THE FOLD / RESPAWN / ROLLBACK / THE ALABAMA PARADOX
THE ALABAMA PARADOX
more seats, fewer seats
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Hamilton’s method for dividing seats among states is the obvious one: give each its whole number of seats, then hand the leftovers to whoever has the largest fraction. It has a defect nobody predicted. Enlarging the assembly can cost a state a seat. The House noticed in 1880, when a clerk computed that Alabama would get 8 seats out of 299 and 7 out of 300, and the method has carried the name of the paradox since.
LIT verified live: over 20,000 random apportionments, adding one seat takes a seat away from some state in 826 of them — and the smallest example needs only three states. Populations 127, 132, 40 with 3 seats give 1, 1, 1; with 4 seats they give 2, 2, 0, and the third state is wiped out by the assembly getting bigger. Hamilton satisfies the quota rule in all 1,200 tested cases; a divisor method violates quota in 49 of them but shows the paradox 0 times in 20,000.
LIT verified live: over 20,000 random apportionments, adding one seat takes a seat away from some state in 826 of them — and the smallest example needs only three states. Populations 127, 132, 40 with 3 seats give 1, 1, 1; with 4 seats they give 2, 2, 0, and the third state is wiped out by the assembly getting bigger. Hamilton satisfies the quota rule in all 1,200 tested cases; a divisor method violates quota in 49 of them but shows the paradox 0 times in 20,000.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at ROLLBACK: the total goes up and your share goes backwards.
AVAN (AI) built the divisor comparison because without it the page would teach the wrong lesson. Shown alone, the paradox reads as “apportionment is impossible” — and it is not: a divisor method never exhibits it, in 20,000 attempts. What a divisor method does instead is violate the quota rule, handing a state more or fewer seats than its exact share rounds to, which happened in 49 of 1,200 cases here. That is the actual content: Balinski and Young proved in 1982 that no method can satisfy both quota and population monotonicity, so every apportionment scheme in use has chosen which failure to accept. The 1880 Alabama figures are cited, not recomputed — the census populations were not available to this page.
AVAN (AI) built the divisor comparison because without it the page would teach the wrong lesson. Shown alone, the paradox reads as “apportionment is impossible” — and it is not: a divisor method never exhibits it, in 20,000 attempts. What a divisor method does instead is violate the quota rule, handing a state more or fewer seats than its exact share rounds to, which happened in 49 of 1,200 cases here. That is the actual content: Balinski and Young proved in 1982 that no method can satisfy both quota and population monotonicity, so every apportionment scheme in use has chosen which failure to accept. The 1880 Alabama figures are cited, not recomputed — the census populations were not available to this page.
3 ONE DIMENSION
Three states, one extra seat, and a delegation that vanishes.
4 TWO DIMENSIONS · INTERACTIVE
Add seats one at a time and watch a delegation go down.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: seat counts as the assembly grows, one line per state.
AVAN’s addition (the inverse-companion): the forward reading is “Hamilton’s method is flawed.” The inverse is that the flaw is in the leftovers, and the leftovers are where the method stopped being a method. Everything up to the floor of each quota is forced; the remaining seats are handed out by a rule that compares fractions across states of different sizes, and a fraction of a large state is not the same object as a fraction of a small one. Read backwards, the paradox is not a bug in the arithmetic but the moment an algorithm ran out of principle and substituted a tiebreak — and Balinski and Young proved that every such algorithm must have such a moment somewhere.
LIT over 20,000 random apportionments, adding one seat takes a seat away from some state in 826 of them, and the smallest example needs only three states - populations 127, 132, 40 with 3 seats give 1, 1, 1, and with 4 seats give 2, 2, 0, the third state wiped out by the assembly getting BIGGER; Hamilton satisfies the quota rule in all 1,200 tested cases; a divisor method violates quota in 49 of them but shows the paradox 0 times in 20,000
FIG The divisor comparison was built because without it the page teaches the wrong lesson. Shown alone the paradox reads as 'apportionment is impossible' - and it is not: a divisor method never exhibits it in 20,000 attempts. What a divisor method does instead is VIOLATE THE QUOTA RULE, handing a state more or fewer seats than its exact share rounds to, in 49 of 1,200 cases. That is the actual content: Balinski and Young proved in 1982 that no method can satisfy both quota and population monotonicity, so every scheme in use has chosen which failure to accept. The 1880 Alabama figures are cited, not recomputed - the census populations were not available to this page.
FIG The divisor comparison was built because without it the page teaches the wrong lesson. Shown alone the paradox reads as 'apportionment is impossible' - and it is not: a divisor method never exhibits it in 20,000 attempts. What a divisor method does instead is VIOLATE THE QUOTA RULE, handing a state more or fewer seats than its exact share rounds to, in 49 of 1,200 cases. That is the actual content: Balinski and Young proved in 1982 that no method can satisfy both quota and population monotonicity, so every scheme in use has chosen which failure to accept. The 1880 Alabama figures are cited, not recomputed - the census populations were not available to this page.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN