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CHAPTER 1
Mathematical Preliminaries
We begin by reviewing basic mathematical facts and notation that we shall use
repeatedly in the book. The basic results we need relate to binary relations and
their extensions and representations, and to solutions to systems of inequalities.
1.1
BASIC DEFINITIONS AND NOTATIONAL CONVENTIONS
1.1.1
Relations
Let X be a set. An n-ary relation on X is a subset of Xn. A binary relation on
X is a subset of X × X. When R is a binary relation on X, we write (x,y) ∈R
as x R y. When R is an n-ary relation, we also write R(x1,...,xn) instead of
(x1,...,xn) ∈R.
Given a binary relation R, deﬁne its strict part, or asymmetric part, PR by
(x,y) ∈PR iff (x,y) ∈R and (y,x) ̸∈R. Deﬁne its symmetric part, or indifference
relation, IR by (x,y) ∈IR iff (x,y) ∈R and (y,x) ∈R.
Two elements x,y ∈X are unordered by R if (x,y) /∈R and (y,x) /∈R. Two
elements are ordered by R when they are not unordered by R. We say that a
binary relation B is an extension of R if R ⊆B and PR ⊆PB. A binary relation
B is a strict extension of R if it is an extension, and in addition there is a pair
that is unordered by R but ordered by B. Finally, a binary relation is complete
if it leaves no pair of elements unordered. That is, R is complete if for all x and
y, x R y or y R x (or both).
The following are standard properties of binary relations. A binary relation
R is:
• transitive if, for all x, y, and z, x R y and y R z imply that x R z;
• quasitransitive if, for all x, y, and z, x PR y and y PR z imply that x PR z;
• reﬂexive if x R x for all x;
• irreﬂexive if (x,x) /∈R for all x;
• symmetric if, for all x and y, (x,y) ∈R implies that (y,x) ∈R;
• antisymmetric if, for all x and y with x ̸= y, (x,y) ∈R implies that
(y,x) /∈R.
Observe that if a binary relation is complete, then it is reﬂexive.


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2
Mathematical Preliminaries
The following are types of binary relations: A binary relation is a(n)
• weak order if it is complete and transitive;
• linear order if it is complete, transitive, and antisymmetric;
• equivalence relation if it is reﬂexive, symmetric, and transitive;
• partial order if it is reﬂexive, transitive, and antisymmetric.
1.1.2
Partially ordered sets
If ≥is a partial order on X, an upper bound on A ⊆X (with respect to ≥) is
an element x ∈X such that x ≥y for all y ∈A; a lower bound on A ⊆X is an
element x ∈X such that y ≥x for all y ∈A. A least upper bound on A is an
upper bound x on A such that if y is an upper bound on A then y ≥x. A greatest
lower bound on A is a lower bound x on A such that if y is a lower bound on A
then x ≥y. We write x ∨y, called the join of x and y, for the least upper bound
on the set {x,y}; and x ∧y, called the meet of x and y, for the greatest lower
bound on the set {x,y}. A set X, with a partial order ≥on X, is a lattice if, for
all x and y in X, x ∨y and x ∧y are deﬁned in X with respect to ≥.
Let X, with the partial order ≥, be a lattice. A function u : X →R is
supermodular if for all x,y ∈X,
u(x) + u(y) ≤u(x ∨y) + u(x ∧y),
and submodular if −u is supermodular.
If ≥is a partial order on X, a chain is a subset Y ⊆X such that for all x,y ∈Y,
either x ≥y or y ≥x. The following result, known as Zorn’s Lemma, is one of
the many equivalents of the axiom of choice in set theory. It allows one to
extend many induction-style proofs to sets of arbitrary cardinality. We shall
use it several times.
Zorn’s Lemma
Let ≥be a partial order on a set X, and suppose that for
every chain Y ⊆X there is an upper bound of Y in X (with respect to ≥). Then
there exists x∗∈X such that, for all x ∈X, x ≥x∗implies x = x∗.
An element like x∗in the statement of Zorn’s Lemma, with the property that
x ≥x∗implies x = x∗, is a maximal element for ≥in X.
1.1.3
Euclidean spaces
We use R to denote the real numbers, Q to denote the rational numbers, and
Z to denote the integers. When X = R, we have the following familiar binary
relation: > ⊆R2 deﬁned as x > y if there is some z ̸= 0 such that x = y + z2.
The relation ≥is deﬁned by x ≥y if either x = y or x > y. We often write x > y
as y < x, and x ≥y as y ≤x. For any x,y ∈R with x < y, the open interval
{z ∈R : x < z < y} is denoted by (x,y).
Let X = Rn; xi ∈R is the ith component of the vector x, for i = 1,...,n.
The inner product of two vectors x and y is x · y = n
i=1 xiyi. Deﬁne the binary


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1.2 Preference and utility
3
relation ≥by x ≥y if xi ≥yi for i = 1,...,n. Deﬁne > and ≫by x > y if x ≥y
and x ̸= y; and x ≫y if xi > yi for i = 1,...,n. Denote by Rn
+ the set of x ∈Rn
with x ≥0; and by Rn
++ the set of x ∈Rn with x ≫0.
Note that ≥on Rn is a partial order, while ≥on R is a linear order. The set
Rn with the order ≥is a lattice, and for any x = (xi)n
i=1 and y = (yi)n
i=1 we have
x ∨y = (max{xi,yi})n
i=1 and x ∧y = (min{xi,yi})n
i=1.
The usual (Euclidean) norm on Rn is ∥x∥=
n
i=1 x2
i . The interior of a set
A ⊆Rn is the set of x ∈A for which there is ε > 0 such that if ∥x−y∥< ε then
y ∈A. The interior of A is denoted by int A. We say that x is interior to A if
x ∈int A.
We use standard notational conventions for functions and correspondences.
Let X and Y be sets. We denote by u : X →Y a function with domain X
and range a subset of Y. A correspondence is a function ϕ : X →2Y, which
we denote by ϕ : X ⇒Y. When X ⊆Rn, Y = R, x ∈X, and u : X →Y
is differentiable (the deﬁnition of which is omitted), then ∇u(x) denotes the
gradient of u at x.
We use 1 to denote an indicator function: for E ⊆X, 1E : X →R is the
function 1E(x) = 1 if x ∈E and 1E(x) = 0 if x /∈E; 1x denotes 1{x}.
The notation for indicator functions is also used as an indicator vector.
When E ⊆{1,...,n}, 1E is the vector in Rn that has a 1 in the ith component if
i ∈E, and has a 0 in the ith component if i /∈E. The vector 1i is called the ith
unit vector. Finally, we write 1 for 1{1,...,n}, the vector all of whose components
are 1.
1.2
PREFERENCE AND UTILITY
A (rational) preference relation on X is a binary relation ⪰⊆X × X that is
a weak order. A preference relation that is a linear order is called a strict
preference or a strict preference relation.
When we use the ⪰notation for a binary relation, we write ≻for P⪰, its
strict part, and ∼for its symmetric part I⪰. Note that when ⪰is a preference
relation then ≻is asymmetric and transitive, while ∼is an equivalence relation.
1.2.1
Properties of preferences
For a preference relation ⪰on X ⊆Rn, we can deﬁne the following standard
properties. We say that ⪰is:
• locally nonsatiated if for all x and ε > 0, there is some y ∈X such that
∥x −y∥< ε and y ≻x;
• monotonic if x ≫y implies x ≻y;
• strictly monotonic if x > y implies x ≻y;
• continuous if the sets {z ∈X : z ⪰x} and {z ∈X : x ⪰z} are relatively
closed in X for every x.


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4
Mathematical Preliminaries
The set {z ∈X : z ⪰x} is the upper contour set of ⪰at x: the set of elements
that are at least as good as x. The strict upper contour set of ⪰at x is {z ∈X :
z ≻x}. And the set {z ∈X : x ⪰z} is the lower contour set of ⪰at x.
In the case when X ⊆Rn is a convex set, it is often useful to understand the
relation between a preference and convex combinations of elements in X. Say
that a binary relation ⪰is
• convex if λ ∈(0,1), y ⪰x, and z ⪰x implies that λy + (1 −λ)z ⪰x;
• strictly convex if λ ∈(0,1), y ⪰x, z ⪰x, and y ̸= z implies that λy +
(1 −λ)z ≻x.
So a preference relation ⪰is convex if the upper contour set of ⪰at x is a
convex set, for all x ∈X.
1.2.2
Utility
Preference relations are often described through a utility function u : X →R by
x ⪰y iff u(x) ≥u(y). We say that u represents ⪰. In that case we say that
u is locally nonsatiated, monotonic, or strictly monotonic if the preference
relation it represents has those properties. A utility function is quasiconcave
if the preference relation it represents is convex; and strictly quasiconcave if
the preference relation it represents is strictly convex. Additionally, we say
that u is continuous or concave if it satisﬁes those properties according to its
ordinary deﬁnitions. A smooth utility is one for which all partial derivatives of
all orders exist.
We have the following general theorem.
Theorem 1.1
A preference relation has a utility function that represents it iff
there exists an (at most) countable set Z ⊆X with the following property: for
all x,y ∈X for which x ≻y, there exists z ∈Z for which x ⪰z ⪰y.
Proof. First, we show that if there exists such a utility representation, then
there exists a countable set Z ⊆X satisfying the property in the statement of
the theorem.
We shall exhibit a two-part construction. Let A = u(X) ⊆R, a nonempty
set of real numbers. Consider two countable sets of open intervals. The ﬁrst,
denoted by I, is the set of all open intervals with rational endpoints. The
second, denoted by I′, is the set of open intervals (x,y) with x,y ∈A and
(x,y) ∩A = ∅. We show that I′ is countable by the following argument: For
(x,y) ∈I′, choose q(x,y) ∈Q such that x < q(x,y) < y.1 Note that, since any two
intervals in I′ are disjoint, the mapping (x,y) →q(x,y) is one-to-one. Since Q
is countable, I′ is countable.
Now, for each I ∈I, if I∩A ̸= ∅, pick a ∈I∩A and denote this by aI. Denote
the set A = {aI : I ∈I,I ∩A ̸= ∅}, and note that this set is at most countable
1 This does not require the axiom of choice. Enumerate Q = {q1,q2,...}, and choose q(x,y) to be
the element of Q for which x < q(x,y) < y associated with the lowest index in the enumeration.


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1.3 Order pairs, acyclicity, and extension theorems
5
because I is countable. Deﬁne A′ to be the set of all x for which (x,y) ∈I′, for
some y. Note that A′ is at most countable because I′ is.
Using the axiom of choice, for each a ∈A∪A′, choose one element x(a) ∈X
such that u(x(a)) = a. Let Z = {x(a) : a ∈A ∪A′}. It is clear that Z is at most
countable and satisﬁes the requisite property.
Conversely, suppose that there exists an at most countable set Z satisfying
the property in the hypothesis. Suppose that Z is countable, and label it
as {zk}∞
k=1. Deﬁne the number uk = 1/2k. Clearly, ∞
k=1 uk < ∞. For all
x ∈X, deﬁne u(x) = 
{k:x≻zk} uk −
{k:zk≻x} uk. To verify that it is indeed a
utility representation, ﬁrst suppose that x ⪰y. Then {k : y ≻zk} ⊆{k : x ≻
zk}, so that 
{k:y≻zk} uk ≤
{k:x≻zk} uk, and {k : zk ≻x} ⊆{k : zk ≻y}, so
that 
{k:zk≻x} uk ≤
{k:zk≻y} uk. We conclude that 
{k:y≻zk} uk −
{k:zk≻y} uk ≤

{k:x≻zk} uk −
{k:zk≻x} uk, or u(y) ≤u(x).
For the strict preference, suppose that x ≻y. By the property of Z, there
exists some zk such that x ⪰zk ⪰y. Then by completeness and transitivity,
either x ≻zk or zk ≻y. If the former, then {j : y ≻zj} ⊊{j : x ≻zj}, so
that 
{j:y≻zj} uj < 
{j:x≻zj} uj. If the latter, then {j : zj ≻x} ⊊{j : x ≻zj}, so
that 
{j:zj≻x} uj < 
{j:zj≻y} uj. In either case, we conclude that 
{j:y≻zj} uj −

{j:zj≻y} uj < 
{j:x≻zj} uj −
{j:zj≻x} uj, or u(x) > u(y).
The proof is similar in the case where Z is ﬁnite.
1.3
ORDER PAIRS, ACYCLICITY, AND EXTENSION THEOREMS
An order pair on X is a pair of binary relations ⟨R,P⟩such that P ⊆R. If R is a
binary relation, then ⟨R,PR⟩is an order pair, but we shall encounter order pairs
⟨R,P⟩in which P may not be the strict part of R. An order pair ⟨R′,P′⟩is an
order pair extension of ⟨R,P⟩if R ⊆R′ and P ⊆P′. (Our previous deﬁnition of
when a binary relation B is an extension of R corresponds to ⟨R,PR⟩being an
order pair extension of ⟨B,PB⟩.)
An order pair ⟨R,P⟩is acyclic if there is no sequence x1,x2,...,xL such that
x1 R x2 R ... R xL,
and xL P x1. A sequence in the situation above is called a cycle of ⟨R,P⟩, and L
is the length of the cycle. A single binary relation R is acyclic if the order pair
⟨R,R⟩is acyclic.
Observation 1.2
If ⟨R,P⟩is acyclic, then ⟨R ∪(x,x),P⟩is acyclic for any
x ∈X. Hence, any acyclic ⟨R,P⟩has an acyclic extension ⟨R′,P′⟩in which R′ is
reﬂexive.
The transitive closure of a binary relation R is the binary relation RT deﬁned
by x RT y if there is a sequence x1,x2,...,xL such that
x = x1 R x2 R ... R xL = y.


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6
Mathematical Preliminaries
Equivalently, RT is the smallest transitive relation containing R, or
RT=

{R′ : R′ is transitive and R ⊆R′}.
Thus, ⟨R,P⟩is acyclic iff there is no x and y with x RT y and y P x. Note also
that ⟨R,P⟩is acyclic iff ⟨RT,P⟩is acyclic.
Acyclicity captures a basic property of rational preference. If a binary
relation ⪰is complete and ≻is the strict part of ⪰, then ⪰is transitive (and
hence a rational preference relation) iff ⟨⪰,≻⟩is acyclic. As we shall see,
acyclicity is essentially what is left of rationality when ⪰is only partially
observed, in the sense that we observe some R ⊊⪰.
Many problems in revealed preference theory are extension exercises, in
the sense that we are given an order pair ⟨R,P⟩and we need to ﬁnd some
well-behaved order pair extension of ⟨R,P⟩. The problem then amounts to
comparing elements that are left uncompared in R; the following lemma is
a basic result on adding comparisons.
Lemma 1.3 (Extension Lemma).
Let ⟨R,P⟩be an acyclic order pair, and
x,y ∈X be two distinct elements, unordered by R. Then one of the following
statements holds true:
I) ⟨R ∪{(x,y)},P ∪{(x,y)}⟩is acyclic.
II) ⟨R ∪{(y,x)},P ∪{(y,x)}⟩is acyclic.
III) ⟨R ∪{(x,y),(y,x)},P⟩is acyclic.
Proof. Deﬁne a binary relation S by letting w S z if there is a sequence
x1,x2,...,xL such that
w = x1 R x2 R ... R xL = z,
and for which xl P xl+1 for some l = 1,...,L−1. Note that if x S y, then none of
the pairs (xl,xl+1) can equal (x,y) or (y,x) because x and y are unordered by R.
Suppose that (III) is not true: we show that one of the other statements
must hold. Because ⟨R,P⟩is acyclic, if ⟨R∪{(x,y),(y,x)},P⟩has a cycle, either
⟨R ∪{(x,y)},P⟩has a cycle, or ⟨R ∪{(y,x)},P⟩has a cycle. Note that if ⟨R ∪
{(x,y)},P⟩has a cycle, then y S x; and if ⟨R ∪{(y,x)},P⟩has a cycle, then x S y.
This implies that if (III) is not true then x S y or y S x.
Suppose that x S y. We show that (I) follows. Since ⟨R,P⟩is acyclic, any
cycle of ⟨R ∪{(x,y)},P ∪{(x,y)}⟩must involve (x,y). But x S y implies that a
cycle with the same initial and terminal points could be constructed in ⟨R,P⟩.
So there can be no cycles in ⟨R ∪{(x,y)},P ∪{(x,y)}⟩.
Similarly, y S x implies (II).
We present two fundamental results on extending a binary relation. The ﬁrst is
called Szpilrajn’s Theorem, and deals with extending a partial order to a linear
order. The second is a basic and useful result on extensions of acyclic order
pairs to preference relation.


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1.3 Order pairs, acyclicity, and extension theorems
7
Theorem 1.4 (Szpilrajn).
Suppose that ⪰is a partial order. Then there is a
linear order ⪰′ which extends ⪰.
Proof. We present a proof for the case when X is ﬁnite. The general result
relies on Zorn’s Lemma: see the proof of Theorem 1.5. We shall use the
notation from the proof of Lemma 1.3.
Since ⪰is a partial order, it is antisymmetric. Consider any antisymmetric
order R and its strict part P. Suppose x and y are unordered according to R. If
we apply Lemma 1.3 to ⟨R,P⟩, then in fact (I) or (II) must hold. The reason is
that if x and y are distinct elements, and there is a sequence x1,x2,...,xL such
that
x = x1 R x2 R ... R xL = y,
then antisymmetry implies that some R may be replaced by P, and hence
x S y. As a consequence, if (II) does not hold, there must exist a sequence
x1,x2,...,xL as above, which means that x S y. In turn, x S y implies (I), by the
argument in the proof of Lemma 1.3.
The proof is completed by induction. We can deﬁne a sequence (⪰n)N
n=0 of
binary relations as follows. Let ⪰0 = ⪰. Now suppose we have given ⪰n−1 such
that ⟨⪰n−1,≻n−1⟩is acyclic and ⪰n−1 is antisymmetric. Suppose there is a pair
x,y unordered according to ⪰n−1. Applying Lemma 1.3, either (I) or (II) must
hold. Without loss, suppose ⟨⪰n−1 ∪{(x,y)},≻n−1 ∪{(x,y)}⟩is acyclic. Deﬁne
⪰n = ⪰n−1 ∪{(x,y)} and note that ≻n−1 ∪{(x,y)} is its strict part. Further, ⪰n
retains antisymmetry. Since X is ﬁnite, there is N such that ⪰N is complete,
and by construction transitive since ⟨⪰N,≻N⟩is acyclic.
The following result is perhaps the ﬁrst “revealed preference theorem” in the
book. Its signiﬁcance will become clear in the next few chapters.
Theorem 1.5
Let ⟨R,P⟩be an order pair. There is a preference relation ⪰
such that ⟨⪰,≻⟩is an order pair extension of ⟨R,P⟩iff ⟨R,P⟩is acyclic.
Proof. It is obvious that if ⪰exists, then ⟨R,P⟩is acyclic. We proceed to
prove the converse. Suppose that ⟨R,P⟩is acyclic. By Observation 1.2, we
can assume without loss of generality that R is reﬂexive.
The proof proceeds by ﬁrst showing that there is a transitive, though not
necessarily complete, relation ˆR for which ⟨ˆR,PˆR⟩is an order pair extension
of ⟨R,P⟩. Then we use Lemma 1.3 in an induction argument to show that the
relation ˆR can be completed.
In the ﬁrst step, deﬁne ˆR = (R)T, the transitive closure of R. It is obvious that
R ⊆ˆR. Further, suppose that x P y. Then, by acyclicity of ⟨R,P⟩, it follows that
y (R)T x is impossible, so that x PˆR y.
Now, let W be the collection of all acyclic order pairs ⟨R∗,P∗⟩that satisfy
(a) that ⟨R∗,P∗⟩is an order pair extension of ⟨R,P⟩, and (b) that P∗is the strict
part of R∗, P∗= PR∗. By the ﬁrst step of the proof, W ̸= ∅. Order W by
pointwise set inclusion; so that ⟨R1,P1⟩≤⟨R2,P2⟩if R1 ⊆R2 and P1 ⊆P2.
That is, ⟨R1,P1⟩≤⟨R2,P2⟩if ⟨R2,P2⟩is an order pair extension of ⟨R1,P1⟩.


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8
Mathematical Preliminaries
We shall use Zorn’s Lemma. Let ⟨Rλ,Pλ⟩λ∈ be a chain in W. We claim that
⟨
λ∈ Rλ,
λ∈ Pλ⟩∈W. Clearly it is an extension of ⟨R,P⟩. It is also clearly
acyclic; for, suppose that there is a cycle x1

λ∈ Rλ

x2 ...xn

λ∈ Rλ

x1,
with at least one instance of 
λ∈ Pλ. Then by deﬁnition, there is λ ∈
for which x1Rλx2 ...xnRλx1, with at least one instance of Pλ. Finally, it is
clear that 
λ∈ Pλ is the strict part of 
λ∈ Rλ, for if (x,y) ∈
λ∈ Pλ and
(y,x) ∈
λ∈ Rλ, there is λ ∈ for which x Pλ y and y Rλ x, contradicting our
hypothesis. And if (x,y) ∈
λ∈ Rλ but (y,x) ̸∈
λ∈ Rλ, then there is λ ∈
for which x Rλ y but not y Rλ x, so that x Pλ y.
By Zorn’s Lemma, there exists a maximal acyclic order pair ⟨˜R, ˜P⟩for which
˜P is the strict part of ˜R and which extends ⟨R,P⟩. Since R is reﬂexive, so
is ˜R. We claim that ˜R is complete. If not, then there are x,y ∈X with x ̸= y
which are unranked. By Lemma 1.3, there is an acyclic order pair extension of
⟨˜R, ˜P⟩and satisfying the relevant property, which also clearly extends ⟨R,P⟩,
contradicting maximality of ⟨˜R, ˜P⟩. The desired relation is then ˜R.
Theorem 1.5 has a direct application to a family (⟨Rλ,Pλ⟩)λ∈ of order pairs.
Corollary 1.6
There is a preference relation ⪰such that for all λ ∈, ⟨⪰,≻⟩
is an order pair extension of ⟨Rλ,Pλ⟩iff ⟨
λ∈ Rλ,
λ∈ Pλ⟩is an acyclic order
pair.
We will say that an order pair ⟨R,P⟩is quasi-acyclic if there is no sequence
x1,x2,...,xL such that
x1 P x2 P ... P xL,
and xL R x1.
The following provides a result related to Theorem 1.5, but guaranteeing
quasitransitive preference instead of transitive preference.
Lemma 1.7
There is a complete, quasitransitive relation ⪰such that ⟨⪰,≻⟩
is an order pair extension of ⟨R,P⟩iff ⟨R,P⟩is quasi-acyclic.
Proof. As in the proof of Theorem 1.5, one direction is obvious. Suppose
instead that ⟨R,P⟩is quasi-acyclic. Deﬁne x ⪰y iff it is not the case that y PT x.
We claim that ⪰is the appropriate relation.
We ﬁrst show that x ≻y iff x PT y. If x ≻y, then since y ⪰x is false, it follows
that x PT y is true. If, instead, x PT y is true, then by acyclicity, y PT x is false
(this uses the property that P ⊆R). As a consequence, we obtain x ⪰y. And
since x PT y, we know that y ⪰x is false, so that x ≻y.
Now, note that if x R y, it follows that y PT x is false (by quasi-acyclicity),
so that x ⪰y. Further, if x P y, then by the preceding paragraph, we know that
x ≻y. This proves that ⟨⪰,≻⟩is an extension of ⟨R,P⟩.
Further, ⪰is complete. Suppose by means of contradiction that it is not: then
there is a pair x,y which are unordered according to ⪰. By deﬁnition, it must be
that x PT y and y PT x. But then we have x PT x, contradicting quasi-acyclicity
(this uses the hypothesis that P ⊆R). And ≻is transitive as ≻= PT, which is
by deﬁnition transitive.


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1.4 Cyclic monotonicity
9
Finally, we establish one more result. It should be clear by now that, in
an order pair ⟨R,P⟩, P might not equal PR, the strict part of R. Starting from
Theorem 1.5, however, we have established extensions to pairs in which the
second order indeed equals the strict part of the ﬁrst order. This motivates the
question of when such extensions are possible.
Say that an order pair ⟨R,P⟩is asymmetric if there are no x,y for which x P y
and y R x. An asymmetric order pair may have cycles, but it has no cycles of
length 2.
Lemma 1.8
There is a complete relation ⪰such that ⟨⪰,≻⟩is an order pair
extension of ⟨R,P⟩iff ⟨R,P⟩is asymmetric.
Proof. One direction is obvious. For the other direction, deﬁne x ⪰y iff it is
not the case that y P x. Then it is clear that this relation is complete, as P ⊆R
and ⟨R,P⟩is asymmetric (so that it is impossible that x P y and y P x). To see
that ⟨⪰,≻⟩is an extension of ⟨R,P⟩: whenever x R y, we have y P x is false,
and consequently that x ⪰y. And whenever x P y, then x R y, so that x ⪰y. And
since x P y, by deﬁnition, (y,x) /∈⪰. Thus x ≻y.
1.4
CYCLIC MONOTONICITY
Let X ⊆Rn be nonempty, and ρ : X ⇒Rn be a correspondence. Say that ρ is
cyclically monotone (or that it satisﬁes cyclic monotonicity) if, for every ﬁnite
sequence x1,...,xL, with L > 1 and xL = x1, and every choice of zi ∈ρ(xi) for
i = 1,...,L −1, it holds that
L−1

i=1
zi · (xi+1 −xi) ≥0.
The property of cyclic monotonicity is important in revealed preference
theory. It is a multidimensional generalization of monotonicity: suppose that
n = 1. Then a function ρ : X →R is monotone decreasing if (y −x) and
(ρ(y) −ρ(x)) are of opposite signs; put differently, if
0 ≤(y −x)(ρ(x) −ρ(y)) = ρ(x)(y −x) + ρ(y)(x −y).
By analogy with the case when n = 1, we can say that a correspondence ρ :
X ⇒Rn is monotone when for all x,y ∈X and all zx ∈ρ(x), zy ∈ρ(y),
0 ≤zx · (y −x) + zy · (x −y).
The notion of cyclic monotonicity generalizes the idea of monotonicity to
account for all ﬁnite cycles in X, not only cycles of length two.
Theorem 1.9
Suppose that one of the following two hypotheses is satisﬁed:
• For all x ∈X, ρ(x) ̸= ∅.
• |{x ∈X : ρ(x) ̸= ∅}| < +∞and 0 <
		
x∈X ρ(x)
		 < +∞.


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10
Mathematical Preliminaries
Then the following statements are equivalent:
I) ρ is cyclically monotone.
II) There is a function u : X →R such that for all x,y ∈X and all zx ∈
ρ(x),
u(y) −u(x) ≤zx · (y −x).
III) There is a function u : X →R such that for all y ∈X,
u(y) = inf{u(x) + zx · (y −x) : x ∈X,zx ∈ρ(x)}.
Proof. It is obvious that (III) implies (II). To show that (II) implies (I), note
that for any sequence x1,...,xL, with xL = x1, it holds that 0 = L−1
l=1 (u(xl+1)−
u(xl)).
To prove the theorem, we shall prove that (I) implies the existence of a
function u that satisﬁes the properties stated in (II) and in (III). Let ρ : X ⇒Rn
be cyclically monotone. Fix an arbitrary x∗∈X for which ρ(x∗) ̸= ∅(this exists
under either of the two hypotheses). Fix z∗∈ρ(x∗).
Let x ∈X. Here is how we deﬁne u(x). Denote by x the set
of
all
ﬁnite
sequences
((x1,z1),(x2,z2),...,(xM−1,zM−1),(xM)),
with
M ≥1, x1 = x∗, xM = x, zi ∈ρ(xi) and z1 = z∗. Think of sequences
((x1,z1),(x2,z2),...,(xM−1,zM−1),(xM)) as paths connecting x∗and x = xM for
which there is zi ∈ρ(xi) for all xi, except for xM. Deﬁne u(x) as follows:
u(x) = inf{
M−1

m=1
zm · (xm+1 −xm) : ((x1,z1),...,(xM)) ∈x}.
The function u : X →R is then well deﬁned under either of the two
hypotheses. Under the ﬁrst hypothesis, for any sequence ending in xM, we may
ﬁx zM ∈ρ(xM) and note that for ((x1,z1),...,(xM))M
m=1 ∈x,
M−1

m=1
zm · (xm+1 −xm) + zM · (x∗−xM) ≥0,
as ρ is cyclically monotone. Therefore, zM · (x −x∗) is a lower bound on
M−1
m=1 zm ·(xm+1 −xm) for ((x1,z1),...,(xM))M
m=1 ∈x, and thus the inﬁmum in
the deﬁnition of u(x) is deﬁned in R. Under the second hypothesis, it is enough
to observe that by cyclic monotonicity we may without loss of generality
restrict to the subset of x which has no repetitions of elements, which by
assumption is ﬁnite. Hence, the inﬁmum becomes a minimum. Note that, under
either case, u(x∗) = 0.
We ﬁnish the proof by showing that the function u satisﬁes the properties in
(II) and (III). Fix x,y ∈X and zx ∈ρ(x). For any ((x1,z1),...,(xM)) ∈x, by
deﬁnition of u(y),
u(y) ≤zx · (y −x) +
M−1

m=1
zm · (xm+1 −xm);


--- Page 11 ---
1.4 Cyclic monotonicity
11
and hence u(y) −zx · (y −x) is a lower bound on the set in the right hand side
of the deﬁnition of u(x). Thus u(y) −zx · (y −x) ≤u(x). This establishes (II).
To prove (III), note that (II) implies that u(y) is a lower bound on {u(x) +
z · (y −x) : x ∈X,z ∈ρ(x)}. Fix ε > 0. By deﬁnition of u(y) there is some
((x1,z1),...,(xM)) ∈y such that u(y) + ε > M−1
m=1 zm · (xm+1 −xm). Let ˆx =
xM−1 and ˆz = zM−1. Then
u(y) + ε > ˆz · (y −ˆx) +
M−2

m=1
zm · (xm+1 −xm) ≥ˆz · (y −ˆx) + u(ˆx);
which ﬁnishes the proof.
When X is convex, the theorem has the important implication that the
function u is concave. We record this fact as follows:
Corollary 1.10
Let X ⊆Rn be a convex set. Under either of the hypotheses
of Theorem 1.9, the following statements are equivalent:
I) ρ is cyclically monotone.
II) There is a concave function u : X →R such that for all x,y ∈X and
zx ∈ρ(x)
u(y) −u(x) ≤zx · (y −x).
III) There is a concave function u : X →R such that for all y ∈X,
u(y) = inf{u(x) + z · (y −x) : x ∈X,z ∈ρ(x)}.
Proof. The deﬁnition u(y) = inf{u(x)+z·(y−x) : x ∈X,z ∈ρ(x)} in statement
(III) implies that u is concave, as it is the lower envelope of afﬁne functions.
The result follows because the function constructed in the proof of statements
(II) and (III) of Theorem 1.9 is the same.
Versions of these results exist even without either of the hypotheses of
Theorem 1.9. However, one must allow for the possibility that u is no longer
real-valued.
The importance of (II) in Corollary 1.10 should be clear from the following
result.
Proposition 1.11
Let X ⊆Rn be convex, and u : X →R be a concave
function. Then, for all x ∈int (X) there is p ∈Rn such that
u(y) ≤u(x) + p · (y −x),
for all y ∈X.
The proof of Proposition 1.11 is an application of the separating hyperplane
theorem, and is omitted.
For any function u : X →R and x ∈X, if p ∈Rn has the property that
u(y) ≤u(x) + p · (y −x), for all y ∈X, then p is a supergradient of u at x.
Proposition 1.11 says that if u is concave then it has a supergradient at any


--- Page 12 ---
12
Mathematical Preliminaries
interior point of its domain. The set of all supergradients of u at x is called the
superdifferential of u at x.
When u is concave and differentiable at x the only supergradient at x is the
gradient of u at x. So the superdifferential of u at x is ρ(x) = {∇u(x)}.
1.5
THEOREM OF THE ALTERNATIVE
Many results in revealed preference theory can be formulated using some
version of the Theorem of the Alternative, or Farkas’ Lemma. We state a
version (Lemma 1.12) that may involve either real or rational coefﬁcients and
solutions. The usefulness of such a result is that it allows one to state a problem
involving real solutions to a system of linear inequalities, and ﬁnd a revealed
preference axiom from the rational solution to the alternative linear system.
For a matrix B, we denote by Bi its ith row.
Lemma 1.12
Let F be either the set of real numbers R, or the set of rational
numbers Q. Let B be a (M1 + M2) × K matrix with entries in F. Consider the
systems of inequalities S1 and S2:
S1 :

Bi · θ ≥0,i = 1,...,M1
Bi · θ > 0,i = M1 + 1,...,M1 + M2
S2 :

η · B = 0
η ≥0,
where θ is the K-dimensional unknown in S1 and η, of dimension (M1 + M2),
is the unknown in S2. Then S1 has a solution θ ∈FK iff S2 has no solution
η ∈FM1+M2 with ηi > 0 for some i ∈{M1 + 1,...,M1 + M2}.
Lemma 1.13
(Integer–Real Farkas) Let {Ai}M
i=1 be a ﬁnite collection of
vectors in QK. Then one and only one of the following statements is true:
I) There exists y ∈RK such that for all i = 1,...,L, Ai · y ≥0 and for all
i = L + 1,...,M, Ai · y > 0.
II) There exists z ∈ZM
+ such that M
i=1 ziAi = 0, where M
i=L+1 zi > 0.
Proof. Both (I) and (II) cannot simultaneously hold. To see why, suppose that
there exist y and z as stated in (I) and (II). Then Ai ·y ≥0 for all i = 1,...,L and
Ai·y > 0 for all i = L+1,...,M. Consider M
i=1 ziAi·y. Since M
i=1 ziAi = 0, we
know that M
i=1 ziAi ·y = 0. Furthermore, since there is some j ∈{L+1,...,M}
for which zjAj·y > 0, and for all i, ziAi·y ≥0, we conclude that M
i=1 ziAi·y > 0,
a contradiction.
We now establish that if (II) does not hold, (I) holds. Note that (II) holds
iff there is z ∈QM
+ satisfying the statement in (II). (The reason being that M is
ﬁnite and that z satisﬁes the statement in (II) iff Nz satisﬁes it, for any positive
integer N.) By Lemma 1.12 if (II) does not hold, there exists y ∈QK such that


--- Page 13 ---
1.6 Chapter references
13
for all i = 1,...,L, Ai·y ≥0 and for all i = L+1,...,M, Ai·q > 0. Since QK ⊆RK,
y ∈RK.
Finally, the following nonhomogeneous version, which is easily proved
using Lemma 1.12, often comes in useful.
Lemma 1.14
Let B be an M × K matrix with real entries and let γ be an
M-vector with real entries. Consider the systems of inequalities S1 and S2:
S1 : Bi · θ ≥γi,i = 1,...,M
S2 :
⎧
⎪⎨
⎪⎩
η · B = 0
η ≥0
η · γ > 0
where θ is the K-dimensional unknown in S1 and η, of dimension M, is the
unknown in S2. Then S1 has a solution θ ∈RK iff S2 has no solution η ∈RM.
1.6
CHAPTER REFERENCES
Szpilrajn’s Theorem (Theorem 1.4) is due to Szpilrajn (1930). Theorem 1.5
generalizes Szpilrajn’s Theorem, and is due to Richter (1966) and Hansson
(1968). It appears in the form stated here in Suzumura (1976b). Sen (1969)
popularized the notion of quasitransitivity. The original proofs of Theorem 1.1
were based on a construction due to Cantor (1895), which shows that any two
countable dense linearly ordered sets without endpoints are order-isomorphic,
and ﬁrst appears in economics in Debreu (1954).
Theorem 1.9 and Corollary 1.10 are standard results in convex analysis:
see Rockafellar (1997). The idea of cyclic monotonicity and the fact that
it characterizes subsets of superdifferentials ﬁrst appears in Rockafellar
(1966). Rockafellar also observes that monotonicity is equivalent to cyclic
monotonicity for X = R. The condition of cyclic monotonicity appears
frequently in the mechanism design literature as well. Rochet (1987) is a
direct generalization of Rockafellar’s result. Since monotonicity is necessary
and sufﬁcient for cyclic monotonicity in one dimension, one might conjecture
that eliminating cycles of length n is necessary and sufﬁcient for cyclic
monotonicity in n −1 dimensions. This turns out to be false. An example
appears in Asplund (1970) (a mapping which rotates every x ∈R2 by π
n is
cyclically monotonic of order n but not n + 1). See also Bartz, Bauschke,
Borwein, Reich, and Wang (2007). However, there are characterizations of
monotonic correspondences which naturally generalize the Rockafellar result;
see Krauss (1985) or Fitzpatrick (1988).
Theorems of the Alternative, such as Lemma 1.12, make an early appearance
in Farkas (1902); classic references in economics are Kuhn and Tucker (1956)
and Gale (1960b). The version here for the real numbers appears ﬁrst in


--- Page 14 ---
14
Mathematical Preliminaries
Motzkin (1936), and can be derived from the more general formulation
due to Slater (1951). The rational version is essentially Theorem 3.2 of
Fishburn (1973b). A general result along these lines is Theorem 1.6.1 in Stoer
and Witzgall (1970). A more recent general reference is Schrijver (1998).
Lemma 1.14 can be found, for example, in Gale (1960b), and Lemma 1.13
in Chambers and Echenique (2014b).
