Is first-past-the-post (plurality voting) neither an ordinal nor a cardinal method?

Question

Wikipedia says:

Cardinal methods (based on cardinal utility) and ordinal methods (based on ordinal preferences, also called ranked voting) are two main categories of modern voting systems, along with plurality voting.

It also says:

Approval Voting [which is a cardinal method] can also be compared to plurality voting, without the rule that discards ballots that vote for more than one candidate.

But strictly speaking, in the non-trivial case of more than two candidates, is first-past-the-post neither ordinal nor cardinal?

Answer 1

Short Answer:

First Past the Post is an ordinal method.

Long Answer:

There are 2 components to a voting method.

1. How voter preferences are expressed on a ballot.
2. How ballots are aggregated to produce a result.

The question of cardinal vs ordinal is only concerned with that first aspect.

• Cardinal ballots allow a voter to arbitrarily score candidates independent of each other, even permitting multiple candidates to be given the same score.

• Ordinal ballots force the voter to provide an ordering.

The FPTP ballot is an ordinal one which omits all information except the top preference. It could easily be replaced with a full ordinal ballot with a complete ordering of preferences. Later in the process, when votes are being tabulated, all preference information other than the top ranking would simply be discarded. This would make the process more verbose, but would not in any way alter the outcome of the election.

The FPTP ballot could not be replaced with a cardinal one in the same way.

Answer 2

It sorta depends on the theoretical framework adopted. Brams and Fishburn (2002) put plurality in the "nonranked methods", i.e in same bin as (their favorite) approval voting:

The multicandidate elect-one social choice functions in this broad class are divided into nonranked one-stage procedures, nonranked multistage procedures, ranked voting methods, and positional scoring rules. Nonranked methods include plurality check-one voting and approval voting, where each voter casts either no vote or a full vote for each candidate. On ballots for positional scoring methods, voters rank candidates from most preferred to least preferred.

But that's not the only way to construe it. By a bit more mathematical artifice, it's possible to assume there exist even a linear ordering in the vote of each participant, but that the aggregating social choice function disergards anything but the first choice on each such (virtual) ballot. While this is a bit less natural with respect to the nature of actual plurality ballots, it does have one advantage, namely that it's possible to consider plurality as a "scoring rule" (with the obvious weights (1, 0, 0, ..., 0)), using the same general notion of "scoring rule" as for a more discriminating count(s), e.g. Borda count (which would have the scoring rule (m-1, m-2, ..., 0), where m is the number of alternatives). This is the approach taken by Zwicker (2016).

Using Fishburn's (1977) classification, Zwicker has something more interesting to say (about plurality). In Fisburn's classification there are three classes

• C1: (roughly speaking) tournaments induced by Condorced extensions, e.g. Coplan's rule or sequential majority comparisons; these need only unweighted information from the pairwise majority tournament.

• C2: methods need weighted [pairwise majority] tournament information, for example Borda's or Simpson's (aka minimax) method. Despite the fact that the latter is a Condorced extension while the former is not, they are in the same Fishburn class from this information-requirement perspective. Zwicker and later Saari however have refined this further as by decomposing the weighted tournament in cycles and co-cycles, and then

  Condorcet extensions use the information in both components, while the Borda count discards cycle [information] and imposes a version of pairwise majority rule based on cocycle[s] alone

• C3: everything else. And surprisingly, plurality voting is included in this class. Zwicker writes that

  one should balk at a suggestion that Borda needs less information than plurality. Borda, for example, needs all the information in the rank vector ρ(x) = (ρ1(x), ρ2(x), . . . , ρm(x)) of an alternative x (where ρj (x) denotes the number of voters who rank x in j th position), while plurality does not. Thus while Fishburn’s classification is a particularly useful approach to informational bases, it is not the only such approach.

  Alas it doesn't look anyone has refined class C3.

One interesting point from Saari, in a much more friendly presentation of Klamler, are the Saari triangles (as they're now called). These illustrate some basic issues/notions with scoring rules for 3 candidates, based on a (projected) simplex representation the voting:

With this I hope to give you some idea why putting Borda and plurality "on the same map" is a useful exercise.

Zwicker also bends approval voting to his framework:

Moreover, if we relax the SCF [social choice function] notion by allowing voters to express indifference among alternatives, approval voting becomes an SCF—in fact it coincides both with Borda and with all Condorcet extensions, when these rules are suitably adapted to handle ballots with many indifferences.

There's nothing preventing this view (using a partial/weak order, and considering indifference among all non-top choices) from being applied to plurality voting as well, except that there's perhaps not much to be gained in doing so.

Answer 3

Neither, leaning towards cardinal.

While I agree with eclipz905 above that "[t]he question of cardinal vs ordinal is only concerned with ... [h]ow voter preferences are expressed on a ballot", I disagree with their claim that FPTP is ordinal.

In FPTP, voters have 1 vote. If we imagine this vote split into, say, 5 parts (5/5 = 1), with voters able to give fractions of their vote to different options, we arrive at cumulative voting. It is not possible to meaningfully divide an ordinal number (e.g. "1st"). I believe FPTP is more like cumulative voting (with voters giving all voting fractions to one option), than ranked voting (with voters only giving a "1st" option).

If we consider FPTP a type of cumulative voting, which is not ordinal, is it therefore cardinal? Cardinal voting could be thought of as including any method where the voter expresses themself using cardinal numbers, 'numbers of quantity' (e.g. "3", compared to ordinal number "3rd"), in which case FPTP and cumulative would be cardinal methods. However there is no support for this definition of cardinal voting.

Rather, the definition of cardinal voting revolves around cardinal utility, and is almost exclusively described as 'allowing the independent rating of each option'. Clearly, FPTP and cumulative both fall outside this definition.

This suggests to me either that:

1. FPTP is a cardinal method, but theorists have been too narrow in their definition of what constitutes cardinal, or
2. there is a '3rd category' of voting one might call 'allocational' which lacks formal recognition.

Answer 4

The reason for using cardinal (rating) or ordinal (ranking) methods is that they contain enough information to do an 'instant runoff' (which can assure a majoritarian result). Plurality voting lacks such information, as you can only choose one candidate per position. It is as much ordinal as it is cardinal, so it is neither.

IRV (instant runoff voting) is usually associated with ordinal methods, but any cardinal ballot choices can be converted to ordinal choices (but not vice versa).

Answer 5

in response to doug's answer that FPTP is not ordinal but instead is unique:

Accepting eclipz' statement, "[t]he question of cardinal vs ordinal is only concerned with ... [h]ow voter preferences are expressed on a ballot"

and presenting the idea of an "Allocated" ballot.

We now have 3 ways to fill a ballot. in ordinal, each voter fully ranks the ballot. in cardinal, each voter fully scores the ballot. in allocational, each voter fully scores the ballot with a max sum.

now, as ordinal/cardinal refers to the tally process and not the ballot itself. I'll start by saying that plurality and any other "method" is not ordinal or cardinal per say but is valid or invalid for those ballots but could be valid for multiple.

so for Plurality with Ordinal ballots you remove all of the "lower than first" preferences and tally up the first place rankings. This is valid.

with Cardinal ballots, candidates may share the same rank. Plurality cannot reduce these ballots to a single candidate.

with allocated ballots, candidates may still share the same rank. So Plurality is also invalid to tabulate these ballots.

So Plurality is only valid in the case of Ordinal ballots. "Plurality is Ordinal"