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The Math Behind Proportional STAR — Quotas, Apportionment, and Fair Division

The "graduate seminar" companion to STAR-PR. None of this is needed to run or advocate for proportional STAR — you mainly need the quota and the spend-a-quota / reweighting intuition. But if you want to know why it's provably fair, you walk into apportionment theory, optimization, and fair-division (cooperative game) theory. This is a different branch of math than Condorcet — that's graph theory; this is apportionment + optimization.

→ The method pages: STAR-PR · STV vs STAR-PR · Glossary: Proportional STAR · Level: 301 · deep dive — Curriculum 301.1 (proportional STAR), 301.5 (limits & theory)

The one mental shift: from "the favorite" to "a fair division of seats"

Single-winner asks who is the favorite? Proportional representation asks how do we split k seats so each group of voters gets seats in proportion to its size? Every idea below is about making "in proportion" precise and computable.

1. Quotas — the unit of "one seat's worth of support"

  • Hare quota = votes ÷ seats. Droop quota = ⌊votes ÷ (seats + 1)⌋ + 1.
  • Proportional STAR's official method, Allocated Score (the engine's allocated), repeatedly elects the top-scoring candidate and then "spends" one quota of the ballots that supported them most. Quota arithmetic is the beating heart — learn it first.

2. Apportionment / divisor methods — the deep root

The reweighting tricks in score-PR are classic apportionment in disguise:

  • Jefferson / D'Hondt (divisors 1, 2, 3, …) ↔ Reweighted Range Voting (rrv): after each winner a ballot's weight becomes 1 / (1 + its already-spent support). That 1, ½, ⅓, … is D'Hondt.
  • Webster / Sainte-Laguë (divisors 1, 3, 5, …) — a "more proportional" alternative.
  • Hamilton / largest-remainder — the quota-and-remainder approach.
  • Balinski–Young theorem — no apportionment rule can satisfy quota and avoid all paradoxes (Alabama, population). The PR analogue of Arrow's impossibility.

3. Reweighting / vote-spending — the mechanic that makes it proportional

The whole game: when a candidate wins, the ballots that elected them must be partly used up, so a majority can't sweep every seat. The three flavors the engine ships:

Method (engine) How it spends / deweights The math it is
Allocated Score (allocated) — Proportional STAR remove one quota of the winner's strongest ballots quota + (fractional) surplus allocation
Sequentially Spent Score (sss) each ballot has a budget it spends on winners budget / flow, surplus scaling
Reweighted Range Voting (rrv) deweight ballots by how satisfied they already are D'Hondt divisor on scores

(The three links run the same 63 ballots through each method — the cleanest way to see what the reweighting rule actually changes.)

The key structural fact: representation has diminishing returns — your 2nd and 3rd winners "count less" (harmonic weights 1, ½, ⅓, …). That concavity — a fancy word for diminishing returns, where each extra unit gives less benefit than the last — is why the method is proportional instead of majoritarian.

Concretely (3 seats; 60 voters back slate A,B,C; 40 back X,Y,Z):

  • Linear value (no diminishing — this is Bloc / majoritarian): 60 > 40 for every seat, so the 60 sweep all three → 3–0, and the minority gets nothing.
  • Concave value (diminishing, via D'Hondt divisors): the majority's strength is 60/1, 60/2, 60/3 = 60, 30, 20; the minority's is 40/1, 40/2 = 40, 20. Rank them — 60 → 40 → 30 — so the seats fall majority, minority, majority = 2–1, mirroring the 60:40 split.

The reweighting is the concavity: deweighting a group's ballots after it wins a seat is exactly what makes its next seat "count less," letting an unrepresented group compete. Remove the diminishing returns and the majority sweeps everything.

4. What "fair" means formally — the axiomatic layer (modern, ~2015+)

  • PSC (Proportionality for Solid Coalitions) — the classic STV-era guarantee.
  • JR → PJR → EJR (Justified / Proportional / Extended Justified Representation) — a hierarchy of set-based guarantees that any large-enough cohesive group gets its share.
  • The core (cooperative game theory) — no group can break off to a committee they'd all prefer in proportion to their size. The strongest notion.
  • KP-transform (Kotze–Pereira) — converts score ballots into approval layers, bridging score-PR to the proportional-approval theory (PAV) where these guarantees are proven.

5. Optimization & complexity

  • Thiele methods (maximize a concave satisfaction sum; PAV = harmonic weights) and Phragmén methods (minimize the maximal voter "load") — the two classical 1890s families.
  • Chamberlin–Courant — maximize each voter's single best representative.
  • Most optimal versions are NP-hard — which is exactly why the practical methods (allocated, sss, rrv) are sequential/greedy approximations. Greedy works here because the satisfaction functions are submodular (diminishing returns again) — a beautiful structural reason the cheap methods come with guarantees.

What to learn, in order of payoff

  1. Apportionment theory (quotas; D'Hondt / Sainte-Laguë; Balinski–Young) — directly explains the reweighting.
  2. Fair division / cooperative game theory (the core, proportionality) — what "proportional" actually means.
  3. Axiomatic multiwinner social choice (PSC, JR / PJR / EJR) — the modern guarantees.
  4. Optimization + submodularity + complexity — why greedy methods exist and are good.
  5. A little linear algebra / probability for the reweighting bookkeeping.

The honest caveat

To use or advocate STAR-PR you need (a) the quota and (b) the spend-a-quota / deweighting intuition — both already covered, with worked numbers, in the STAR-PR page. The JR/EJR/core/Thiele/Phragmén machinery is the graduate seminar: it proves why the method is fair, but you can run and explain allocated / sss / rrv without it.

The single thing to internalize: electing a winner "uses up" a quota of the ballots that supported them — that one move is what turns scoring into proportional scoring.