Matpic
mathematics, in pictures
Concept

Moving knife

A division procedure in which a knife sweeps continuously across a good and a participant calls stop when the piece passed over is worth a set share. It gives exact proportional divisions, but it needs continuous attention rather than a finite sequence of questions.

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

Three people, a trimmed piece, and the nine comparisons that settle it. The four stages of the Selfridge–Conway division drawn along a cake, with the full three-by-three matrix of each person's value of each share.

Three people and a trimmed piece

For two people, one cut and one choice deliver a division nobody would swap out of. For three, the same promise costs a trimming, a residue and a choosing order contrived so that an advantage once given cannot be taken back — and the verdict is not three numbers but a whole three-by-three matrix.

applied · Fair division Even and Paz's halving, for 4 people. The recursive halving procedure run on a cake valued differently by 4 people: each round's marks and cuts, and the final pieces, each worth at least a 1/4 share to its owner. 8 marks are made.

How many cuts a fair share costs

Every person can be guaranteed a share of a cake worth at least one n-th by their own measure, and the oldest rule that does it asks about n²/2 questions. Splitting the people into halves and the cake at a median mark asks about n log n — and a theorem says nothing can ask fewer. Fairness has a price, and it can be counted.

applied · Fair division

Named alongside it

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

Divide and chooseFair divisionProportionalityValuation measureComplexityEnvy-freenessEnvy matrixLower boundRecursionSelfridge conway

All concepts