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Satisfaction Approval Voting (SAV) — one vote, split among everyone you approve

Level: 301 · deep dive

The same Yes/No ballot as Approval, a different way of adding it up. Under bloc Approval (AV) every approved candidate gets a whole vote; under SAV each voter has one vote, split evenly among the candidates they approved. Approve four, and each gets ¼. Proposed by Steven Brams and D. Marc Kilgour in 2010, and — under the name equal and even cumulative voting — used for real, in Peoria, Illinois, since 1991. Every number on this page is reproduced with Martin Lackner's abcvoting and checked against an independent implementation of the paper's own formula.

Source: Brams, S. J. & Kilgour, D. M. (2010), Satisfaction Approval Voting, MPRA Paper 22709 — prepared for the Midwest Political Science Association, Chicago, 22–25 April 2010. Section and proposition numbers below are the paper's.

The rule

A voter's satisfaction score is the fraction of their approved candidates who are elected. SAV elects the size-k committee that maximises the sum of all voters' satisfaction scores:

s(S) = Σ_i  |V_i ∩ S| / |V_i|          (V_i = voter i's approval set, S = the committee)

That looks like it needs checking every possible committee. It doesn't — the paper's Proposition 1 shows the objective is additive, so:

Under SAV, the k winners are simply the k candidates with the highest individual satisfaction scores, where a candidate's score is Σ 1/|V_i| over the voters who approved them.

So the whole method is one pass: a voter approving n candidates gives each of them 1/n of a vote; add the columns; the top k win. That is exactly as summable and as hand-countable as plain Approval — the only change is that the ballot is worth 1 vote total rather than 1 vote per mark.

what one voter contributes to each approved candidate
AV (bloc Approval) 1 — approve more, and you push harder for all of them
SAV 1/n — one vote, divided; approving more dilutes each
PAV 1/j where j counts the voter's approved winners — diminishing returns on representation, not on marks

The PAV row is the distinction worth holding onto. SAV divides by how many you approved; PAV divides by how many of yours won. That one difference is why PAV is proportional and clone-independent while SAV is neither.

The worked example (§2, Proposition 2)

Four candidates, ten voters, two seats. Four voters approve a two-person slate; the other six bullet-vote:

voters approve
4 Ada, Ben
3 Cleo
3 Dev
Ada Ben Cleo Dev elects
AV — one vote per mark 4 4 3 3 Ada, Ben
SAV — one vote, split 4×½ = 2 4×½ = 2 3×1 = 3 3×1 = 3 Cleo, Dev

The two methods elect disjoint committees from identical ballots — that is the paper's Proposition 2, and this is its proof. The slate voters' marks were worth a full vote each to AV and half a vote each to SAV, and half a vote was not enough.

Which answer is better depends on what you are counting. Brams & Kilgour count voters left with nobody: {Cleo, Dev} gives six voters a representative, {Ada, Ben} only four. On that measure SAV wins, and that is the paper's whole argument — AV "can fail to reflect the diversity of interests in the electorate."

Runnable, with the full six-rule comparison: the disjointness case, and live on BetterVoting under three methods — see Worked examples below.

Where SAV sits on the spectrum

The ABC-rule spectrum runs from utilitarian (AV: maximise total approvals) to egalitarian (CC: cover as many voters as possible). SAV is not a point on the Thiele dial at all — a Thiele method weights by how many of your approved candidates won, and SAV weights by how many you approved, which is a property of the ballot rather than of the outcome. It sits beside the family, aimed at the same coverage goal as CC but reached by a much cheaper computation (CC is NP-hard; SAV is one addition pass).

That cheapness is the trade. SAV usually moves toward coverage, but it does not optimise for it, and the paper is candid about the gap — see the next two sections.

What SAV buys you (Proposition 5)

Seventeen voters, three candidates, two seats:

voters approve
5 Ash, Bree
5 Ash, Cole
4 Bree
3 Cole

AV counts Ash 10, Bree 9, Cole 8 and elects {Ash, Bree} — leaving the three Cole-only voters with nobody. SAV scores Ash 5, Bree 6½, Cole 5½ and elects {Bree, Cole}, which represents all seventeen voters and is the smallest set that can. Ash, approved by more voters than anyone, is elected by neither measure of merit that matters here — every one of Ash's supporters already has a second choice in the committee.

Note what did the work: the ten slate voters each split their vote, so Ash — approved by all ten of them and nobody else — collected only 5. Bree and Cole each combined half-votes from the slate with whole votes from bullet voters, and the bullet voters decided it.

Runnable: the coverage case.

What SAV costs you — the authors' own counter-examples

A paper that only showed its method winning would not be worth citing. Brams & Kilgour prove the other direction too:

  • Proposition 3 — an AV outcome can be more representative than SAV's. With 2: a, 5: ab, 6: cde and two seats, every AV committee ({a,c}, {a,d}, {a,e}) represents all 13 voters; SAV elects {a,b}, representing 7. The six cde voters split three ways and each of their candidates collected only 6×⅓ = 2.
  • Proposition 4 — a committee can beat both. With 4: ab, 4: acd, 3: ade, 1: e, AV and SAV agree on {a,d} (11 of 12 voters represented) while {a,e} represents all 12.
  • Proposition 6 — SAV, AV and the greedy coverage heuristic can all miss. With 3: ab, 3: ac, 2: b, 1: c, all three return {a,b}, while {b,c} is the minimal representative set.

The honest summary is the one the paper gives: SAV "generally" — not always — represents more voters than AV, and maximising total satisfaction is a proxy for coverage, not a guarantee of it.

Clones, spoilers, and the incentive that runs backwards

This is where SAV is most interesting for this library, because it inverts the usual spoiler story. Twelve voters, two seats, everyone bullet-voting:

before the split after a splits into clones a₁, a₂ (its 5 supporters approve both)
profile 5: a, 4: b, 3: c 5: a₁a₂, 4: b, 3: c
AV {a, b} — represents 9 of 12 {a₁, a₂} — represents 5 of 12
SAV {a, b} — represents 9 of 12 {b, c} — represents 7 of 12

Under AV, running a second clone is a winning strategy: the five-voter faction takes both seats and two-thirds of the electorate ends up unrepresented. Under SAV the same move is self-defeating — a₁ and a₂ land on 2½ each and both lose. Brams & Kilgour make the point directly: because SAV divides one vote among all of a voter's approvals, it discourages clones, whereas AV "creates the greatest incentive to form clones."

So SAV's vote-splitting is not an accident of the design; it is the design, and it points the strategic incentive at coalition-building rather than at candidate-multiplying.

The flip side is the one the critics press, and it is equally real: a faction that cannot coordinate — that runs three candidates and whose voters sincerely approve all three — is punished for it, whatever its size. Under SAV a voting bloc has to know how many seats it is entitled to and mark exactly that many. Neither AV nor SAV is proportional; they fail in opposite directions, and PAV is the rule that fails in neither, because it divides by winners rather than by marks.

Semi-proportional, and what that means precisely

Applied to party lists (§5), SAV apportions seats by the Jefferson/d'Hondt method with a quota constraint — a genuine, if large-party-favouring, proportional rule. Applied to individual candidates, that guarantee does not carry over, because the divisor becomes the number of candidates a voter marked rather than the size of their party. This is the precise sense in which SAV is called semi-proportional: proportional over parties, and proportional over candidates only to the extent that voters coordinate their marks. It is the same bargain SNTV and cumulative voting strike.

In the wild

  • Peoria, Illinois has elected its five at-large council seats by equal and even cumulative voting since 1991, adopted in settlement of a 1987 Voting Rights Act suit (FairVote — an advocacy source, cited here for the factual history). A voter selects up to five candidates and their five votes are divided equally among them: pick two and each gets 2.5. Dividing 5 votes instead of 1 scales every score by the same constant, so Peoria's rule ranks candidates exactly as SAV does — with the one difference that Peoria caps approvals at the number of seats, where SAV as defined imposes no cap.
  • The Game Theory Society, 2003 (§3) — 161 voters elected 12 of 24 council members by AV. Recomputing those ballots under SAV swaps 2 of the 12 winners: the candidates who finished 10th and 12th are displaced by those who finished 13th and 14th. The SAV committee leaves 2 voters unrepresented; the AV committee leaves 5. The authors flag the obvious caveat themselves — voters might well have marked different ballots had SAV been the announced rule, so this is a recount, not a prediction.

Strategy

The decision-theoretic analysis (§4) enumerates the 19 contingencies in which one voter can change a 3-candidate, 2-seat outcome. The result: every strategy is undominated except approving of your least-preferred candidate. Approving your top two performs about as well as bullet-voting, even though doing so halves the weight behind your favourite. So SAV does not push voters toward bullet-voting the way the vote-splitting arithmetic might suggest — which is the paper's answer to the natural first objection.

Claim-check: the Wikipedia retelling doesn't add up

Wikipedia's SAV article illustrates the method with a version of §2 dressed in 1824 clothing — Adams, Clay and Webster as Whigs, against Crawford and Jackson. It is a corrupted copy of the example above, and three of its numbers are wrong. Recorded here because the page is many readers' first contact with SAV, and because the underlying result (Proposition 2) is perfectly sound and deserves a correct telling.

Wikipedia says what the arithmetic gives
"there are 10 voters" the blocs it then lists are 6 + 3 + 3 = 12
"despite winning a full majority of the vote, the Whigs receive no seats" 6 of 12 is exactly half, not a majority. In Brams & Kilgour's original the slate bloc is 4 of 10 — 40%, and no majority is claimed
"Had one of the three candidates dropped out, the remaining two would have received 4 votes and swept both seats" each survivor gets 6/2 = 3 — a four-way tie with Crawford and Jackson at 3, and no sweep. (Had two dropped out, the survivor would score 6 and take one seat.) "4 votes" is the original's number for a bloc of 4 voters, left behind when someone inflated the bloc to 6

The mangling is mechanical: the original's 4 voters: ab became 6 voters: Adams, Clay, Webster. That preserves the satisfaction score that makes the example work (4×½ = 2, and 6×⅓ = 2 — the slate candidates still finish below the bullet-voted rivals at 3) while breaking the voter count, the majority claim, and the dropout arithmetic around it.

One framing point beyond the arithmetic. Wikipedia presents this outcome as a failure — a "wipeout" caused by "spoiler effects." Brams & Kilgour present the same committee on the same ballots as the point: it represents six voters where AV's represents four. Both readings are defensible and they are counting different things; what is not defensible is presenting either one as the arithmetic. The genuine criticism of SAV is sharper and narrower than the article's, and the authors state it themselves in Propositions 3, 4 and 6.

Run it here

sav is one of abcvoting's rules, so any approval YAML in this repo can be counted under it:

python 06_Other/abcvoting_tabulation_engine/abc_tabulation.py \
  04_Approval/02_Examples/multiwinner/cases/approval_sav_disjoint_c4_b10_brams_kilgour.yaml \
  --rules av,sav,pav,seqpav,cc,seqphragmen

The LH engine's voting_method: Approval_Multi_Winner is the AV half of every comparison on this page — it counts one vote per mark. SAV, PAV and the rest come from the abcvoting engine.

Worked examples in this repo

Case Seats The lesson Live
SAV vs AV — disjoint committees 2 Proposition 2 on the paper's own ballots: AV elects {Ada, Ben}, SAV elects {Cleo, Dev}, no overlap. Bloc STAR and Ranked Robin both agree with AV — so SAV is the lone dissenter, not AV the outlier BV2271 · results ↗
SAV covers everyone AV leaves out 2 Proposition 5: AV's {Ash, Bree} strands three voters; SAV's {Bree, Cole} is the minimal set representing all 17 — and PAV agrees with SAV here, while sequential PAV does not BV2272 · results ↗

Each live election carries the same electorate under Approval, STAR and Ranked Robin (BetterVoting has no SAV tabulator). Within each election all three agree — which is the useful negative result: SAV's answer is not one that the methods in ordinary use ever reach.

References