There is a set, and a set which is a sigma-algebra of subsets of. There are partners. Every partner has a personal value measure. This function determines how much each subset of is worth to that partner. Let a partition of to measurable sets:. Define the matrix as the following matrix: This matrix contains the valuations of all players to all pieces of the partition. Let be the collection of all such matrices : The Dubins–Spanier theorems deal with the topological properties of.
This was already proved by Dvoretzky, Wald, and Wolfowitz.
Corollaries
Consensus partition
A cake partition to k pieces is called a consensus partition with weights if: I.e, there is a consensus among all partners that the value of piece j is exactly. Suppose, from now on, that are weights whose sum is 1: and the value measures are normalized such that each partner values the entire cake as exactly 1: The convexity part of the DS theorem implies that: PROOF: For every, define a partition as follows: In the partition, all partners value the -th piece as 1 and all other pieces as 0. Hence, in the matrix, there are ones on the -th column and zeros everywhere else. By convexity, there is a partition such that: In that matrix, the -th column contains only the value. This means that, in the partition, all partners value the -th piece as exactly. Note: this corollary confirms a previous assertion by Hugo Steinhaus. It also gives an affirmative answer to the problem of the Nile provided that there are only a finite number of flood heights.
A cake partition to n pieces is called a super-proportional division with weights if: I.e, the piece allotted to partner is strictly more valuable for him than what he deserves. The following statement is Dubins-Spanier Theorem on the existence of super-proportional division The hypothesis that the value measures are not identical is necessary. Otherwise, the sum leads to a contradiction. Namely, if all value measures are countably-additive and non-atomic, and if there are two partners such that, then a super-proportional division exists.I.e, the necessary condition is also sufficient.
Sketch of Proof
Suppose w.l.o.g. that. Then there is some piece of the cake,, such that. Let be the complement of ; then. This means that. However,. Hence, either or. Suppose w.l.o.g. that and are true. Define the following partitions:
: the partition that gives to partner 1, to partner 2, and nothing to all others.
: the partition that gives the entire cake to partner and nothing to all others.
Here, we are interested only in the diagonals of the matrices, which represent the valuations of the partners to their own pieces:
In, entry 1 is, entry 2 is, and the other entries are 0.
In , entry is 1 and the other entires are 0.
By convexity, for every set of weights there is a partition such that: It is possible to select the weights such that, in the diagonal of, the entries are in the same ratios as the weights. Since we assumed that, it is possible to prove that, so is a super-proportional division.
Utilitarian-optimal division
A cake partition to n pieces is called utilitarian-optimal if it maximizes the sum of values. I.e, it maximizes: Utilitarian-optimal divisions do not always exist. For example, suppose is the set of positive integers. There are two partners. Both value the entire set as 1. Partner 1 assigns a positive value to every integer and partner 2 assigns zero value to every finite subset. From a utilitarianpoint of view, it is best to give partner 1 a large finite subset and give the remainder to partner 2. When the set given to partner 1 becomes larger and larger, the sum-of-values becomes closer and closer to 2, but it never approaches 2. So there is no utilitarian-optimal division. The problem with the above example is that the value measure of partner 2 is finitely-additive but not countably-additive. The compactness part of the DS theorem immediately implies that: In this special case, non-atomicity is not required: if all value measures are countably-additive, then a utilitarian-optimal partition exists.
Leximin-optimal division
A cake partition to n pieces is called leximin-optimal with weights if it maximizes the lexicographically-ordered vector of relative values. I.e, it maximizes the following vector: where the partners are indexed such that: A leximin-optimal partition maximizes the value of the poorest partner ; subject to that, it maximizes the value of the next-poorest partner ; etc. The compactness part of the DS theorem immediately implies that:
Further developments
The leximin-optimality criterion, introduced by Dubins and Spanier, has been studied extensively later. In particular, in the problem of cake-cutting, it was studied by Marco Dall'Aglio.
