1. Setting

2. Statements

3. Corollaries
     Consensus partition    Super-proportional division  

4. Sketch of Proof
     Utilitarian-optimal division    Leximin-optimal division  

5. Further developments

6. See also

7. References

The Dubins–Spanier theorems are several theorems in the theory of fair cake-cutting. They were published by Lester Dubins and Edwin Spanier in 1961.^[1] Although the original motivation for these theorems is fair division, they are in fact general theorems in measure theory.

Setting

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 (for the same value measures, the same , and different partitions):

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

Statements

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

• is a compact set;
• is a convex set.

This was already proved by Dvoretzky, Wald, and Wolfowitz. ^[2]

Corollaries

Consensus partition

A cake partition to k pieces is called a consensus partition with weights (also called exact division) 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:^[1]{{rp|5}}

If all value measures are countably-additive and nonatomic,

then a consensus partition exists.

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.

Super-proportional division

A cake partition to n pieces (one piece per partner) 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 {{rp|6}}

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.
• (for ): 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 (for ), 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 (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 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 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:^[1]{{rp|7}}

If all value measures are countably-additive and nonatomic,

then a utilitarian-optimal division exists.

In this special case, non-atomicity is not required: if all value measures are countably-additive, then a utilitarian-optimal partition exists.^[1]{{rp|7}}

Leximin-optimal division

A cake partition to n pieces (one piece per partner) 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 (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:^[1]{{rp|8}}

If all value measures are countably-additive and nonatomic,

then a leximin-optimal division exists.

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.^[3]

See also

• Lyapunov vector-measure theorem^[4]
• Weller's theorem

References

• Damien (play)
• Damien Pottinger
• Damien Ricketson
• Damien Robinson
• Damien Robitaille
• Damien Ryan
• Damien Saez
• Damien Sargue
• Damien Sin
• Damien smith
• Damien Smith
• Damien Smith (artist)
• Damien Smith (journalist)
• Damien Stone
• Damien Sully
• Damien The Leper Society
• Damien Thomas
• Damien Thorn
• Damien Tixier
• Damien Top
• Damien Traille
• Damien Trotter
• Damien Wayans
• Damien Wayne
• Damien Wayne Echols