Dubins–Spanier theorems

Setting

There is a set U , and a set U which is a sigma-algebra of subsets of U .

There are n partners. Every partner i has a personal value measure V i : U → R . This function determines how much each subset of U is worth to that partner.

Let X a partition of U to k measurable sets: U = X 1 ⊔ ⋯ ⊔ X k . Define the matrix M X as the following n × k matrix:

This matrix contains the valuations of all players to all pieces of the partition.

Let M be the collection of all such matrices (for the same value measures, the same k , and different partitions):

The Dubins–Spanier theorems deal with the topological properties of M .

Statements

If all value measures V i are countably-additive and nonatomic, then:

This was already proved by Dvoretzky, Wald, and Wolfowitz.

Consensus partition

A cake partition X to k pieces is called a consensus partition with weights w 1 , w 2 , … , w k (also called exact division) if:

I.e, there is a consensus among all partners that the value of piece j is exactly w j .

Suppose, from now on, that w 1 , w 2 , … , w k 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 j ∈ { 1 , … , k } , define a partition X j as follows:

In the partition X j , all partners value the j -th piece as 1 and all other pieces as 0. Hence, in the matrix M X j , there are ones on the j -th column and zeros everywhere else.

By convexity, there is a partition X such that:

In that matrix, the j -th column contains only the value w j . This means that, in the partition X , all partners value the j -th piece as exactly w j .

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.

Super-proportional division

A cake partition X to n pieces (one piece per partner) is called a super-proportional division with weights w 1 , w 2 , . . . , w n if:

I.e, the piece allotted to partner i is strictly more valuable for him than what he deserves.

Let w 1 , w 2 , . . . , w n be weights whose sum is 1, and assume that the value measures are normalized such that each partner values the entire cake as exactly 1.

A necessary condition for the existence of super-proportional divisions is that the value measures are not identical. PROOF: if the value measures are identical, then the sum of values of the pieces is exactly 1. Hence, it is not possible that all partners receive more than their fair share (if one gets more, another must get less).

The convexity part of the DS theorem implies that:

I.e, the necessary condition is also sufficient.

PROOF: Suppose w.l.o.g. that V 1 ≠ V 2 . Then there is some piece of the cake, Z ⊆ U , such that V 1 ( Z ) > V 2 ( Z ) . Let Z ¯ be the complement of Z ; then V 2 ( Z ¯ ) > V 1 ( Z ¯ ) . This means that V 1 ( Z ) + V 2 ( Z ¯ ) > 1 . However, w 1 + w 2 ≤ 1 . Hence, either V 1 ( Z ) > w 1 or V 2 ( Z ¯ ) > w 2 . Suppose w.l.o.g. that the former is true.

Define the following partitions:

Here, we are interested only in the diagonals of the matrices M X j , which represent the valuations of the partners to their own pieces:

By convexity, for every set of weights z 1 , z 2 , . . . , z n there is a partition X such that:

It is possible to select the weights z i such that, in the diagonal of M X , the entries are in the same ratios as the weights w i . Since we assumed that V 1 ( Z ) > w 1 , it is possible to prove that ∀ i ∈ 1 … n : V i ( X i ) > w i , so X is a super-proportional division.

Utilitarian-optimal division

A cake partition X to n pieces (one piece per partner) 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 U is the set of positive integers. There are two partners. Both value the entire set U as 1. Partner 1 assigns a positive value to every integer and partner 2 assigns zero value to every finite subset. From a utilitarian point 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 X to n pieces (one piece per partner) is called leximin-optimal with weights w 1 , w 2 , . . . , w n 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 (relative to his weight); subject to that, it maximizes the value of the next-poorest partner (relative to his weight); etc.

The compactness part of the DS theorem immediately implies that:

Further developments