The lone divider procedure is a procedure for proportional cake-cutting. It involves a heterogenous and divisible resource, such as a birthday cake, and npartners with different preferences over different parts of the cake. It allows the n people to divide the cake among them such that each person receives a piece with a value of at least 1/n of the total value according to his own subjective valuation. The procedure was developed by Hugo Steinhaus for n=3 people. It was later extended by Harold W. Kuhn to n>3, using the . A description of the cases n=3, n=4 appears in and the general case is described in.
Description
For convenience we normalize the valuations such that the value of the entire cake is n for all agents. The goal is to give each agent a piece with a value of at least 1. Step 1. One player chosen arbitrarily, called the divider, cuts the cake into n pieces whose value in his/her eyes is exactly 1. Step 2. Each of the other n-1 partners evaluates the resulting n pieces and says which of these pieces he considers "acceptable", i.e, worth at least 1. Now the game proceeds according to the replies of the players in step 3. We present first the case n=3 and then the general case.
Steinhaus' procedure for the case ''n''=3
There are two cases.
Case A: At least one of the non-dividers marks two or more pieces as acceptable. Then, the third partner picks an acceptable piece ; the second partner picks an acceptable piece ; and finally the divider picks the last piece.
Case B: Both other partners mark only one piece as acceptable. Then, there is at least one piece that is acceptable only for the divider. The divider takes this piece and goes home. This piece is worth less than 1 for the remaining two partners, so the remaining two pieces are worth at least 2 for them. They divide it among them using divide and choose.
The procedure for any ''n''
There are several ways to describe the general case; the shorter description appears in and is based on the concept of envy-free matching - a matching in which no unmatched agent is adjacent to a matched piece. Step 3. Construct a bipartite graphG = in which each vertex in X is an agent, each vertex in Y is a piece, and there is an edge between an agent x and a piece y iff xvaluesy at least 1. Step 4. Find a maximum-cardinality envy-free matching in G. Note that the divider is adjacent to all n pieces, so |NG|= n ≥ |X| . Hence, a non-empty envy-free matching exists. Step 5. Give each matched piece to its matched agent. Note that each matched agent has a value of at least 1, and thus goes home happily. Step 6. Recursively divide the remaining cake among the remaining agents. Note that each remaining agent values each piece given away at less than 1, so he values the remaining cake at more than the number of agents, so the precondition for recursion is satisfied.
