Cardinal Voting

As I have studied voting methods over the years, I have come to favor cardinal (or evaluative) methods, where voters give each candidate a grade on some grading scale.

  • With Score Voting, voters grade the candidates on a numeric scale from, say, 0 to 5.
  • When using Majority Judgment, they use evaluative words for the grades (e.g. Excellent, Very Good, Good, Fair, Poor, Reject).
  • And in Approval Voting, voters give one of only two grades, approve or disapprove, to each of the candidates.

In each of these cardinal methods, voters should analyze and evaluate each candidate independently. Hence a voter can award the same grade to multiple candidates (in contrast to ranking-based voting methods where each rank must be used exactly once). Without even discussing how the winner is chosen with these different methods, I have come to believe that the form of the ballot itself matters a great deal. Asking voters to evaluate each candidate independently gives us the best basis for achieving elections with lower animosity, higher participation, and better outcomes than the moribund political stalemate we have fallen into.


Interestingly enough, the standard canon of voting theory (also known as "social choice theory") ignores cardinal voting methods, analyzing everything through the framework of rankings only. From what I can tell, there are three reasons for this:

  1. Much of the research in voting theory has been done by economists. In general, economists disregard cardinal utilities and focus on ordinal utilities. Here is the argument from one of my favorite economists:

    [B]y the early twentieth century, most economists had abandoned the idea of a measurable (or cardinal) unit of utility. Instead, they explained the actions of consumers and producers by reference to a ranking of desired outcomes (called ordinal preferences)...

    To understand the modern view of utility, an analogy will help: People make use of the concept of friendship all the time. It makes sense for Tina to say, "Alice is my best friend, while John is my second-best friend." Such a statement implies that Alice is a better friend than John and that Tina has more friendship with Alice than with John. However, it is nonsensical to ask, "What is the percentage by which Alice's friendship is higher than John's?" This is because friendship is an ordinal, not a cardinal concept. In the same way, economists use utility as a concept to rank the outcomes that acting individuals desire. The higher an outcome is on a preference ranking, the more utility it renders to the individual.

    ~ Robert P. Murphy, Choice (2015), p30.

    Economists have even stronger arguments against comparing cardinal utilities between individuals, and their first reaction to cardinal voting methods is that they do exactly this.

  2. Perhaps because of reason 1, cardinal methods were not even seriously considered until relatively recently. Ordinal methods were vigorously debated in France in the late 1700s. Approval Voting, on the other hand, was first popularized in the 1970s, Score Voting around the year 2000, and Majority Judgment in 2008. As a result, the ranked methods have had much more time to build up literature and mindshare. Cardinal methods are now fighting an uphill battle to catch up.

  3. Furthermore, voting methods are often analyzed with game theory, which generally assumes that everyone is maximally manipulative and has perfect knowledge of how everyone else will vote. In that case, voters will use their grades in whatever way necessary to achieve any advantage, however small, even to get one disliked candidate elected over another who they prefer slightly less. Under these assumptions, only the voters' rankings end up mattering, and the common paradoxes that apply to ranked-voting methods can be replicated with all cardinal methods.

A Thought Experiment

While these arguments are not without merit, they are actually not conclusive for me. Here is a thought experiment to show that cardinal methods can deliver convincing decisions in some cases where ranked methods fail:

Consider the 2000 U.S. Presidential election. The two major candidates were Al Gore and George W. Bush, with Ralph Nader also on the ballot. The electorate was almost exactly split between Gore and Bush. To simplify the election, we suppose (perhaps accurately) that every Gore voter preferred Nader to Bush. We also imagine that every Bush voter liked Nader better than Gore (this may be a stretch, but bear with me). Then the ranked ballots for the election would have looked like this:

50% : Bush > Nader > Gore

50% : Gore > Nader > Bush

Most ordinal voting methods would pronounce this an exact tie between all three candidates (though instant runoff voting dismisses Nader immediately and considers the tie to be only between Bush and Gore). These ordinal methods essentially assume that Nader falls exactly in the middle of Bush and Gore for each voter. This is what the ballots seem to indicate, but if we asked the voters, would they say that since there are only three choices, their esteem of Nader must be balanced precisely between the other two? Or would each voter have some feeling whether Nader is closer to their favorite candidate or their least favorite candidate?

Imagine a world where almost all voters in the Bush and Gore camps say Nader is a close second to whoever is their first choice. In this case, Nader is the perfect compromise candidate and he should absolutely be the winner.

On the other hand, we may be in a universe where almost everybody in both camps strongly dislikes Nader, considering him barely better then their least-favorite choice. If this were the case, we should indeed reject Nader and make the difficult choice between Bush and Gore.

But should we not at least ask the voters? If this information can make Nader the clear winner or the clear loser, and if it is easy enough for voters to give, why would we not make use of it?

We could ask the question as suggested above, "In your estimation, would you say Nader is closer to your favorite candidate or your least favorite candidate?", but perhaps a clearer version is suggested by the framework of von Neumann and Morgenstern: [1]

Would you rather have a 100% chance of Nader or a 50% chance of Bush and a 50% chance of Gore?

It is likely that most voters could make that decision without much trouble, and if it turns out that almost everyone prefers the former, then Nader should be the clear winner.

Finer Grading Scale?

I assume that most people are comfortable reasoning about 50/50 situations, even odds, a coin flip. Could voters meaningfully answer questions about more complicated odds? I believe most people could make decisions involving one-in-four odds, but only the most quantitative individuals (and seasoned gamblers) would be able to determine their preferences down to one or even five percent. A sensible number of grading levels for human evaluators will be somewhere between these extremes.

This is why I am in favor of cardinal voting methods with 2-6 grades. It is clear that the voters are likely to have a rough idea of the size of the gaps between candidates in their ranking. Ranked methods do not ask for this valuable information, thus they cannot use it to determine the election outcome. Yet, in many cases this information could prove decisive. Cardinal methods get more meaningful information from voters and, as a result, have the potential to deliver better outcomes.

[1]One objection to the von Neumann-Morgenstern framework, that it assumes everyone is risk-neutral, is moot in this case because in an election, all actors are already essentially involved in a probabilistic scenario.