Every essay — page 22
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Applied
A rule for choosing, stated exactly, and what it forces on whoever adopts it.
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.
Patience instead of a contract
Commitment had to assume an announcement binds. Play the same game again tomorrow and the assumption is unnecessary — the future does the binding. What it costs is that nearly every outcome becomes an equilibrium, so a theory that could not choose between two now cannot choose between infinitely many.
The reading that is almost right
Every account of simultaneous choice so far has assumed the payoffs are known to both choosers and known to be known. Replace that with each chooser seeing a private reading off by a little, and a band of equilibria closes to a single point — so the assumption nobody states decides the answer.