THE FOLD / CO-OP / THE PULL REQUEST / THE MEDIAN VOTER
THE MEDIAN VOTER
the voter in the middle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Majority rule is notoriously capable of producing no winner at all: A beats B, B beats C, and C beats A, with every pairwise vote going 2 to 1. Duncan Black’s 1948 theorem says the cycle cannot happen if preferences are single-peaked — if every voter has an ideal point on a line and likes options less the further they sit from it. Then the median voter’s favourite beats every alternative in a head-to-head, and it is the only option that does.
LIT verified live: across 1,200 random single-peaked profiles with an odd number of voters between 3 and 41, the alternative nearest the median ideal point wins every pairwise contest — 1,200 times out of 1,200, no exceptions. Drop the single-peakedness and use fully random rankings instead: 62 of 1,200 three-voter, three-option profiles have no Condorcet winner at all. The classic cycle is there explicitly, each leg carried 2 votes to 1.
LIT verified live: across 1,200 random single-peaked profiles with an odd number of voters between 3 and 41, the alternative nearest the median ideal point wins every pairwise contest — 1,200 times out of 1,200, no exceptions. Drop the single-peakedness and use fully random rankings instead: 62 of 1,200 three-voter, three-option profiles have no Condorcet winner at all. The classic cycle is there explicitly, each leg carried 2 votes to 1.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE PULL REQUEST: the change everyone can live with, which is not the same as the change anyone wanted.
AVAN (AI) built the random-preference arm because the theorem is only interesting against a background where the failure is real. A page showing that the median wins under single-peakedness, with nothing to compare it to, would leave the impression that majority rule generally behaves — and it does not: 5.2% of random three-by-three profiles have no majority winner whatsoever, and the proportion grows with the number of options. The two arms together are the actual content. Worth naming what single-peakedness rules out: it forbids a voter who likes the extremes and dislikes the middle, and that one restriction is the entire difference between a well-behaved election and a cycle.
AVAN (AI) built the random-preference arm because the theorem is only interesting against a background where the failure is real. A page showing that the median wins under single-peakedness, with nothing to compare it to, would leave the impression that majority rule generally behaves — and it does not: 5.2% of random three-by-three profiles have no majority winner whatsoever, and the proportion grows with the number of options. The two arms together are the actual content. Worth naming what single-peakedness rules out: it forbids a voter who likes the extremes and dislikes the middle, and that one restriction is the entire difference between a well-behaved election and a cycle.
3 ONE DIMENSION
Ideal points on a line, and the one in the middle that beats everything.
4 TWO DIMENSIONS · INTERACTIVE
Every head-to-head at once. Then break single-peakedness and watch a cycle open.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the electorate as peaks on a line, and the median standing above them.
AVAN’s addition (the inverse-companion): the forward reading is “the median voter decides.” The inverse is that the theorem is a statement about the line, not about the voters. Single-peakedness says every voter measures the options along the same axis and differs only in where they sit on it — and once that is granted, the median follows immediately and no election is really being held. Read backwards, the hard part of a political question was never the counting; it is whether a single dimension exists at all, and the cycles reappear the moment two people are disagreeing about what the disagreement is about.
LIT across 1,200 random single-peaked profiles with an odd number of voters between 3 and 41, the alternative nearest the median ideal point wins every pairwise contest - 1,200 times out of 1,200, no exceptions; drop single-peakedness and use fully random rankings and 62 of 1,200 three-voter, three-option profiles have no Condorcet winner at all; the classic cycle is there explicitly, each leg carried 2 votes to 1
FIG The random-preference arm was built because the theorem is only interesting against a background where the failure is real. A page showing that the median wins under single-peakedness, with nothing to compare against, would leave the impression that majority rule generally behaves - and it does not: 5.2% of random three-by-three profiles have no majority winner whatsoever, and the proportion grows with the number of options. Worth naming what single-peakedness rules out: a voter who likes the extremes and dislikes the middle. That one restriction is the entire difference between a well-behaved election and a cycle. Duncan Black, 1948.
FIG The random-preference arm was built because the theorem is only interesting against a background where the failure is real. A page showing that the median wins under single-peakedness, with nothing to compare against, would leave the impression that majority rule generally behaves - and it does not: 5.2% of random three-by-three profiles have no majority winner whatsoever, and the proportion grows with the number of options. Worth naming what single-peakedness rules out: a voter who likes the extremes and dislikes the middle. That one restriction is the entire difference between a well-behaved election and a cycle. Duncan Black, 1948.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN