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Applied
A rule for choosing, stated exactly, and what it forces on whoever adopts it.
Five rules and one dial
Adams, Webster and Jefferson are usually taught as three rules for rounding a share. They are one rule with a number in it, and turning that number from nought to one moves seats from the smallest region to the largest, one at a time.
The rule with no favourites
Over four hundred instances, Jefferson's method gives the largest region a third of a seat more than its exact share and the smallest a third of a seat less. Adams reverses both. Webster's average is a hundredth of a seat, and that is not luck.
Two out of three, and never all three
Stay inside every region's quota, never take a seat away when the house grows, never take one from a region that grew faster. Each pair is achievable. All three together are not, and the proof is that no rule anywhere manages it.
Choosing what unfair means
Ask whether moving one seat between two regions would make them more equal, and the answer depends on what "equal" is measured in. Three measures, three different answers, and each of the classical methods is the one no transfer can improve for exactly one of them.
Seats to parties and places at once
The ladder's first five rungs give seats to regions in proportion to one list of populations, and prove that no rule does it perfectly. Ask for seats to regions and to parties simultaneously and the object stops being a list — and the impossibility that closed the subject does not apply.
The corners are whole assignments
A table of shares can be written as a lottery over whole assignments, which the anchor's first rung demonstrates on one example. The general statement is that the corners of the set of such tables are exactly the whole assignments, and that single fact is why the whole subject is easy.
A price for every person and task
The cheapest assignment can be found without comparing it to any other. Attach a number to each person and each task so that no pair's two numbers exceed its cost, and if the numbers add to an assignment's total, that assignment is cheapest — proved, by an argument that never mentions the alternatives.
One table, two lotteries
A table of shares says what fraction of each task each person does. It does not say how — the same table is a mixture of whole assignments in many different ways, and the differences are exactly what the people being assigned would care about.
Where the corners stop being whole
Everything on this ladder rests on one property — the relaxation of the assignment problem has whole-numbered corners. Add a single edge that closes an odd cycle and the property fails, a corner appears with a half in every coordinate, and the problem changes character completely.
No rule escapes the doctrinal paradox
A court whose members each hold a consistent position can reach an inconsistent verdict by majority. The anchor's first rung exhibits one such case, which invites the hope that a better rule would avoid it — and every rule that responds to the votes at all fails somewhere.
Deciding the premises or the conclusion
A body that cannot be both decisive and coherent has to choose which. The two live options are to vote on the reasons and let the verdict follow, or to vote on the verdict and let the reasons look after themselves — and they reach opposite answers on exactly the profiles the impossibility identifies.
Four ways out, and what each costs
An impossibility theorem lists conditions and says no rule has them all. That leaves exactly as many escapes as there are conditions, each of them a real institution — a dictator, a two-stage procedure, a supermajority, a restricted agenda — and each escape's price can be counted rather than argued about.