Only criteria range voting obeys, can be good criteria for single-winner voting systems to obey

Warren D. Smith, March 2018.

Abstract. The imprecise claim in the title is made precise, and proved as a theorem. Although this theorem has some limitations it nevertheless seems adequate to upend a large amount of voting-system literature.

It is not down in any map; true places never are.    – Herman Melville (1819-1891), Moby Dick.

The title's provocative claim sounds on the surface like some propagandistic nonsense "FairVote" would come up with about Instant Runoff Voting. Indeed, infamously, "FairVote" once declared

The only voting methods that should be weighed seriously for governmental elections are methods that do not violate the "later no harm" criterion (plurality voting and forms of runoff elections and instant runoff voting) or only do so indirectly (such as Condorcet voting methods).

which was completely ridiculous, and solely motivated by FairVote's backwards-directed desire to "prove" Instant Runoff the greatest-ever possible voting method. So might it be that I, as a known advocate of range voting, am again merely working backwards to prove my desired conclusion?

Definition 1: For us "Range voting" shall mean the following voting method. Each voter provides as her vote, a set of real number scores, each in [0,1], one for each candidate. The candidate with greatest score-sum, is elected.

Well, rather remarkably given the late date, this all is not nonsense – it is a theorem. And perhaps the best explanation I can give for why it was previously overlooked, was because it was too obvious. However, we must admit that

1. It is a "theorem" with a rather large asterisk attached about its applicability (or that of its underlying definitions) to the real world – we shall explain that in due time),
2. To achieve that, we need to make some precise definitions (or at least partial definitions) of "single-winner voting systems," "good," and "criteria," which the reader might not adore. I am perfectly happy with the first. With the second, I only provide (and only need) a partial definition, but to the extent it goes, I am ok with it modulo the later asterisk. Finally to handle the third, we actually shall coin the new word "scriterion," note the "s," to signify we are speaking of a specific technical notion of what voting-system criteria the Theorem is speaking about. It does not speak about others. Our Theorem then potentially has various-strength versions, with the stronger ones permitting wider "scriteria" notions. (We'll initially start with a simple one, but some strengthenings will be given later. Probably other strengthenings also could be devised.)
3. Note that our Theorem asserts only a unidirectional implication: any scriterion disobeyed by range voting, cannot be a good scriterion. It does not assert that any scriterion obeyed by range voting is a good scriterion.

Despite all those warnings and limitations, our Theorem remains quite remarkably simple and powerful. First of all, it affirms the pre-eminent position range voting holds among single-winner voting methods. Second, many voting methods books have been published over the decades, discussing many voting systems criteria, but none of them mentioned our Theorem. And despite its limitations, this Theorem still is like a giant armed with a shillelagh, smashing cruddy criteria right and left. Plenty of voting system criteria listed in those books, both

1. meet our technical requirements to be "scriteria," and
2. now are instantly reduced to rubble.

The "asterisk" attached to our Theorem can be quite bothering, but for practical purposes most of the time does not bother us – that shillelagh still is doing a hell of a lot of smashing, and few should be willing to dispute that.

THEOREM-WITH-ASTERISK: Only scriteria obeyed by range voting, can be good criteria for single-winner voting systems to obey.

Definition 2: A "voting system" shall mean a function F mapping a set v of votes to the name w of a winner: w=F(v). "Votes" are arbitrary information packets (e.g. bitstrings).

But for superhumans, TUV would be feasible, and would actually be the unique best possible voting system in the sense it always would elect whatever option maximized society-wide utility-sum. (Zero regret.)

Therefore, only scriteria obeyed by TUV can be good scriteria for single-winner voting systems to obey. Because any insistence on any other scriterion, would immediately exclude the unique best voting system for superhumans, as a matter of foundational demand. That would be absurd and would immediately brand the insistor as an idiot.

The preceding paragraph can be viewed as a partial definition, adequate for the purposes of this Proof, of "good." We hereby brand any scriterion failing to meet that goodness-requirement, as definitely "bad."

Now for TUV and range voting, F_TUV is the same function as F_range, except that F_range is restricted to a subdomain.

Any scriterion obeyed by F_TUV therefore automatically is obeyed by F_range.

Any scriterion disobeyed by F_range. therefore automatically is disobeyed by F_TUV and hence cannot be claimed to be a "good" criterion.
Q.E.D.

Example 1: Consider this form of the Condorcet-winner criterion: "whenever a candidate C exists who would defeat each rival pairwise in a simple majority vote, then C must win." Evidently this is a bad scriterion, since it is disobeyed by range voting and TUV. And indeed Fishburn already had made arguments against Condorcet in 1974:

But range voting actually does obey the following scriterion, which is a somewhat inequivalent wording of the Condorcet criterion: "whenever a candidate C exists who would defeat each rival X pairwise in a vote with the same voting system but with all candidates besides C & X erased from all ballots, then C must win."

Example 2: Consider this form of the Majority-top criterion: "if >50% of the votes rate some particular candidate C unique-top, then C must win." Again, this evidently is a bad scriterion, since it is disobeyed by range voting and TUV. Unrelated arguments had also previously been made against majority-top, based on the conjecture that any country using a system obeying it is at substantial risk of falling irreversibly into 2-party domination.

Range voting does obey this somewhat inequivalent wording of majority-top: "if a majority of voters want to force X's election, they can." This new wording, however, does not satisfy the requirements of the definition of "scriterion," nor even the below kth order variant for any particular k≥1.

Example 3: Nevin Brackett-Rozinsky's "infinity criterion" (which TUV passes, but range voting fails) "A voting method should not impose any finite bound on a voter's scores for candidates." This would be a fine counterexample to our Theorem if we permitted as "criteria" arbitrary English descriptions. However, the Theorem only permits "scriteria" and this isn't a scriterion. (It would have been a scriterion if we replaced "should not impose any" with "must impose a" while demanding candidate-rating-style ballots, in which case the Theorem would not have branded this as a "bad" scriterion, nor would it have been a counterexample.)

Example 4: Brams 2015's criterion forbidding the "paradox of grading systems" is a "bad scriterion" according to our definitions, since TUV disobeys it.

Example 5: "Later no harm" does not meet the demands of our above definition of a "scriterion." However, that problem is solved (proving LNH also is bad) if we strengthen our Theorem by also allowing this wider

Definition 4 of "2nd-order scriterion": A second-order scriterion S₂ is a set of 2-tuples of ordered pairs (x,y). A voting system F "obeys" it if the set of 2-tuples

of ordered pairs (v,w) with w=F(v), is a subset of S₂.

We similarly may define "kth-order scriterion" for any k≥1: just replace "S₂" by "S_k" and "2-tuple" by "k-tuple."

Definition 5 of "subdomain-invariant criterion": Any predicate concerning a voting system which, if true for voting system A, necessarily also is true for voting system B whenever B's defining function is the same as A's restricted to a subdomain.

Strengthenings: The Theorem (and its proof) still work if we replace the word "scriterion" in its statement by "kth-order scriterion for any finite k≥1" or by "subdomain-invariant criterion." Also (pointed out by Ciaran Dougherty): the theorem statement's "obeyed by range voting" can be strengthened to "simultaneously obeyed by range, approval, and plurality voting."

But Dougherty's strengthening comes with the further asterisk that a lot of criteria in voting books, are pretty much just not applicable to (or at least uninteresting if you try to apply them to) approval or especially to plurality. For example, if the criterion were discussing rank orders...

References

Steven J. Brams: The paradox of grading systems, Public Choice 165, 3-4 (Dec. 2015) 193-210.

Peter Fishburn: Paradoxes of Voting, Amer. Political Science Review 68 (1974) 537-546.

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