Matpic
mathematics, in pictures
Concept

Efficiency

The property of an allocation that no alternative makes someone better off without making someone else worse off. It is also called Pareto optimality, and it is the condition that fairness rules are most often forced to trade against.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Drop one condition, and something else satisfies the rest. A column for each of the four conditions, holding a sharing rule that breaks that one and keeps the other three, with the split each rule gives on a stated four-player game.

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.

applied · Shapley value Maximising the product of two people's values. The frontier of value pairs from dividing 4 goods between two people, with the points maximising the product, the sum and the smaller value. The product's maximum is (65.0, 54.2) and is envy-free.

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.

applied · Fair division Credit for one prediction, three ways of leaving an input out. Groups of bars, one group for each way of filling in the inputs that are not known, each bar one input's share of the difference between the prediction and the starting value.

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.

applied · Shapley value An exchange box, and a core that shrinks as the market grows. Edgeworth box with endowment (3,1)/(1,3), utilities xy, contract curve the diagonal; core for 1, 2, 5 replicas: 1.7321–2.2679, 1.9161–2.0839, 1.9722–2.0278.

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.

applied · The core

Named alongside it

The objects these essays reach for when they reach for this one.

AllocationConvexityCooperative gameMarginal contributionShapley valueSymmetryAdditivityAxiomatic characterisationCoalitionConditional expectationCoreDuality

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