35 Evaluative Voting

In the previous chapters, we assumed that the input to a voting method or a social welfare function is the voters’ rankings of the candidates. However, Arrow’s theorem highlights the limitations of relying solely on rankings. One response to these challenges is to allow voters to provide more information about their opinions of the candidates beyond a simple ranking.

Evaluative voting methods offer an alternative approach by enabling voters to rate or score candidates using a predefined scale or language of evaluation. This approach provides voters with the flexibility to express the intensity of their preferences and to evaluate each candidate independently. As a result, evaluative voting methods can capture a different perspective on voter preferences compared to ranking-based methods.

This chapter introduces several prominent evaluative voting methods that have gained attention in the literature and real-world applications.

35.1 Approval Voting

Approval Voting is a simple method where each voter selects a subset of candidates that they “approve” of. The candidate with the most total approvals is declared the winner, with ties between candidates receiving the same number of approvals.

Equivalently, we can think of each voter assigning either a 1 (approve) or 0 (not approve) to each candidate. The candidate with the highest total score wins. Here’s an example with three candidates and five voters:

\[\begin{array}{|c|c|c|c|} \hline \# Voters & a & b & c \\ \hline 2 & 1 & 0 & 1 \\ \hline 2 & 0 & 0 & 1 \\ \hline 1 & 1 & 1 & 0 \\ \hline \end{array}\]

Calculating the total approvals:

Candidate \(a\): 3 approvals (from two voters in the first group and one in the third).
Candidate \(b\): 1 approval (from one voter in the third group).
Candidate \(c\): 4 approvals (from all voters in the first two groups).

Therefore, \(c\) wins with the most approvals.

It is important to note that Approval Voting asks voters for fundamentally different information than ranked voting methods. While we can assume voters prefer candidates they approve to those they don’t approve, we cannot determine a voter’s approval set just from their ranking. One natural assumption connecting preferences and approvals is that if a voter approves of candidate \(x\), then they must also approve of any candidate they prefer to \(x\).

For example, a voter that has the ranking \(a\ b\ c\) might have any of the following approvals:

Only their top choice (\(a\)).
Their top two choices (\(a\) and \(b\)).
All candidates (\(a\), \(b\), and \(c\)).
Or even none of the candidates.

Approval Voting’s simple binary choice - approve or not approve - can be viewed as a special case of a more general approach where voters evaluate candidates by assigning grades or scores. This raises a natural question: why restrict voters to just two possible grades? When voters can assign more than two grades, we need to specify how to determine which candidates have the best “social grade” - that is, how to aggregate individual grades for a candidate into a collective grade for each candidate.

35.2 Score Voting

In Score Voting (also called Range Voting), voters assign each candidate a numerical grade from a fixed set, such as the integers from 0 to 10. The method then aggregates these individual grades by taking their average - the candidate with the highest average grade wins.

For example, suppose eight voters grade two candidates \(a\) and \(b\) as follows:

Candidate \(a\) receives grades: \(7, 7, 8, 8, 9, 9, 9, 10\).
Candidate \(b\) receives grades: \(9, 9, 9, 9, 9, 10, 10, 10\).

Since \(b\)’s average grade (\(9.375 = \frac{9+9+9+9+9 + 10 + 10 + 10}{8}\)) is higher than \(a\)’s average grade (\(8.375 = \frac{7+7+8+8+9+9+9 + 10}{8}\)), candidate \(b\) wins.

Like Approval Voting, Score Voting allows voters to express support for multiple candidates. However, it provides more flexibility by letting voters distinguish between different levels of support rather than just approve/disapprove.

35.3 Majority Judgment

Majority Judgment is similar to Score Voting but uses a different method to aggregate grades. Instead of using the average grade, it selects the candidate with the highest median grade as the winner.

The method works as follows:

Voters assign each candidate a grade from a fixed set (for example, integers from 0 to 10).
For each candidate, find their median grade.
The candidate with the highest median grade wins.

For cases with an even number of voters, Majority Judgment uses the “lower median.” For example, if a candidate receives the grades: \[7, 7, 8, 8, 11, 11, 11, 13.\]

The lower median is 8, since it’s the lower of the two middle values (8 and 11).

To see how this works, consider an example with eight voters grading candidates \(a\) and \(b\):

Candidate \(a\) receives grades: \(7, 7, 8, 8, 9, 9, 9, 10\).
Candidate \(b\) receives grades: \(9, 9, 9, 9, 9, 10, 10, 10\).

Since \(b\)’s median grade (9) is higher than \(a\)’s median grade (8), candidate \(b\) wins.

35.4 Average vs. Median

The choice between using the average (Score Voting) or the median (Majority Judgment) to aggregate grades is fundamental. The average is sensitive to outliers, while the median is more robust to extreme grades. This can lead to different outcomes, as shown in these examples:

Example 1: Consider this election with five voters and three candidates:

\[\begin{array}{|l|c|c|c|c|} \hline \# Voters & a & b & c \\ \hline 1 & 4 & 3 & 1 \\ 1 & 4 & 3 & 2 \\ 1 & 2 & 0 & 3 \\ 1 & 2 & 3 & 4 \\ 1 & 1 & 0 & 2 \\ \hline Average & 2.6 & 1.8 & 2.4 \\ Median & 2 & 3 & 2 \\ \hline \end{array}\]

Using averages, candidate \(a\) wins with the highest average grade (2.6). However, using medians, candidate \(b\) wins with the highest median grade (3).

Example 2: Here’s another example (p. 282 from the Majority Voting book) with two candidates and \(2k+1\) voters:

\[\begin{array}{|l|c|c|c|c|} \hline \# Voters & a & b \\ \hline k & 20 & 20 \\ 1 & 19 & 20 \\ k &19 & 0 \\ \hline Average & \textrm{slightly under 19.5} & \textrm{slightly over 10} \\ Median & 19 & 20 \\ \hline \end{array}\]

Using averages, candidate \(a\) wins with a mean grade of approximately 19.5. Using medians, candidate \(b\) wins with a median grade of 20. This example shows how extreme low grades can significantly affect the average while having no impact on the median.