The Dubins–Spanier theorems are several theorems in the theory of
fair cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without ...
. They were published by
Lester Dubins and
Edwin Spanier
Edwin Henry Spanier (August 8, 1921 – October 11, 1996) was an American mathematician at the University of California at Berkeley, working in algebraic topology. He co-invented Spanier–Whitehead duality and Alexander–Spanier cohomology ...
in 1961.
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
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity ...
and
nonatomic, then:
*
is a
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
;
*
is a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
.
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
'' (also called
exact division Exact division, also called consensus division, is a partition of a continuous resource ("cake") into some ''k'' pieces, such that each of ''n'' people with different tastes agree on the value of each of the pieces. For example, consider a cake whic ...
) 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:
[
:::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
Hugo Dyonizy Steinhaus ( ; ; January 14, 1887 – February 25, 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz Un ...
. 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 A strongly-proportional division (sometimes called super-proportional division) is a kind of a fair division. It is a division of resources among ''n'' partners, in which the value received by each partner is strictly more than his/her due share of ...
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.
* (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
In ethical philosophy, utilitarianism is a family of normative ethical theories that prescribe actions that maximize happiness and well-being for all affected individuals.
Although different varieties of utilitarianism admit different charact ...
-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
In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity ...
.
The compactness part of the DS theorem immediately implies that:[
:::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.][
]
Leximin-optimal division
A cake partition to ''n'' pieces (one piece per partner) is called ''leximin
In mathematics, leximin order is a total preorder on finite-dimensional vectors. A more accurate, but less common term is leximin preorder. The leximin order is particularly important in social choice theory and fair division.
Definition
A ve ...
-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:[
:::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.
See also
* Lyapunov vector-measure theorem
* Weller's theorem Weller's theorem is a theorem in economics. It says that a heterogeneous resource ("cake") can be divided among ''n'' partners with different valuations in a way that is both Pareto-efficient (PE) and envy-free (EF). Thus, it is possible to divide a ...
References
{{DEFAULTSORT:Dubins-Spanier theorems
Fair division
Theorems in measure theory