The Alabama paradox in Proportional STAR¶

Five gardeners. Four candidates. Add a seat to the committee, and someone who had one loses it.

Level: 301 · deep dive

→ the method: STAR-PR · what proportionality does and doesn't promise: what "proportional" actually means · the theory underneath: the math behind proportional STAR

The election¶

A community garden is electing a committee. Five members score four candidates 0–5:

	Aster	Basil	Clover	Dahlia
Gardener 1	3	3	2	4
Gardener 2	5	4	0	2
Gardener 3	1	0	4	3
Gardener 4	0	5	5	3
Gardener 5	5	5	0	0

Count these ballots with Allocated Score — the Equal Vote Coalition's recommended Proportional STAR tabulation, and the method BetterVoting runs as STAR_PR:

Two seats → Basil and Dahlia. Three seats → Aster, Basil and Clover.

Nothing changed but the number of seats. Dahlia was on a two-person committee and is off a three-person one. Nobody changed a ballot. Nobody withdrew. The committee got bigger and she lost her place on it.

This is the Alabama paradox — a failure of house-size monotonicity. It is named for the 1880 US census, where Alabama was allotted 8 seats in a 299-seat House and 7 in a 300-seat one.

Why it happens¶

The quota is the whole story. Allocated Score works out what one seat costs — the Hare quota, voters ÷ seats — and spends that many voters' ballots on each winner.

Seats	Quota	A seat costs
2	5 ÷ 2 = 2.5	half the electorate
3	5 ÷ 3 ≈ 1.67	a third of it

Changing the seat count changes the price of a seat, which changes which voters get spent on the first winner, which changes who is left to decide the second, and so on. The rounds are a chain, and the seat count is an input at every link. There is no reason the chain should end in a superset of where it ended before — and here it doesn't.

Is that unfair?¶

Worth arguing honestly, because it is not obvious and the answer is not simply "yes".

The case that it is unfair. Dahlia's support did not shrink. The electorate did not change its mind. A rule under which enlarging a body can eject a sitting member offends a plain intuition about what more seats should mean — more representation, not a reshuffle. If a real council expanded from two seats to three and an incumbent lost her place while nobody's vote changed, "the arithmetic did it" would be a hard thing to say out loud at the meeting.

The case that it is not. Proportional representation never promised Dahlia a seat; it promised that cohesive groups of voters get representation in proportion to their size. At two seats, a seat is worth half the electorate and Dahlia is the best answer to "who represents the second half?" At three seats, the question is a genuinely different one — "who represents each third?" — and the answer is a different committee. Those are not the same question with one more answer bolted on; they are different questions. The paradox looks like a broken promise only if you assumed a promise that was never made.

Where that leaves it. Both readings are defensible, and which one governs is a design choice rather than a mathematical one. What is not defensible is being surprised by it after adopting the method. Anyone recommending Proportional STAR for a body whose size might change — a council that may expand, a board that adds a seat, a committee sized by turnout — should know this can happen and decide in advance whether they mind.

It is not a bug, and it is not avoidable¶

This is a theorem, not an implementation defect. Pukelsheim's Coherence Theorem (Proportional Representation, ch. 9) proves that an apportionment method is house-size monotone if and only if it is a divisor method. Allocated Score is a quota method. The paradox is a structural consequence of guaranteeing quota, and it is the other half of the Balinski–Young trade: you may have the quota guarantee or house-size monotonicity, and not both.

Measured across tie-free random electorates by pr_alabama_paradox.py:

Method	Family	Alabama paradox
Allocated Score (`allocated`)	quota	37.8%
Sequentially Spent Score (`sss`)	quota	20.8%
Reweighted Range Voting (`rrv`)	divisor	0.0%

RRV's zero is not a sampling limit — it is the theorem. It is the method that cannot do this, and the price it pays is failing the Hare Quota Criterion: a quota-sized faction cannot always force itself a seat. Neither method is simply better. That is the trade, stated as a number instead of an intuition.

Read the 37.8% carefully. Those ballots are independent uniform scores — an impartial-culture-style model, which is known to manufacture more paradoxes than real electorates contain (why). It means "easy to construct, not a curiosity." It does not mean one election in three.

Run it yourself¶

Identical ballots in both files; only num_winners: differs.

.venv/bin/python STARVote_LH_tabulation_engine/starvote_larry_hastings.py 03_STAR_PR/03_Criteria/alabama_paradox/cases/alabama_2seats.yaml

Case	Seats	Elects	File
Two seats	2	Basil, Dahlia	page · yaml
Three seats	3	Aster, Basil, Clover	page · yaml