Approval Voting in the theory literature — the case, the critiques, and what "approve" means¶

The academic treatment of Approval says something the advocacy pages on both sides tend to skip: the deepest disagreement isn't whether Approval works, it's what a checkmark means. Three incompatible answers are in circulation, and which one you hold decides whether Approval is nearly strategy-free or entirely strategic — before any simulation is run. This page walks the standard survey treatment: the arguments made for Approval, the criticisms made against it, which criticisms survive scrutiny, and one elegant result — on a restricted domain, Approval is Borda is Condorcet.

→ Overview: Approval Voting (the ballot and the count) · critique companion: honest limits · the advocacy side's own academic case, claim-checked: Hamlin & Hua (2023) · run it: the Approval examples · Black Curtain. Source and its lean: the last section. Curriculum: 301.5.

Where the method came from¶

Formally, an approval ballot is just a subset of the candidates — the ones this voter approves. A candidate's approval score is the number of voters whose subset contains them, and the winner is whoever scores highest. That's the entire definition, and it is equivalent to the familiar phrasing: vote for as many as you like, most votes wins.

The provenance is unusually recent for a voting method. Brams and Fishburn's Approval Voting (2007) credits the idea to five different groups who arrived at it independently during the 1970s — no single inventor, which is itself a mild argument that the rule is a natural thing to land on. The multi-author reference is Laslier and Sanver's edited collection (2010), which is also where the critiques live; both books are on the rated & score methods shelf.

The six arguments made for it¶

Restated from the survey (which frames them as the case for Approval as an improvement on Choose-One in political elections), with where this library can check each one:

#	The argument	Where it lands here
1	Simplicity. The ballot is barely more complicated than a plurality ballot and the counting rule is conceptually transparent — so it's an easier sell to the public.	Agreed, and it's the core of the stepping-stone case.
2	It fixes plurality's worst failure — one candidate at the minority end of a spectrum defeating several who split the majority end.	The spoiler effect and the vote-splitting set; see the real election below.
3	Better odds the winner has majority support, which makes a governing "mandate" easier to claim.	Odds, not a guarantee — Approval has no majority criterion.
4	No wasted votes, so minor-party candidates finally show their true level of support.	The strongest of the six, and hard to argue with: nothing on an Approval ballot punishes you for marking a long shot.
5	Likely to elect the Condorcet winner when one exists (Beaujard et al. 2014, who also argue Approval favors "consensual" candidates near the middle of a multidimensional issue space — a generalization of argument 2).	Likely, not always — the counterexample is in this repo and is worked at the bottom of this page.
6	Relatively resistant to strategic manipulation.	The most contested of the six — and, as the next two sections show, the claim can't even be evaluated until you fix what "approve" means.

The real election behind argument 2. The survey's example is the 1980 U.S. Senate race in New York: Alfonse D'Amato won with a plurality under 45%, Elizabeth Holtzman finished close behind, and Jacob Javits — who had lost the Republican primary to D'Amato and stayed in on the Liberal line — took roughly a tenth of the vote from the same end of the spectrum. The claim is that had even a small share of Javits's voters also approved Holtzman, she would have won. That is vote-splitting in its textbook form. This repo has not modelled that election — no ballot data exists to model it with, and the library's rule is to build a model and label it one rather than invent real ballots. The mechanism itself is runnable here in miniature: the split-voting set.

The five criticisms — and which of them survive¶

#	The criticism	Verdict
1	There is an ambiguity at its heart — little agreement on, or understanding of, what it means to approve a candidate.	Stands, and it's the deep one. Balinski and Laraki (2010) consider it fatal. Laslier's counter-report is that voters experience the flexibility as a relief — an answer to the plurality dilemma of whether to vote for the best candidate or the best one with a chance. → next section. Its sharpest published form is Saari & Van Newenhizen's indeterminacy argument: worked in full, with Hillinger's inversion.
2	It over-restricts expressivity, forcing voters to compress a ranking into two levels and to declare pairs equivalent that they don't actually feel are equivalent.	Stands. It's honest limits §1, stated by the literature rather than by STAR advocates.
3	It violates "one person, one vote."	Weak — the survey reads this as an argument of convenience rather than conviction. See one person, one vote: Approval passes the mathematical version of the standard (the Test of Balance).
4	It's unfair — voters who approve more candidates get more influence.	No basis. The rebuttal is pure symmetry: recast Approval in terms of disapproval and the identical argument says the voter who names more non-approved candidates is the one gaining an advantage. An objection that flips with an arbitrary relabelling isn't an objection.
5	Some arguments for Approval are rigged by methodology — in particular, resistance-to-manipulation results depend on how you set up a comparison between preference ballots and approval ballots.	Stands, and cuts both ways. This is the claim-check habit stated in a neutral source: when a method is compared to another on a ballot type only one of them uses, check who chose the translation.

Two of these — 3 and 4 — are the ones usually shouted, and they're the two that don't hold up. That asymmetry is worth remembering in a debate: the strong criticisms of Approval are about expressivity and meaning, not about fairness.

The real fault line: what does "approve" mean?¶

Criticism 1 isn't a quibble — it's a genuine three-way split among researchers about what a voter is doing when they check a box. The survey lays out three views:

A compressed ranking. The voter really has a ranking (possibly with ties), and the approval ballot forces several distinct levels of liking down into exactly two.
A dichotomous primitive. There is no hidden ranking. The voter simply likes or dislikes each candidate, and is genuinely indifferent among those in each group.
A ranking plus a line. The voter has both a ranking and a meaningful dividing line between the candidates they like and those they don't — a true zero. Cardinal utilities can sit under view 1 as well, and under view 3 if utilities may be negative.

These are not shades of the same position. They imply different answers to whether a given ballot is even sincere.

Why the strategy argument never settles¶

Each view produces a different verdict on argument 6 — before anyone runs a simulation:

If approval is a primitive (view 2): each voter has exactly one sincere ballot, and no incentive whatever to cast a different one. Approval comes out looking maximally honest, essentially by construction.
If there's an underlying ranking (view 1): it is never to your advantage to approve a candidate without also approving everyone you like at least as much — so a ballot is just a cut through your own ranking, and the whole decision reduces to, in the survey's phrase, "where to draw the line." But if that line carries no intrinsic meaning, there is no basis for calling one ballot sincere and another insincere — you may as well say every Approval ballot is strategic, or that none is.
If the voter has a true zero (view 3): drawing the line anywhere else is insincere by definition — and such a voter can have a real incentive to do exactly that.

So "Approval is resistant to strategy" is downstream of a philosophical premise, not only of simulation methodology. That is criticism 5 with teeth, and it's why this argument recurs forever.

Where voters actually draw the line — two findings the survey cites, and both match what honest limits §2–3 describes from the practical side:

Duddy et al. (2013): drawing the line at your mean utility maximizes a measure of total separation between the approved and unapproved groups — a defensible "honest" threshold, if you want one.
Laslier (2009): the strategically best line sits near the utility you assign to the expected winner — and voters tend to behave that way in practice. Which means Approval outcomes move with the polls, exactly as this library's threshold critique claims, now with a citation instead of an assertion.

Approval = Borda = Condorcet — on one restricted domain¶

Here is the elegant part, and it is genuinely surprising the first time you meet it.

Translate each approval ballot into a weak ranking: everything approved sits above everything not approved, with indifference inside each group. That confines the ballots to the domain of dichotomous preferences — weak rankings with exactly two non-empty indifference classes. On that domain, three things that normally disagree collapse into one:

Approval = Borda. Applying the ordinary net-preference and symmetric-Borda definitions directly to these weak rankings reproduces the approval count exactly. (The equivalent bookkeeping: when a ballot expresses indifference, award each candidate in a tied group the average of the scoring weights that group spans.) Whether other scoring rules also collapse to Approval depends on the convention used for indifference — for some, like k-approval, the adaptation that would make it work looks artificial.
Every profile has a Condorcet winner. Define "x beats y" as more voters strictly prefer x to y than the reverse. On dichotomous ballots that is precisely "more voters approve x than approve y" — which is transitive and always has a maximum. No cycles are possible.
Therefore Approval agrees with every Condorcet method on this domain. One can fairly say that on dichotomous preferences, Approval reconciles Borda and Condorcet — the two poles that disagree nearly everywhere else.

The structural reason is one this library already has a page for: the disagreement between Borda and Condorcet lives entirely in the cyclic component of the weighted tournament, and on dichotomous ballots that component is always zero. See the cycle–cocycle decomposition — this is the same theorem viewed from the approval end, and Copeland vs Borda — margins matter is what it looks like when the cyclic part is not zero.

The caveat that matters in a real election¶

Two conditions are doing quiet work above, and both are worth stating out loud:

The majority relation is defined by net strict preference — more voters preferring x to y than y to x — not by "more than half of all voters rank x above y." With many indifferences those are different relations, and the second one behaves badly.
If the approval ballots were produced by compressing real rankings, the Condorcet winner of the uncompressed rankings can be somebody else. The theorem is about a domain, not a property you can carry into an election where voters are doing the compressing.

Caveat 2 is not hypothetical, and this repo has it on file. In Black Curtain #1, five voters score three candidates; three of them love Cal and loathe Ann, two the reverse, and every voter rates Bob near the top. On the underlying scores the engine reports a clean Condorcet winner:

[Condorcet Winner]
  Condorcet Winner: Cal — matches the STAR winner

Now let those same five voters compress to approval at "a 3 or better is an approval" — the same election as an Approval count:

--- Approval Voting (single winner) ---
 Tabulating 5 ballots (any non-zero score = approval).

   Bob -- 5 (100%) -- Elected
   Cal -- 3 (60%)
   Ann -- 2 (40%)

[Approval Distribution] (how many candidates each ballot approved)
   10 approvals across 5 ballots — average 2.0 of 3 (range 2–2).
     approved 2: 5 ballots

Both results are correct, and together they are the theorem and its limit in five ballots:

	Underlying scores/rankings	The same voters, compressed to approval
Cal vs Bob	Cal wins 3–2 (voters 1–3 score Cal above Bob)	Bob wins 2–0, with 3 Equal Support — voters 1–3 approve both, so they express no preference
Bob vs Ann	Bob 3–2	Bob 3–0, with 2 Equal Support
Condorcet winner	Cal	Bob
Method winner	Cal (STAR, RCV-IRV, Choose-One)	Bob (Approval — and STAR, and every Condorcet method)

On the compressed ballots Bob really is the Condorcet winner, exactly as the theorem promises — the dichotomous domain has no cycles, and Approval finds its winner. But the preference that decided the original election, Cal over Bob, was thrown away by the three voters who approved both. Argument 5 ("likely to elect the Condorcet winner") survives as a statement about tendencies; it is not a guarantee, and the gap is the compression itself.

Both columns are engine output, not arithmetic done here: the right-hand one is case 01b, the same five approval ballots read pairwise. Worked in full, with both matrices side by side and what the Equal Support column is doing: When compression moves the Condorcet winner.

What to take from all this¶

The strong criticisms of Approval are expressivity (criticism 2) and meaning (criticism 1). The popular ones — "unfair to voters who approve more," "violates one person one vote" — do not hold up, and conceding that makes the real critique land harder.
"Is Approval strategy-resistant?" is not purely an empirical question. Answer "what does approving mean?" first; the strategy verdict follows from that answer.
The Borda = Condorcet result is real and beautiful, and it is about a restricted domain. Quote it as a property of dichotomous preferences, never as "Approval elects the Condorcet winner."
All of it is consistent with this library's own honest limits page — which is worth noting, because that page was written from the STAR side and this material was not.

Source¶

William S. Zwicker, "Introduction to the Theory of Voting," in Handbook of Computational Social Choice (Brandt, Conitzer, Endriss, Lang & Procaccia, eds., Cambridge University Press, 2016) — the approval-voting section, including the arguments and criticisms above, the three readings of "approve," and the Approval = Borda = Condorcet result. Lean: neutral; the standard academic reference, and the same chapter this repo leans on for May's theorem, the SWF/SCF distinction, what a method reads and the cycle–cocycle decomposition.
Works it cites, and which this page names: Brams & Fishburn, Approval Voting (2007) and Laslier & Sanver (eds.), Handbook on Approval Voting (2010) — both on the books shelf; Balinski & Laraki (2010) on the ambiguity as a fatal flaw (see majority judgment); Beaujard et al. (2014) on Condorcet efficiency and consensual candidates; Duddy et al. (2013) on the mean-utility threshold; Laslier (2009) on the expected-winner threshold.

file: approval_in_the_literature.md¶