IIA because the removal of a losing candidate from the ballot changed the winner of the election.

  8. The NBC sitcom The Office ran for nine years and has been one of the most popular streamed television shows of all time. One of the trademarks of the show was that characters would often break the fourth wall to communicate with the audience just by staring directly into the camera. In fact, there is a website dedicated to "The Office stares" where you can watch over 700 of these stares! Suppose that 36 fans were asked which character had the best "The Office stare." Use the ballot summary in below to answer each question.

    Number of Ballots911763
    (J) Jim Halpert (John Krasinski)12424
    (P) Pam Beesly-Halpert (Jenna Fischer)41243
    (D) Dwight Schrute (Rainn Wilson)23312
    (M) Michael Scott (Steve Carell)34131
    1. Determine the winner of the election by the pairwise comparison method.
    2. Determine the winner of the election by the pairwise comparison method if Michael Scott is removed from the ballot.
    3. Does this election violate IIA?

    Révèle la réponse
    1. Construct and analyze a pairwise comparison matrix:

      Jim Halpert wins with 2 points.

    2. If Michael Scott is removed, the summary of ranked ballots becomes:
      Number of Ballots911763
      (J) Jim Halpert (John Krasinski)12323
      (P) Pam Beesly-Halpert (Jenna Fischer)31132
      (D) Dwight Schrute (Rainn Wilson)23211

      Construct and analyze a pairwise comparison matrix:

      Pam wins with \(1\frac{1}{2}\) points.

    3. Yes, this violates the IIA, because the winning candidate was hurt by the elimination of a losing candidate.

Symbols used here

n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\emptyset,\ |A|
empty set, cardinality
The set with no elements; the number of elements of A.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow
not, and, or, implies, iff
Logical connectives.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
O(n^2),\ \Theta,\ \Omega
big-O notation
Grows no faster than n² (up to a constant), for large n.
a \bmod n
remainder
What is left after dividing a by n.

How to: Fairness in Voting Methods

  1. Compare and contrast fairness of voting using majority criterion.
  2. Compare and contrast fairness of voting using head-to-head criterion.
  3. Compare and contrast fairness of voting using monotonicity criterion.
  4. Compare and contrast fairness of voting using irrelevant alternatives criterion.
  5. Apply Arrow’s Impossibility Theorem when evaluating voting fairness.
  6. Which of these voting systems satisfy the majority criterion and which do not?
  7. Is it always “fair” for a voting system to satisfy the majority criterion?
  8. Which restaurant is the winner using the plurality voting method?

Questions people ask

What makes mathematics "discrete"?

It deals with separate, countable objects — integers, graphs, statements — rather than continuous quantities. No limits, no infinitesimals; instead induction, counting and logic.

How does a proof by induction work?

Show the statement for the first case, then show that whenever it holds for n it holds for n + 1. Like dominoes: the first falls, and each knocks over the next.

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Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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