36 Grading vs. Ranking

In previous chapters, we studied two types of voting methods: those using ranked ballots, where voters order candidates from most to least preferred, and those using graded ballots, where voters assign scores or grades to each candidate independently. These represent fundamentally different approaches to capturing voter preferences.

While grading systems might seem more expressive than ranking systems by allowing voters to indicate how much they like or dislike each candidate, they come with their own challenges and paradoxes. In this chapter, we examine two fundamental issues that arise with grading systems. First, we explore the paradox of grading systems, where grade-based methods can select winners that most voters consider inferior. Second, we discuss the preference intensity problem, which asks how voting methods should account for the strength of voters’ preferences, not just their direction.

36.1 Paradox of Grading Systems

Consider this example with three voters grading two candidates on a scale from 0 to 20:

\[\begin{array}{|l|c|c|c|c|} \hline \# Voters & $a$ & $b$ \\ \hline 1 & 20 & 11 \\ 1 & 9 & 0 \\ 1 & 9 & 10 \\ \hline Median & 9 & 10 \\ \hline \end{array}\]

According to Majority Judgment, candidate $b$ wins with a median grade of 10 compared to $a$’s median grade of 9. However, two out of three voters prefer $a$ to $b$ based on their grades!

This discrepancy becomes even more striking if we change the number of voters while keeping the same pattern of grades:

\[\begin{array}{|l|c|c|c|c|} \hline \# Voters & a & b \\ \hline 50 & 20 & 11 \\ 50 & 9 & 0 \\ 1 & 9 & 10 \\ \hline Median & 9 & 10 \\ \hline \end{array}\]

Now candidate $b$ still wins under Majority Judgment despite being preferred by only 1 voter out of 101.

We have a similar issue with Score Voting. Consider an election with three candidates ($a$, $b$, and $c$) where voters can assign grades from 0 to 5. Here’s how five voters grade the candidates:

\[\begin{array}{|l|c|c|c|c|c|} \hline \# Voters & a & b & c \\ \hline 1 & 5 & 0 & 0 \\ 4 & 0 & 1 & 1\\ \hline Average & 1 & 4/5 & 4/5 \\ \hline \end{array}\]

The Score Voting winner is candidate $a$ with a mean grade of 1, even though the vast majority of voters (4 out of 5) prefer $b$ and $c$ to $a$.

These examples illustrate what Brams and Potthoff describe as the Paradox of Grading Systems: a conflict between evaluating candidates based on their overall grades and comparing them directly in pairs. Interestingly, increasing the number of grades to let voters make finer distinctions between candidates can introduce new challenges in combining individual preferences into a collective decision.

36.2 Preference Intensity Problem

The Paradox of Grading Systems, discussed in the previous section, connects closely to a key issue in voting theory that has been extensively debated in Political Science: the preference intensity problem. This problem arises when we move beyond considering which candidate voters prefer to examining how strongly they hold these preferences.

Consider an election where 51% of voters slightly prefer candidate $a$ to candidate $b$, while 49% strongly prefer $b$ to $a. Since candidate $a$ has majority support, they would typically win the election. But should we automatically select $a$ simply because they are the majority winner? What if the 49% who prefer $b$ feel much more strongly about their choice than the 51% who favor $a$? Should the election result remain the same?

Many would argue (at least in this scenario) that there is a compelling reason to elect $b$, despite the majority ranking $a$ above $b$. But how far should we prioritize intensity of preferences? For example, if 80% of voters slightly prefer $a$ to $b$, while 20% hold an extremely strong preference for $b$ over $a$, should we still override the majority? How do we balance the majority’s preferences against the strength of the minority’s feelings?

This issue becomes even more pressing when we consider the potential harm to minority groups. For example, if electing candidate $a$ would cause significant harm to a minority group, should the majority’s preferences still take precedence? How can we balance the importance of respecting the majority’s will with the ethical responsibility to safeguard minority interests?

This problem suggests several important points:

Not all decisions should be made by voting: Some issues may require deliberation and consensus-building rather than simple majority rule.
Structural solutions may be necessary: When systematic minorities exist, mechanisms like fair districting (see https://mggg.org/) or power-sharing arrangements might better ensure equitable outcomes.
Challenges of capturing preference intensity: Traditional voting methods capture only the direction of preferences, not their strength. While grading systems like Score Voting attempt to address this by allowing voters to express the intensity of their preferences, they raise new challenges, as discussed in the previous section.