Shapley–Folkman lemma

In any vector space (or algebraic structure with addition), ${\displaystyle X}$, the Minkowski sum of two non-empty sets ${\displaystyle A,B\subseteq X}$ is defined to be the element-wise operation ${\displaystyle A+B:=\{x+y\mid x\in A,~y\in B\}.}$ (See also.^[12]) For example

${\displaystyle \{0,1\}+\{0,1\}=\{0+0,0+1,1+0,1+1\}=\{0,1,2\}}$

This operation is clearly commutative and associative on the collection of non-empty sets. All such operations extend in a well-defined manner to recursive forms ${\displaystyle \sum _{n=1}^{N}Q_{n}=Q_{1}+Q_{2}+\ldots +Q_{N}.}$ By the principle of induction it is easy to see that^[13]

${\displaystyle \sum _{n=1}^{N}Q_{n}=\{\sum _{n=1}^{N}q_{n}\mid q_{n}\in Q_{n},~1\leq n\leq N\}.}$

Convex hulls of Minkowski sums

Minkowski addition behaves well with respect to taking convex hulls. Specifically, for all subsets ${\displaystyle A,B\subseteq X}$ of a real vector space, ${\displaystyle X}$, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls. That is,

${\displaystyle \mathrm {Conv} (A+B)=\mathrm {Conv} (A)+\mathrm {Conv} (B).}$

And by induction it follows that

${\displaystyle \mathrm {Conv} (\sum _{n=1}^{N}Q_{n})=\sum _{n=1}^{N}\mathrm {Conv} (Q_{n})}$

for any ${\displaystyle N\in \mathbb {N} }$ and non-empty subsets ${\displaystyle Q_{n}\subseteq X}$, ${\displaystyle 1\leq n\leq N}$.^[14][15]

Statements of the three main results

Notation

${\displaystyle D,N\in \{1,2,3,...\}}$ are positive integers.

${\displaystyle D}$ is the dimension of the ambient space ${\displaystyle \mathbb {R} ^{D}}$.

${\displaystyle Q_{1},...,Q_{N}}$ are nonempty, bounded subsets of ${\displaystyle \mathbb {R} ^{D}}$. They are also called "summands". ${\displaystyle N}$ is the number of summands.

${\displaystyle Q=\sum _{n=1}^{N}Q_{n}}$ is the Minkowski sum of the summands.

${\displaystyle x}$ is an arbitrary vector in ${\displaystyle \mathrm {Conv} (Q)}$.

Shapley–Folkman lemma

Since ${\displaystyle \mathrm {Conv} (Q)=\sum _{n=1}^{N}\mathrm {Conv} (Q_{n})}$, for any ${\displaystyle x\in \mathrm {Conv} (Q)}$, there exist elements ${\displaystyle q_{n}\in \mathrm {Conv} (Q_{n})}$ such that ${\displaystyle \sum _{n=1}^{N}q_{n}=x}$. The Shapley–Folkman lemma refines this statement.

For example, every point in ${\displaystyle [0,2]=[0,1]+[0,1]=\mathrm {Conv} (\{0,1\})+\mathrm {Conv} (\{0,1\})}$ is the sum of an element in ${\displaystyle \{0,1\}}$ and an element in ${\displaystyle [0,1]}$.^[7]

Shuffling indices if necessary, this means that every point in ${\displaystyle \mathrm {Conv} (Q)}$ can be decomposed as

${\displaystyle x=\sum _{n=1}^{D}q_{n}+\sum _{n=D+1}^{N}q_{n}}$

where ${\displaystyle q_{n}\in \mathrm {Conv} (Q_{n})}$ for ${\displaystyle 1\leq n\leq D}$ and ${\displaystyle q_{n}\in Q_{n}}$ for ${\displaystyle D+1\leq n\leq N}$. Note that the reindexing depends on the point ${\displaystyle x}$.^[16]

The lemma may be stated succinctly as

${\displaystyle \mathrm {Conv} \left(\sum _{n=1}^{N}Q_{n}\right)\subseteq \bigcup _{I\subseteq \{1,2,\ldots N\}:~|I|=D}\left(\sum _{n\in I}\mathrm {Conv} (Q_{n})+\sum _{n\notin I}Q_{n}\right).}$

The converse of Shapley–Folkman lemma

In particular, the Shapley–Folkman lemma requires the vector space to be finite-dimensional.

Shapley–Folkman theorem

Shapley and Folkman used their lemma to prove the following theorem, which quantifies the difference between ${\displaystyle Q}$ and ${\displaystyle \mathrm {Conv} (Q)}$ using squared Euclidean distance.

For any nonempty subset ${\displaystyle S\subseteq \mathbb {R} ^{D}}$ and any point ${\displaystyle x\in \mathbb {R} ^{D},}$ define their squared Euclidean distance to be the

$${\displaystyle d^{2}(x,S)=\inf _{y\in S}\|x-y\|^{2}.}$$

${\displaystyle S,S'\subseteq \mathbb {R} ^{D},}$

$${\displaystyle d^{2}(S,S')=\inf _{x\in S,y\in S'}\|x-y\|^{2}.}$$

Note that ${\displaystyle d^{2}(x,S)=(\inf _{y\in S}\|x-y\|)^{2},}$ so we can simply write ${\displaystyle d^{2}(x,S)=d(x,S)^{2},}$ where ${\displaystyle d(x,S)=\inf _{y\in S}\|x-y\|.}$ Similarly, ${\displaystyle d^{2}(S,S')=d(S,S')^{2}.}$

For example, ${\displaystyle d^{2}([0,2],\{0,1,2\})=1/4=d([0,2],\{0,1,2\})^{2}.}$

The squared Euclidean distance is a measure of how "close" two sets are. In particular, if two sets are compact, then their squared Euclidean distance is zero if and only if they are equal. Thus, we may quantify how close to convexity ${\displaystyle Q}$ is by upper-bounding ${\displaystyle d^{2}(\mathrm {Conv} (Q),Q).}$

For any bounded subset ${\displaystyle S\subset \mathbb {R} ^{D},}$ define its circumradius ${\displaystyle rad(S)}$ to be the infimum of the radius of all

$${\displaystyle rad(S)\equiv \inf _{x\in R^{N}}\sup _{y\in S}\|x-y\|}$$

where we use the notation ${\displaystyle \sum _{\max D}}$ to mean "the sum of the ${\displaystyle D}$ largest terms".

Note that this upper bound depends on the dimension of ambient space and the shapes of the summands, but not on the number of summands.

Shapley–Folkman–Starr theorem

Define the inner radius ${\displaystyle r(S)}$ of a bounded subset ${\displaystyle S\subset \mathbb {R} ^{D}}$ to be the infimum of ${\displaystyle r}$ such that, for any ${\displaystyle x\in \mathrm {Conv} (S)}$, there exists a ball ${\displaystyle B}$ of radius ${\displaystyle r}$ such that ${\displaystyle x\in \mathrm {Conv} (S\cap B)}$.^[20]

For example, let ${\displaystyle B'\subset B\subset \mathbb {R} ^{D}}$ be two nested balls, then the circumradius of ${\displaystyle B\setminus B'}$ is the radius of ${\displaystyle B}$, but its inner radius is the radius of ${\displaystyle B'}$.

Since ${\displaystyle r(S)\leq rad(S)}$ for any bounded subset ${\displaystyle S\subset \mathbb {R} ^{D}}$, the following theorem is a refinement:

In particular, if we have an infinite sequence ${\displaystyle (Q_{n})_{n=1,2,...}}$ of nonempty, bounded subsets of ${\displaystyle \mathbb {R} ^{D}}$, and if there exists some ${\displaystyle r_{0}\geq 0}$ such that the inner radius of each ${\displaystyle Q_{n}}$ is upper-bounded by ${\displaystyle r_{0}}$, then

$${\displaystyle d^{2}\left(\mathrm {Conv} \left({\frac {1}{N}}\sum _{n=1}^{N}Q_{n}\right),{\frac {1}{N}}\sum _{n=1}^{N}Q_{n}\right)\leq {\frac {Dr_{0}^{2}}{N}}.}$$

Other proofs of the results

There have been many proofs of these results, from the original,^[20] to the later Arrow and Hahn,^[22] Cassels,^[23] Schneider,^[24] etc. An abstract and elegant proof by Ekeland^[25] has been extended by Artstein.^[26] Different proofs have also appeared in unpublished papers.^[2][27] An elementary proof of the Shapley–Folkman lemma can be found in the book by Bertsekas,^[28] together with applications in estimating the duality gap in separable optimization problems and zero-sum games.

Usual proofs of these results are nonconstructive: they establish only the existence of the representation, but do not provide an algorithm for computing the representation. In 1981, Starr published an iterative algorithm for a less sharp version of the Shapley–Folkman–Starr theorem.^[29]

A proof of the results

The following proof of Shapley–Folkman lemma is from.^[30] The proof idea is to lift the representation of ${\displaystyle x}$ from ${\displaystyle \mathbb {R} ^{D}}$to ${\displaystyle \mathbb {R} ^{D+N}}$, use Carathéodory's theorem for conic hulls, then drop back to ${\displaystyle \mathbb {R} ^{D}}$.

Proof of Shapley–Folkman lemma

Since ${\displaystyle x\in \sum _{n=1}^{N}\mathrm {Conv} (Q_{n})}$, there exists a minimal set of indices ${\displaystyle I\subset 1:N}$, such that ${\displaystyle x\in \sum _{n\in I}\mathrm {Conv} (Q_{n})}$, but ${\displaystyle x\not \in \sum _{n\in J}\mathrm {Conv} (Q_{n})}$ for any proper subset ${\displaystyle J\subsetneq I}$.

Since we can always drop to such a set ${\displaystyle I}$, we WLOG assume that ${\displaystyle 1:N}$ is itself a minimal set of indices, that is, ${\displaystyle x}$ cannot be represented as ${\displaystyle \sum _{n\in I}q_{n}}$ where ${\displaystyle I}$ is a proper subset of ${\displaystyle 1:N}$.

For each ${\displaystyle n}$, represent ${\displaystyle q_{n}\in \mathrm {Conv} (Q_{n})}$ as ${\displaystyle q_{n}=\sum _{k=1}^{K}w_{n,k}q_{n,k}}$, where ${\displaystyle K}$ is a large finite number, ${\displaystyle w_{n,k}\geq 0}$, and ${\displaystyle \sum _{k}w_{n,k}=1}$.

Now "lift" the representation ${\displaystyle x=\sum _{n}\sum _{k}w_{n,k}q_{n,k}}$ from ${\displaystyle \mathbb {R} ^{D}}$ to ${\displaystyle \mathbb {R} ^{D+N}}$. Define

$${\displaystyle {\bar {x}}=(x,1,...,1);\quad {\bar {q}}_{n,k}=(q_{n,k},e_{n})}$$

where ${\displaystyle e_{n}}$ is the vector in ${\displaystyle \mathbb {R} ^{N}}$ that has 1 at coordinate ${\displaystyle n}$, and 0 at all other coordinates.

With this, we have a lifted representation

$${\displaystyle {\bar {x}}=\sum _{n}\sum _{k}w_{n,k}{\bar {q}}_{n,k}.}$$

That is, ${\displaystyle {\bar {x}}}$ is in the conic hull of ${\displaystyle \{{\bar {q}}_{n,k}\}_{n\in 1:N,k\in 1:K}}$.

By Carathéodory's theorem for conic hulls, we have an alternative representation

$${\displaystyle {\bar {x}}=\sum _{n}\sum _{k}w'_{n,k}{\bar {q}}_{n,k}}$$

such that ${\displaystyle w'_{n,k}\geq 0}$, and at most ${\displaystyle N+D}$ of them are nonzero. Since we defined

$${\displaystyle {\bar {x}}=(x,1,...,1);\quad {\bar {q}}_{n,k}=(q_{n,k},e_{n})}$$

this alternative representation is also a representation for ${\displaystyle x}$.

Since we assumed that ${\displaystyle x}$ cannot be represented as ${\displaystyle \sum _{n\in I}q_{n}}$ where ${\displaystyle I}$ is a proper subset of ${\displaystyle 1:N}$, we find that for every ${\displaystyle n\in 1:N}$, at least one of ${\displaystyle w'_{n,k}}$ is nonzero.

Since at most ${\displaystyle N+D}$ of ${\displaystyle w_{n,k}'}$ are nonzero, we find that for at most ${\displaystyle D}$ of ${\displaystyle n\in 1:N}$, at least two of ${\displaystyle w_{n,k}'}$ are nonzero.

Thus we obtain a representation

$${\displaystyle {\bar {x}}=\sum _{n}\left(\sum _{k}w'_{n,k}{\bar {q}}_{n,k}\right)}$$

where for at most ${\displaystyle D}$ of ${\displaystyle n}$, the term ${\displaystyle \sum _{k}w'_{n,k}{\bar {q}}_{n,k}}$ is not in ${\displaystyle Q_{n}}$.

The following "probabilistic" proof of Shapley–Folkman–Starr theorem is from.^[23]

We can interpret ${\displaystyle \mathrm {Conv} (S)}$ in probabilistic terms: ${\displaystyle \forall x\in \mathrm {Conv} (S)}$, since ${\displaystyle x=\sum w_{n}q_{n}}$ for some ${\displaystyle q_{n}\in S}$, we can define a random vector ${\displaystyle X}$, finitely supported in ${\displaystyle S}$, such that ${\displaystyle Pr(X=q_{n})=w_{n}}$, and ${\displaystyle x=\mathbb {E} [X]}$.

Then, it is natural to consider the "variance" of a set ${\displaystyle S}$ as

$${\displaystyle Var(S):=\sup _{x\in \mathrm {Conv} (S)}\inf _{\mathbb {E} [X]=x,X{\text{ is finitely supported in }}S}Var[X]}$$

${\displaystyle d(S,\mathrm {Conv} (S))^{2}\leq Var(S)\leq r(S)\leq rad(S)}$

Proof of Shapley–Folkman–Starr theorem

It suffices to show ${\displaystyle Var\left(Q\right)\leq \sum _{\max D}Var(Q_{n})}$.

${\displaystyle \forall x\in \mathrm {Conv} (Q)}$, by Shapley–Folkman lemma, there exists a representation ${\displaystyle x=\sum _{n\in I}x_{n}+\sum _{n\in J}q_{n}}$, such that ${\displaystyle I,J}$ partitions ${\displaystyle \{1,2,...,N\},|I|\leq D,x_{n}\in \mathrm {Conv} (Q_{n}),q_{n}\in Q_{n}}$.

Now, for each ${\displaystyle n\in I}$, construct random vectors ${\displaystyle X_{n}}$ such that ${\displaystyle X_{n}}$ is finitely supported on ${\displaystyle Q_{n}}$, with ${\displaystyle \mathbb {E} [X_{n}]=x_{n}}$ and ${\displaystyle Var[X_{n}]<Var[Q_{n}]+\epsilon }$, where ${\displaystyle \epsilon >0}$ is an arbitrary small number.

Let all such ${\displaystyle X_{n}}$ be independent. Then let ${\displaystyle X=\sum _{n\in I}X_{n}+\sum _{n\in J}q_{n}}$. Since each ${\displaystyle q_{n}}$ is a deterministic vector, we have

$${\displaystyle Var[X]=Var\left[\sum _{n\in I}X_{n}\right]=\sum _{n\in I}Var\left[X_{n}\right]\leq \sum _{n\in I}\left(Var(Q_{n})+\epsilon \right)\leq \sum _{\max D}Var(Q_{n})+D\epsilon .}$$

Since this is true for arbitrary ${\displaystyle \epsilon >0}$, we have ${\displaystyle Var[X]\leq \sum _{\max D}Var(Q_{n})}$, and we are done.

History

The lemma of Lloyd Shapley and Jon Folkman was first published by the economist Ross M. Starr, who was investigating the existence of economic equilibria while studying with Kenneth Arrow.^[1] In his paper, Starr studied a convexified economy, in which non-convex sets were replaced by their convex hulls; Starr proved that the convexified economy has equilibria that are closely approximated by "quasi-equilibria" of the original economy; moreover, he proved that every quasi-equilibrium has many of the optimal properties of true equilibria, which are proved to exist for convex economies.

Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used to show that central results of (convex) economic theory are good approximations to large economies with non-convexities; for example, quasi-equilibria closely approximate equilibria of a convexified economy. "The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote Roger Guesnerie.^[31]

The topic of

The Shapley–Folkman lemma enables researchers to extend results for Minkowski sums of convex sets to sums of general sets, which need not be convex. Such sums of sets arise in economics, in mathematical optimization, and in probability theory; in each of these three mathematical sciences, non-convexity is an important feature of applications.

Economics

In economics, a consumer's preferences are defined over all "baskets" of goods. Each basket is represented as a non-negative vector, whose coordinates represent the quantities of the goods. On this set of baskets, an indifference curve is defined for each consumer; a consumer's indifference curve contains all the baskets of commodities that the consumer regards as equivalent: That is, for every pair of baskets on the same indifference curve, the consumer does not prefer one basket over another. Through each basket of commodities passes one indifference curve. A consumer's preference set (relative to an indifference curve) is the union of the indifference curve and all the commodity baskets that the consumer prefers over the indifference curve. A consumer's preferences are convex if all such preference sets are convex.^[32]

An optimal basket of goods occurs where the budget-line supports a consumer's preference set, as shown in the diagram. This means that an optimal basket is on the highest possible indifference curve given the budget-line, which is defined in terms of a price vector and the consumer's income (endowment vector). Thus, the set of optimal baskets is a function of the prices, and this function is called the consumer's demand. If the preference set is convex, then at every price the consumer's demand is a convex set, for example, a unique optimal basket or a line-segment of baskets.^[33]

Non-convex preferences

However, if a preference set is non-convex, then some prices determine a budget-line that supports two separate optimal-baskets. For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that a zoo's budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zoo-keeper does not want to purchase half of an eagle and half of a lion (or a griffin)! Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.^[34]

When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not connected; a disconnected demand implies some discontinuous behavior by the consumer, as discussed by Harold Hotelling:

If indifference curves for purchases be thought of as possessing a wavy character, convex to the origin in some regions and concave in others, we are forced to the conclusion that it is only the portions convex to the origin that can be regarded as possessing any importance, since the others are essentially unobservable. They can be detected only by the discontinuities that may occur in demand with variation in price-ratios, leading to an abrupt jumping of a point of tangency across a chasm when the straight line is rotated. But, while such discontinuities may reveal the existence of chasms, they can never measure their depth. The concave portions of the indifference curves and their many-dimensional generalizations, if they exist, must forever remain in unmeasurable obscurity.^[35]

The difficulties of studying non-convex preferences were emphasized by Herman Wold^[36] and again by Paul Samuelson, who wrote that non-convexities are "shrouded in eternal darkness ...",^[37][a] according to Diewert.^[38]

Nonetheless, non-convex preferences were illuminated from 1959 to 1961 by a sequence of papers in

Previous publications on non-convexity and economics were collected in an annotated bibliography by Kenneth Arrow. He gave the bibliography to Starr, who was then an undergraduate enrolled in Arrow's (graduate) advanced mathematical-economics course.^[47] In his term-paper, Starr studied the general equilibria of an artificial economy in which non-convex preferences were replaced by their convex hulls. In the convexified economy, at each price, the aggregate demand was the sum of convex hulls of the consumers' demands. Starr's ideas interested the mathematicians Lloyd Shapley and Jon Folkman, who proved their eponymous lemma and theorem in "private correspondence", which was reported by Starr's published paper of 1969.^[1]

In his 1969 publication, Starr applied the Shapley–Folkman–Starr theorem. Starr proved that the "convexified" economy has general equilibria that can be closely approximated by "quasi-equilibria" of the original economy, when the number of agents exceeds the dimension of the goods: Concretely, Starr proved that there exists at least one quasi-equilibrium of prices p_opt with the following properties:

• For each quasi-equilibrium's prices p_opt, all consumers can choose optimal baskets (maximally preferred and meeting their budget constraints).
• At quasi-equilibrium prices p_opt in the convexified economy, every good's market is in equilibrium: Its supply equals its demand.
• For each quasi-equilibrium, the prices "nearly clear" the markets for the original economy: an
  upper bound on the distance between the set of equilibria of the "convexified" economy and the set of quasi-equilibria of the original economy followed from Starr's corollary to the Shapley–Folkman theorem.^[48]

"in the aggregate, the discrepancy between an allocation in the fictitious economy generated by [taking the convex hulls of all of the consumption and production sets] and some allocation in the real economy is bounded in a way that is independent of the number of economic agents. Therefore, the average agent experiences a deviation from intended actions that vanishes in significance as the number of agents goes to infinity".[49]

Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used in economic theory.

The Shapley–Folkman lemma has been used to explain why large

Nonlinear optimization relies on the following definitions for functions:

• The graph of a function f is the set of the pairs of arguments x and function evaluations f(x)

Graph(f) = { (x, f(x) ) }

• The epigraph of a real-valued function f is the set of points above the graph

• A real-valued function is defined to be a convex function if its epigraph is a convex set.^[62]

For example, the

For example, problems of linear optimization are separable. Given a separable problem with an optimal solution, we fix an optimal solution

x_min = (x₁, ..., x _{${\displaystyle N}$})_min

with the minimum value f(x_min). For this separable problem, we also consider an optimal solution (x_min, f(x_min) ) to the "convexified problem", where convex hulls are taken of the graphs of the summand functions. Such an optimal solution is the limit of a sequence of points in the convexified problem

(x_j, f(x_j) ) ∈  Σ Conv (Graph( f_n ) ).^[4][b]

Of course, the given optimal-point is a sum of points in the graphs of the original summands and of a small number of convexified summands, by the Shapley–Folkman lemma.

This analysis was published by

Probability and measure theory

Convex sets are often studied with probability theory. Each point in the convex hull of a (non-empty) subset Q of a finite-dimensional space is the expected value of a simple random vector that takes its values in Q, as a consequence of Carathéodory's lemma. Thus, for a non-empty set Q, the collection of the expected values of the simple, Q-valued random vectors equals Q's convex hull; this equality implies that the Shapley–Folkman–Starr results are useful in probability theory.^[67] In the other direction, probability theory provides tools to examine convex sets generally and the Shapley–Folkman–Starr results specifically.^[68] The Shapley–Folkman–Starr results have been widely used in the probabilistic theory of random sets,^[69] for example, to prove a law of large numbers,^[6][70] a central limit theorem,^[70][71] and a large-deviations principle.^[72] These proofs of probabilistic limit theorems used the Shapley–Folkman–Starr results to avoid the assumption that all the random sets be convex.

A probability measure is a finite measure, and the Shapley–Folkman lemma has applications in non-probabilistic measure theory, such as the theories of volume and of vector measures. The Shapley–Folkman lemma enables a refinement of the Brunn–Minkowski inequality, which bounds the volume of sums in terms of the volumes of their summand-sets.^[73] The volume of a set is defined in terms of the Lebesgue measure, which is defined on subsets of Euclidean space. In advanced measure-theory, the Shapley–Folkman lemma has been used to prove Lyapunov's theorem, which states that the range of a vector measure is convex.^[74] Here, the traditional term "range" (alternatively, "image") is the set of values produced by the function. A vector measure is a vector-valued generalization of a measure; for example, if p₁ and p₂ are probability measures defined on the same measurable space, then the

(p₁ p₂)(ω)=(p₁(ω), p₂(ω)).

Lyapunov's theorem has been used in economics,^[45][75] in ("bang-bang") control theory, and in statistical theory.^[76] Lyapunov's theorem has been called a continuous counterpart of the Shapley–Folkman lemma,^[3] which has itself been called a discrete analogue of Lyapunov's theorem.^[77]

Notes