Every essay — page 21
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Applied
A rule for choosing, stated exactly, and what it forces on whoever adopts it.
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
The easy theory of assignment 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. One such case is easy to build, 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.
A signal both can see
Two choosers who randomise privately can reach a set of outcomes that is smaller, and worse, than the set they reach when a device draws one cell and whispers each of them their half of it. Nothing is enforced and nobody is bound, and the arrangement is stable anyway.
The landscape nobody is looking at
Letting participants move one at a time to whatever is currently better can cycle forever, and on a network of congestible roads it cannot. The reason is a single number attached to each state that falls by exactly what the mover saves.
Two equilibria and no way to choose
A game can have two states nobody wants to leave, one paying more than the other, and the definition of an equilibrium has nothing to say about which happens. The two standard tie-breakers disagree, and the one that wins is usually the worse.
A mixture that is a population
A mixed equilibrium between two choosers is a knife-edge nobody has a reason to stand on. Read the same mixture as a population whose shares grow with how well they do, and it becomes a point every population is carried to — or one every population circles for ever without arriving.
Worth more for being seen first
Moving first sounds like a disadvantage, since the other side gets to see the move and answer it. When the move is a mixture that is announced and believed, it is never a disadvantage, it is worth exactly nothing in a game of pure conflict, and in other games it is worth more than any equilibrium — sometimes by announcing an action that would never be played in secret.
The table inside every quota
Give seats to districts and parties at once, and every cell of the table has a fair share it ought to round from. A table rounding every cell to its floor or its ceiling, with every total exact, always exists. The biproportional method does not always choose one: here it gives a party 2 seats where its fair share is 3.088.
Where the rounding runs out
In two dimensions a table of seats inside every fair share always exists. Add a third family of totals — every district and party split between groups — and it need not. Sixteen halves in a three-by-three-by-three table meet every total, and no whole table does it without a seat where the fair share is nothing, because the halves close a loop of seven.
Five weighings and the question is closed
Searching the triangle of splits can only ever fail to find a stable one, which is not the same as there being none. Weighing five families of coalitions against the whole settles the question outright — and the family that fails is the proof that nothing survives.
An objection one player makes to another
The core lets a coalition object to everybody at once. Narrow it to one player objecting to one other, require every such objection to be met by an equally loud one coming back, and exactly one split survives — with no dictionary order anywhere in the argument.