Efficiency
Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.
None of the four conditions is spare
Four conditions pick out one sharing rule. The half that is usually shown is that they are enough; the other half is that each is needed — drop any one and a different rule satisfies the rest, so the list cannot be shortened.
The product that makes a division fair
Divide goods to make the total happiness as large as possible and the result can be monstrously unfair; make the least happy person as happy as possible and it can waste. Multiply the people's values together and maximise the product instead, and something unexpected happens — nobody envies anybody when goods can be split, and nobody envies by more than one item when they cannot.
What a missing input is worth
A model prices a house at 180 from its size, its garden and its bedrooms, and the question is how much of the price each input is responsible for. Make the inputs the players and the average over orders answers it — once somebody decides what the model says when an input is not known. Three reasonable decisions give bedrooms nothing, nothing, and sixteen, for a model that never reads them.
A market too large to bargain in
Two traders swapping goods can settle anywhere along a stretch of possible deals; nothing forces a price. Bring in a second pair of the same traders and some of those deals can be refused by a group of three. With thirty of each, the only deals no group can refuse lie within a few thousandths of the one a price would produce. Edgeworth guessed in 1881 that competition shrinks bargaining to prices; Debreu and Scarf proved it, and the shrinking can be computed.
The best middle when the shape is unknown
The mean is the best estimate of a centre when the data are normal and worthless when they are Cauchy. The median is best for Laplace data and wastes a third of normal data. Between them lie the trimmed means — drop a share from each end, average the rest — and across seven symmetric shapes the best guarantee comes from trimming about 28 per cent from each end, which never does worse than 81 per cent of the best possible. Peter Bickel proved in 1965 that trimming can never cost more than a known factor against the mean, whatever the shape.
Named alongside it
The objects these essays reach for when they reach for this one.
AllocationConvexityCooperative gameMarginal contributionShapley valueSymmetryAdditivityAxiomatic characterisationCoalitionConditional expectationCoreDuality