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Concept

Subadditivity

The property that the value for a combined piece is at most the sum of the values for its parts, as with a shortest route. Fekete's lemma then makes the value per unit of size converge, which is how limit shapes are proved to exist.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Every order of arrival for three users of one shared capacity, and what each player adds. A table with one row per order in which the players could arrive, giving what each adds to the group already present, and the average of each column as that player's share.

Sharing a cost that is not the sum of its parts

Three users need capacities three, six and twelve of one shared thing, and serving any group costs the largest of them. Averaging what each adds over every order of arrival divides the bill — and for this family the average collapses to a rule anybody could apply by hand.

applied · Shapley value A ball of the square grid with random travel times. A ragged roughly round region of grid cells shaded in bands by the time a signal from the centre first reaches them.

The shape a random ball grows into

Give every road of the square grid a random travel time and ask what can be reached from one point in time t. The region is ragged, and rescaled it converges to a fixed convex shape — but which shape is unknown for every natural law. Computing it shows a curve within a few per cent of a circle for continuous travel times, a flat side where fast roads percolate along a diagonal, a time per step that is still drifting at a hundred and twenty-eight steps, and fluctuations that grow like the distance to the power one third rather than one half.

algebra · Cayley graph

Named alongside it

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

Airport problemCayley graphClosed formConvexityCooperative gameCost-sharingDualityFairnessFirst-passage percolationFluctuationsIncremental costLimit shape

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