Fixed-size legislature

Why must one region lose a legislative seat whenever another region gains one?

Every ten years the census comes back and a handful of American states are told they have gained a seat in the House of Representatives, while another handful are told they have lost one. The 2020 round moved seven seats: Texas gained two, and Colorado, Florida, Montana, North Carolina and Oregon gained one each. Exactly seven other states — California, Illinois, Michigan, New York, Ohio, Pennsylvania and West Virginia — each lost one.

Seven and seven. Not six and eight. Doesn't that exactness bother you? Population change is messy and independent in fifty places at once, so what would have to be true of the world for the gains and the losses to cancel every single time?

Start by imagining the rule that would not produce that balance. Suppose the law said: one representative for every 700,000 people. Then a growing country simply acquires more representatives, and a state that grows gains a seat without anyone losing one. That was roughly how it worked for the first century and a half — the chamber went from 65 members to 435 as the country filled in.

So the cancellation is not a fact about population. Something is holding the total still. What?

A statute. The Permanent Apportionment Act of 1929 fixed the House at 435 members and made reapportionment automatic thereafter. The Constitution itself sets only a floor and a ceiling — at least one representative per state, and no more than one per thirty thousand people — so 435 is a number Congress chose and has kept. (Alaska and Hawaii briefly pushed it to 437 in 1959, and it snapped back at the next apportionment.)

Now watch what that single decision does. Once the total is frozen, the thing being handed out becomes a conserved quantity: whatever the fifty states hold must always sum to the same number. And a conserved quantity cannot be measured absolutely by any one holder. A state's seat count stops reporting how big that state is, and starts reporting what fraction of the country lives there.

Test that against the awkward case. California grew between 2010 and 2020 — it added people — and it still lost a seat. Does that make sense now? The nation grew about 7.4 percent that decade; California grew more slowly. Its share fell, so its slice of a fixed thing fell, even though the thing it was a slice of was itself larger.

One more step, because shares are fractional and seats are whole. Since 1941 the allocation has used the method of equal proportions: give every state its constitutional first seat, then hand out the remaining 385 one at a time, each to whichever state has the highest claim on the next one. The 435th seat goes to somebody. The state that would have taken the 436th simply does not get one. That is the whole event — and notice that the gain and the loss are not two events but one event seen from either end. Which particular state ends up on the wrong side of that line does depend on the rounding method chosen, and the choice among methods is genuinely contested rather than settled mathematics.

Three refinements are worth making, because "zero-sum" is easy to over-apply.

First, the conservation law here is legislated, not physical. Congress could enlarge the chamber by ordinary statute — proposals like the "Wyoming rule," which would size districts to the smallest state's population, do exactly that — and the zero-sum property would immediately dissolve. It feels like a law of nature only because the statute has held for close to a century.

Second, seats are conserved but representation is not. The population keeps growing inside a fixed chamber, so the average district went from roughly 647,000 people after the 2000 census to roughly 761,000 after 2020. Every voter's individual share of a representative shrinks, everywhere, at once. That quantity is not conserved and it moves in only one direction.

Third, conservation holds for the whole, not for pairs. No state loses a seat to another state; the ledger balances only in aggregate, and the temptation to narrate a rivalry between the winner and the loser is a story the arithmetic does not support.

It is also worth noticing where else this propagates. Each state's electoral votes equal its House seats plus two, so the fixed chamber quietly makes presidential electors a conserved quantity as well.

• How the different apportionment methods — Hamilton, Jefferson, Webster, Huntington-Hill — disagree, and the paradoxes (like the Alabama paradox) that got several of them abandoned.
• What a growing chamber would cost and buy: committee workload, floor time, and the size of a district a member can plausibly know.
• Where else a legislated cap turns a measurement into a share: immigration visa caps by country, carbon allowances, medical residency slots.

Every term the collection defines is gathered in the glossary.