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.