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?


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.

  • 1
    @aml I'm aware of approval voting and find it vastly superior to FPTP. It is a different from FPTP and generates different results, specifically because it uses a cardinal ballot.
    – eclipz905
    Oct 2 '18 at 16:29
  • 1
    Yes, but do you agree that FPTP and Approval can use the exact same ballot, simply changing the instructions from 'pick one' to 'pick any'.
    – amI
    Oct 2 '18 at 16:37
  • 1
    @aml If by 'ballot', you mean 'piece of paper', then yes, but only superficially. I'm referring more broadly to the manner in which voters express their preferences. This includes the rules governing what counts as a legal submission. Changing the rules from "select your top choice" to "select as many as you like" is a change from an ordinal to a cardinal ballot.
    – eclipz905
    Oct 2 '18 at 16:52
  • 1
    @ aml Yes, as stated in my answer, FPTP is ordinal. The only reason voters are limited to one choice on their ballot is because the tabulation would just throw away any additional info they provide. Omitting the higher rankings is a timesaving courtesy for the voter, since that info is not used any way.
    – eclipz905
    Oct 2 '18 at 17:16
  • 1
    @aml Instant Runoff Voting and Approval Voting are different methods that produce different results. One of the primary differences between them is that IRV is ordinal, and Approval is cardinal
    – eclipz905
    Oct 2 '18 at 17:21

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:

enter image description here enter image description here enter image description here

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.


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).

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