Approval + Top-Two — the runoff you have to come back for¶

An Approval primary plus a head-to-head general between the two most-approved: a real, field-tested reform package — St. Louis runs on it — and the sharpest little lesson in ballot design, because the runoff must be a second election: re-counting the same 0/1 ballots head-to-head just returns the approval count. STAR is this exact package folded into one ballot.

→ Overview: Approval Voting (how the primary round works). · Companions: honest limits · multi-winner Approval. · Curriculum: 301.5.

The package¶

Approval + Top-Two is a two-round system. Round 1 is a single nonpartisan Approval primary: every candidate on one ballot regardless of party, every voter marks approve / not-approve on each, and the two most-approved advance. Round 2 is an ordinary head-to-head general election, decided by majority. It keeps the familiar structure of the classic two-round runoff (a Choose-One round feeding a top-two round — Georgia's runoffs, French presidential elections) while fixing round 1's defect: with an Approval ballot, supporting your favorite never splits your vote against the compromise you'd accept.

Two real-world flagships:

St. Louis, Missouri adopted it as Proposition D in November 2020 (68% yes) and has elected its mayor, comptroller, and aldermen with it since 2021.
Oregon's "Unified Primary" — Mark and Jon Frohnmayer's proposal (drafted 2011, statewide petition 2014) to nominate by Approval into a top-two general (Wikipedia). The initiative fell short of the ballot, but the campaign founded the Equal Vote Coalition — and the follow-up question, "could the runoff ride along on the ballot itself?", became STAR. That lineage is the last section below.

Why bolt a Top-Two onto Approval?¶

The runoff answers, head-on, the two biggest questions an approval count leaves open (both are in honest limits):

The majority check. An approval total measures breadth — it can crown everyone's lukewarm second choice over a candidate a majority actively prefers (limits §4). The runoff puts the two broadest candidates in front of the whole electorate one-on-one, and the majority settles it.
A lower-stakes approval line. "Where do I draw my line?" is the one hard strategic call an Approval ballot forces (limits §2). A nominating round softens it: your approvals now choose the finalists, not the winner — whatever you decided at the margin, your full preference between the survivors still gets its own dedicated say in round 2.

Why the runoff has to be a second election¶

Here's the instructive part. Suppose you tried to spare everyone the second trip by making the runoff automatic — advance the two most-approved, then re-count the same approval ballots head-to-head. Try it on the flagship Approval example, approval_101 (5 voters, live on BetterVoting — results ↗). The LH engine's report:

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

Ballots:
   columns = Ann, Bob, Cal      (1 = approve; 0 = not approved)
   1,1,0
   0,1,1
   1,1,0
   0,1,0
   1,0,1

   Bob -- 4 (80%) -- Elected
   Ann -- 3 (60%)
   Cal -- 2 (40%)

Finalists: Bob and Ann. Now ask each ballot which finalist it prefers:

Ballot	Approved	In a Bob-vs-Ann runoff it says…
voter 1	Ann ● Bob ●	approved both — Equal Support
voter 2	Bob ● Cal ●	Bob
voter 3	Ann ● Bob ●	approved both — Equal Support
voter 4	Bob ●	Bob
voter 5	Ann ● Cal ●	Ann

The "instant runoff" says Bob 2, Ann 1 — a margin of one vote. The approval count said Bob 4, Ann 3 — a margin of one vote. That's not a coincidence, and no cleverer example escapes it: every ballot that approved both finalists (or neither) lands in Equal Support, and what remains is exactly approved-Bob-only versus approved-Ann-only — which is the approval margin. An automatic top-two runoff computed from 0/1 ballots reproduces the approval result — same winner, same margin, every election. It adds nothing.

What the runoff needs is information the checkmark never recorded: which of two approved candidates the voter actually prefers. Line up one opinion in three ballot styles and the gap is visible — say voter 1's true feeling is "Ann all the way; Bob would be fine":

Candidate	Choose-One	Approval (voter 1's actual ballot)	Score 0–5
Ann	●	●	5
Bob	○	●	3
Cal	○	○	0

The approval ballot records Ann = Bob, so in a runoff this voter has no voice. The second round therefore has to collect new information — either by bringing the voters back for another election (Approval + Top-Two), or by asking a richer question the first time (STAR). Ranked ballots carry the finalist preference too — that's what RCV-IRV's "instant" runoff consults; the 0/1 ballot is the one style with nothing to consult. (The full one-voter/three-ballots anatomy is alternate ballot styles.)

St. Louis in practice¶

The first outing — the March 2021 mayoral primary, four candidates — shows the package working, and shows the second round genuinely measuring something new (Ballotpedia · Wikipedia):

Round 1 (Approval): Tishaura Jones 57%, Cara Spencer 46%, Lewis Reed 39%, Andrew Jones 14%. The shares sum to ~156% by design: 44,538 voters cast 69,607 approvals — about 1.56 approvals per ballot.
Round 2 (head-to-head): Jones 51.7% — Spencer 47.8%.

The approval order held, but an 11-point approval gap shrank to under 4 points head-to-head: the general asked a different question — of these two, which? — and nearly got a different answer. (In the 2025 rematch it ran the other way: Spencer led the approval round 68%–33% and won the general comfortably.) The cost side is just as visible: every cycle needs two campaigns, two election days, and two electorates that never quite match — the March 2021 electorate that picked the finalists was about 13,500 voters smaller than the April one that picked the mayor.

STAR: the same package, one trip¶

STAR is Approval + Top-Two folded into a single ballot — historically, not just conceptually. When the Unified Primary campaign ended in 2014, the Equal Vote Conference in Eugene kept its architecture — a broad, expressive first round, then a top-two majority check — and asked what round-1 ballot would let the runoff run automatically (STAR voting — Wikipedia). The answer was to replace the checkmark with a 0–5 score. The score round finds the two strongest candidates (the primary's job), and because every ballot now records degrees of support, the runoff reads each voter's preference between the finalists off the ballot they already cast (the general's job) — Score Then Automatic Runoff. One trip, one electorate, no second campaign; ballots that scored the finalists equal are counted openly as Equal Support (Two Denominators, One Winner). And unlike the approval echo above, STAR's runoff has real information to consult — it can even overturn the scoring-round leader, which is the whole point of having it (the runoff reversal).

	Choose-One + Top-Two	Approval + Top-Two	STAR
Round-1 ballot	pick one	approve any number (0/1)	score each 0–5
Vote-splitting in round 1	severe	largely solved	solved
Finalists picked by	top two vote counts	top two approval counts	top two score totals
The runoff	second election	second election	automatic — same ballots
What the runoff learns	new head-to-head votes	new head-to-head votes	each ballot's finalist preference (Equal Support counted openly)
Trips to the polls	2	2	1
Electorates	two different turnouts	two different turnouts	identical by construction

The fidelity ladder says the same thing in one line: Approval is Score at 1-bit resolution — and one bit is one bit too few for a runoff.

file: approval_top_two.md¶