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CHAPTER 2
Classical Abstract Choice Theory
We start our development of revealed preference theory by discussing
the abstract model of choice. All revealed preference problems have two
components: data, and theory. Given a family of possible data, and a particular
theory, a revealed preference exercise seeks to describe the particular instances
of data that are compatible with the theory. We shall illustrate the role of each
component for the case of abstract choice. The data consist of observed choices
made by an economic agent. A theory describes a criterion, or a mechanism,
for making choices.
Given is a set X of objects that can possibly be chosen. In principle, X
can be anything; we do not place any structure on X. A collection of subsets
 ⊆2X\{∅} is given, called the budget sets. Budget sets are potential sets of
elements from which an economic agent might choose. A choice function is a
mapping c :  →2X\{∅} such that for all B ∈, c(B) ⊆B. Importantly, choice
from each budget is nonempty.
For the present chapter, choice functions are going to be our notion of data.
The interpretation of a choice function c is that we have access to the choices
made by an individual agent when facing different sets of feasible alternatives.
A particular choice function, then, embodies multiple observations.
The main theory is that of the maximization of some binary relation on X.
The theory postulates that the agent makes choices that are “better” than other
feasible choices, where the notion of better is captured by a binary relation.
The theory will be reﬁned by imposing assumptions on the binary relation: for
example that the relation is a preference relation (i.e. a weak order).
Given notions of data and theory, the problem is to understand when the
former are consistent with the latter. We are mainly going to explore two ways
of formulating this notion of consistency.
We say that a binary relation ⪰on X strongly rationalizes c if for all B ∈,
c(B) = {x ∈B : ∀y ∈B,x ⪰y}.
When there is such a binary relation, we say that c is strongly rationalizable.
The idea behind strong rationalizability is that c(B) should comprise all of
the best elements of B according to ⪰.


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Classical Abstract Choice Theory
In contrast, we may only want to require that c(B) be among the best
elements in B: Say that a binary relation ⪰on X weakly rationalizes choice
function c if for all B ∈,
c(B) ⊆{x ∈B : ∀y ∈B,x ⪰y}.
A utility function u weakly rationalizes the choice function c if the binary
relation ⪰deﬁned by x ⪰y ⇔u(x) ≥u(y) weakly rationalizes c.
Weak rationalizability allows for the existence of feasible alternatives that
are equally as good as the chosen ones, but that were not observed to be chosen.
It should be clear from the deﬁnitions that all choice functions are weakly
rationalizable by the binary relation X × X – meaning the binary relation
deﬁned by x ⪰y for all x and y (or, in other words, that x ∼y for all x,y ∈X). For
the exercise to be interesting, one must impose constraints on the rationalizing
⪰: such constraints can be thought of as a discipline on the revealed preference
exercise. The need to impose various kinds of discipline shall emerge more
than once in this book.
In contrast with weak rationalizability, strong rationalizability does rule
out some choice functions: There are choice functions that are not strongly
rationalizable. We say that strong rationalizability is testable, or that it has
observable implications. Our ﬁrst results, in Section 2.1 to follow, seek to
describe precisely those choice functions that are strongly rationalizable, and
to discuss rationalization by binary relations with particular properties.
2.1
STRONG RATIONALIZATION
Given a choice function c, we can deﬁne its revealed preference pair ⟨⪰c,≻c⟩
by x ⪰c y iff there exists B ∈ such that {x,y} ⊆B and x ∈c(B), and x ≻c y
iff there exists B ∈ such that {x,y} ⊆B, x ∈c(B), and y ̸∈c(B). The binary
relations in ⟨⪰c,≻c⟩give rise to the name “revealed preference theory.” The
idea, of course, is that if an agent’s choices are guided by a preference relation,
then x ∈c(B) and y ∈B when x is at least as good as y according to the agent’s
preferences. Thus x ⪰c y captures those binary comparisons which are revealed
by c to be part of the agent’s preferences.
It is important to recognize that, in general, ≻c need not be asymmetric, and
it need not be the strict part of ⪰c. In general, though, ≻c ⊆⪰c. Thus, it is an
order pair according to our deﬁnition. In fact, the notion of an order pair is
meant to capture precisely the pairs of orders arising in revealed preference
theory.
The theory developed here is based on the revealed preference pair. It
is, however, important to caution that there may be relevant information
in a choice function that is not contained in the revealed preference pair.
The following example presents two choice functions that give rise to the
same revealed preference pair. The ﬁrst is strongly rationalizable (by a
quasitransitive relation), while the other is not.


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2.1 Strong rationalization
17
Example 2.1
Let X = {x,y,z,w}. Consider  = {{x,y,z,w},{y,z,w}}. Deﬁne
a choice function c({x,y,z,w}) = {x,y} and c({y,z,w}) = {y,z}. This choice
function is rationalizable by the reﬂexive binary relation which ranks x ∼y,
x ≻z, y ∼z and each of x, y, and z above w. This relation is quasitransitive.
Now, let ′ = {{x,y},{y,z},{x,z},{x,w},{y,z,w}} where c′({x,y}) = {x,y},
c′({y,z}) = {y}, c′({x,z}) = {x}, c′({x,w}) = {x}, and c′({y,z,w}) = {y,z}. Note
this generates exactly the same revealed preference pair as the preceding, but
c′ is not rationalizable by any relation: The reason is that a rationalizing
relation would have to have y ≻z (from c′({y,z}) = {y}) and y ∼z (from
c′({y,z,w}) = {y,z}).
We begin with a description of strongly rationalizable choice functions. The
following observation, a direct consequence of the deﬁnition, almost gives us
our ﬁrst answer.
Proposition 2.2
A choice function c is strongly rationalizable iff it is strongly
rationalizable by ⪰c.
Proof. Suppose c is strongly rationalizable. Then there exists ⪰which strongly
rationalizes c. Let B ∈, and let x ∈c(B). By deﬁnition, for all y ∈B, x ⪰c y.
On the other hand, suppose that x ⪰c y for all y ∈B. For any y ∈B, since x ⪰c y,
there is By ∈ for which y ∈By and x ∈c(By). Since ⪰strongly rationalizes
c, we conclude that x ⪰y and thus that for all y ∈B, x ⪰y. Then x ∈c(B) as ⪰
strongly rationalizes c.
Given Proposition 2.2, it is easy to formulate a condition that says that
⪰c rationalizes c. The condition is called the V-axiom: A choice function c
satisﬁes the V-axiom if for all B ∈ and all x ∈B, if x ⪰c y for all y ∈B, then
x ∈c(B).
Theorem 2.3
A choice function is strongly rationalizable iff it satisﬁes the
V-axiom.
Proof. By Proposition 2.2, we need to show that the V-axiom is equivalent to
strong rationalizability by ⪰c. But that x ∈c(B) and y ∈B implies x ⪰c y is
just the deﬁnition of ⪰c, so strong rationalizability by ⪰c is the statement that
x ∈c(B) iff for all y ∈B, x ⪰c y, which is just the V-axiom.
Many revealed preference exercises boil down to an extension exercise; in
which, given a revealed preference pair, we seek an order pair extension with
particular properties. Recall that when we write the order pair ⟨⪰,≻⟩, then ≻
is the strict part of ⪰. This may not be the case for ⟨⪰c,≻c⟩.
Theorem 2.4
A binary relation ⪰strongly rationalizes c if ⟨⪰,≻⟩is an order
pair extension of ⟨⪰c,≻c⟩.
Proof. Suppose that ⟨⪰,≻⟩is an order pair extension of ⟨⪰c,≻c⟩. We need to
show that for all B ∈, x ∈c(B) if and only if x ⪰y for all y ∈B. So suppose


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Classical Abstract Choice Theory
that x ∈c(B) and y ∈B. Then by deﬁnition, x ⪰c y. Consequently, x ⪰y. Now
suppose that x ⪰y for all y ∈B. Since c(B) ̸= ∅, if x ̸∈c(B) then there is y ∈B
for which y ≻c x. Then y ≻x, as ⟨⪰,≻⟩is an order pair extension of ⟨⪰c,≻c⟩,
which would contradict that x ⪰y for all y ∈B. So x ∈c(B).
A converse to Theorem 2.4 is available when ⪰is a preference relation:
Theorem 2.5
A preference relation ⪰strongly rationalizes c iff ⟨⪰,≻⟩is an
order pair extension of ⟨⪰c,≻c⟩.
Proof. Given Theorem 2.4, we shall only prove the necessity of order pair
extension.
Suppose that c is strongly rationalizable by some preference relation ⪰. The
deﬁnition of strong rationalization implies that if x ⪰c y, then x ⪰y. Suppose
that x ≻c y. Then there is B for which {x,y} ⊆B, x ∈c(B) and y ̸∈c(B). Since
y ̸∈c(B), completeness of ⪰implies that there is z ∈B with z ≻y. Since x ∈
c(B), we know x ⪰z. By transitivity of ⪰, x ≻y.
We now turn to strong rationalization by a preference relation (a weak order).
The condition should be familiar from our discussion in Chapter 1. Say that
a choice function c is congruent if ⟨⪰c,≻c⟩is acyclic. Our next result is
the fundamental characterization of rationalization by a preference relation;
it follows quite directly from the results we have established in Chapter 1.
Theorem 2.6
A choice function is strongly rationalizable by a preference
relation iff it is congruent.
Proof. By Theorem 2.5, there is a preference relation ⪰rationalizing c iff
there is a preference relation ⪰for which ⟨⪰,≻⟩is an order pair extension of
⟨⪰c,≻c⟩. By Theorem 1.5, this is true iff ⟨⪰c,≻c⟩is acyclic.
It is frequently convenient to work with single-valued choice functions, choice
functions such that for every B ∈, c(B) is a singleton. The following result
says that many natural assumptions one could want to place on such choice
functions are equivalent: one could say observationally equivalent, in the sense
that they strongly rationalize the same choice functions.
Proposition 2.7
Suppose for all B ∈, B is ﬁnite, and |c(B)| = 1. Then the
following statements are equivalent:
I) c is strongly rationalizable by a complete, quasitransitive relation.
II) c is strongly rationalizable by a preference relation.
III) c is strongly rationalizable by a strict preference relation.
IV) ≻c is acyclic.
Proof. That (III) implies (II) and (II) implies (I) are obvious, the latter because
a transitive relation is always quasitransitive. We show that (I) implies (III).
First, note that ⪰c = (≻c ∪=), because c is single-valued. We know that there
is a complete and quasitransitive ⪰which rationalizes c. We will show that ≻c


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2.1 Strong rationalization
19
⊆≻. So suppose that x ≻c y. This means that there is B ∈ for which {x,y} ⊆B,
x ∈c(B) and y ̸∈c(B). Because y ̸∈c(B) and ⪰is complete, there exists y2 ∈B
for which y2 ≻y1 = y. If y2 ̸∈c(B), there is y3 ∈B for which y3 ≻y2. We
can inductively continue this construction, and because ⪰is quasitransitive, ≻
is acyclic, and B is ﬁnite, there is yk such that yk ≻yk−1 ≻...y1 = y, where
yk ∈c(B). By quasitransitivity, yk ≻y. But since c(B) is single-valued, yk = x,
so that x ≻y, which is what we wanted to show.
Now, we let ⪰∗be a strict preference relation for which x ≻y implies
x ≻∗y, which exists by Theorem 1.4 (note that (≻∪=) is a partial order by
quasitransitivity of ⪰). We claim that ⪰∗rationalizes c. So let x ∈c(B), which
means that for all y ∈B where y ̸= x, x ≻c y, so that x ≻y and ﬁnally x ≻∗y.
Conversely, suppose that x ⪰∗y for all y ∈B. Then it must be that x ⪰y for
all y ∈B, as otherwise, there would exist y ∈B for which y ≻x, which would
imply y ≻∗x. Consequently, as ⪰rationalizes c, we have x ∈c(B).
Finally, (I) is equivalent to (IV). It is easy to see that if c is single-valued
and quasitransitive rationalizable, then ≻c is acyclic. Conversely, suppose that
≻c is acyclic. Since c is single-valued, ⪰c= (≻c ∪=). By Lemma 1.7, there is
a quasitransitive ⪰for which ⪰c ⊆⪰and ≻c ⊆≻. The result now follows by
Theorem 2.4.
2.1.1
Weak axiom of revealed preference
Congruence in Theorem 2.6 requires one to rule out cycles of any length (using
the terminology of Chapter 1). A weaker condition only rules out cycles of
length two. We say that a choice function satisﬁes the weak axiom of revealed
preference if whenever x ⪰c y, it is not the case that y ≻c x.1
While the weak axiom of revealed preference has less bite than congruence,
its weakness can be compensated for by a condition on , the domain of the
choice function. Note that the larger is the collection , the more restrictive is
the weak axiom. So one can make up for the weakness of the weak axiom
by demanding a large collection . In particular, if  includes all sets of
cardinality at most three, the weak axiom of revealed preference is equivalent
to rationalizability by a preference relation.
The following pair of results gives the implications of the weak axiom
for choice on arbitrary domains. We then present a result which gives the
implications of the weak axiom when  is rich, in a sense to be made precise.
Theorem 2.8
A choice function c satisﬁes the weak axiom of revealed
preference iff there exists a complete binary relation ⪰which strongly
rationalizes c such that ≻extends ≻c (i.e. ≻c ⊆≻).
1 The weak axiom of revealed preference is an instance of the condition we termed asymmetry
of an order pair in Chapter 1. Indeed, by Lemma 1.8, the weak axiom of revealed preference is
the weakest hypothesis that establishes the existence of a complete relation ⪰such that ⟨⪰,≻⟩
is a order pair extension of ⟨⪰c,≻c⟩.


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Classical Abstract Choice Theory
Proof. First, let ⪰be a complete rationalizing relation such that ≻extends ≻c.
Since ⪰strongly rationalizes c, ⪰c ⊆⪰. Therefore, if x ⪰c y and y ≻c x, we
would have x ⪰y and y ≻x, a contradiction with the deﬁnition of ≻.
On the other hand, suppose c satisﬁes the weak axiom of revealed
preference. By Lemma 1.8, there is a complete ⪰for which ⪰c ⊆⪰and
≻c ⊆≻. By Theorem 2.4, ⪰rationalizes c.
Theorem 2.9
Suppose that  contains all sets of cardinality at most three.
Then c is strongly rationalizable by a preference relation iff it satisﬁes the weak
axiom of revealed preference.
Proof. That a choice function rationalizable by a preference relation satisﬁes
the weak axiom of revealed preference is obvious, as the weak axiom is implied
by congruence. Conversely, suppose that c satisﬁes the weak axiom of revealed
preference. By assumption, for all x,y ∈X, {x,y} ∈. Since c({x,y}) ̸= ∅, this
implies that ⪰c is complete. Next, we claim that it is transitive. For suppose
that x ⪰c y and y ⪰c z. If either x = y, y = z, or x = z, then it is obvious that
x ⪰c z. So suppose they are all distinct. To prove x ⪰c z, we will show that x ∈
c({x,y,z}), for if not, either y ∈c({x,y,z}), in which case y ≻c x (contradicting
the weak axiom); or y ̸∈c({x,y,z}) and z ∈c({x,y,z}), in which case z ≻c y,
again contradicting the weak axiom. Thus x ∈c({x,y,z}), so that x ⪰c z, and ⪰c
is transitive. This shows that ⪰c is a preference relation.
Finally, ≻c is the strict part of ⪰c. Clearly, x ≻c y implies x ⪰c y, and, by the
weak axiom, also implies that y ⪰c x is false. Conversely, if x ⪰c y and y ⪰c x is
false, we know that x = c({x,y}), so that x ≻c y. By Theorem 2.4, c is strongly
rationalizable by ⪰c.
Note that, under the hypotheses of Theorem 2.9, the revealed preference
relation ⪰c strongly rationalizes c (Proposition 2.2). By observing choice
from all sets of cardinality two, one can identify uniquely the rationalizing
preference. For all x and y, {x,y} ∈, so c({x,y}) is deﬁned and nonempty. So
x and y must be ordered by ⪰c.
There are two conditions that are well known to “decompose” the weak
axiom of revealed preference. We give them the names ﬁrst used by Sen (1969),
and we show that they are equivalent to the weak axiom.
Say that c satisﬁes condition α if, for any B,B′ ∈ where B ⊆B′, c(B′) ∩
B ⊆c(B). Say that c satisﬁes condition β if for all B,B′ ∈ where B ⊆B′, if
{x,y} ⊆c(B), then x ∈c(B′) if and only if y ∈c(B′).
The weak axiom refers implicitly to choice from two sets: the set at which x
is chosen over y when we say that x ⪰c y, and the set at which y is chosen over
x when we say that y ≻c x. Conditions α and β talk explicitly about these sets;
α constrains choice from a smaller set, given what was chosen at a larger set,
while condition β goes in the opposite direction. The following result explains
that α and β can be said to decompose the weak axiom.


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2.1 Strong rationalization
21
Theorem 2.10
If a choice function satisﬁes the weak axiom, then it satisﬁes
conditions α and β. Conversely, suppose that either of the two following
properties is satisﬁed:
I)  has the property that whenever B,B′ ∈ have nonempty intersec-
tion, then B ∩B′ ∈
II) For all x,y, {x,y} ∈.
Then if a choice function satisﬁes conditions α and β, it satisﬁes the weak
axiom.
Proof. Let c be a choice function with domain . Suppose that c satisﬁes the
weak axiom. We shall prove that it satisﬁes α and β. Let B,B′ ∈ with B ⊆B′.
Consider ﬁrst condition α. Suppose that x ∈c(B′) ∩B. Then x ⪰c y for all
y ∈B′. In particular, x ⪰c y for any y ∈c(B), as B ⊆B′. If x ̸∈c(B), then there
is z ∈c(B) (which is assumed nonempty), so that z ≻c x, violating the weak
axiom. Hence x ∈c(B).
Second, consider condition β. Let {x,y} ⊆c(B) and B ⊆B′. Then x ⪰c y and
y ⪰c x. Suppose x ∈c(B′). If y ̸∈c(B′), we have y ≻c x, contradicting the weak
axiom. Hence y ∈c(B′). A symmetric argument establishes that if y ∈c(B′),
then x ∈c(B′).
Conversely, suppose c satisﬁes conditions α and β. Suppose, toward a
contradiction, that there are x and y with x ⪰c y and y ≻c x. There are then
A,A′ ∈ with {x,y} ⊆A ∩A′, x ∈c(A), y ∈c(A′) and x /∈c(A′). Let B = A ∩A′
if condition (I) is satisﬁed, and let B = {x,y} if condition (II) is satisﬁed. Then
by condition α, x ∈c(A) and y ∈c(A′) imply that x,y ∈c(B). Now condition β
implies that we cannot have y ∈c(A′) and x /∈c(A′). Hence c satisﬁes the weak
axiom.
Corollary 2.11
Suppose that  contains all sets of cardinality at most
three. Then c is strongly rationalizable by a preference relation iff it satisﬁes
conditions α and β.
Proof. Follows from Theorem 2.9 and Theorem 2.10.
When choice c(B) is a singleton for all B, then condition β has no bite. Our
next result considers choice functions with singleton values. It explains why
some authors reserve the phrase “weak axiom of revealed preference” for what
we have called condition α.
Theorem 2.12
Suppose that  has the property that whenever B,B′ ∈ have
nonempty intersection, then B ∩B′ ∈. Suppose further that for all B ∈,
|c(B)| = 1. Then c satisﬁes the weak axiom of revealed preference iff it satisﬁes
condition α.
Proof. Note that condition β is vacuous under the single-valuedness hypothe-
sis. The result therefore follows directly from Theorem 2.10.


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Classical Abstract Choice Theory
2.1.2
Maximal rationalizability
Strong rationalizability requires that c(B) consist of the “best” alternatives in B
for a binary relation. Instead, one can require that c(B) contain all the elements
of B that cannot be “improved” upon. If the binary relation in question is
complete, the two approaches will be equivalent. In general, however, they
differ.
Say that a binary relation ⪰maximally rationalizes c if
c(B) = {x ∈B : ∄y ∈B,y ≻x}.
That is, c(B) corresponds to a set of undominated elements.
If there exists a binary relation ⪰that maximally rationalizes c, then there
exists a binary relation ⪰′ which strongly rationalizes c. Namely, x ⪰′ y if y ≻x
is false.2 When ⪰is complete, it is easy to see that ⪰′ = ⪰.
The following example shows that there are choice functions that are
strongly rationalizable but not maximal rationalizable.
Example 2.13
Let  = {{x,y},{x,z},{x,y,z}}, and c({x,y}) = {x,y}, c({x,z}) =
{x,z}, and c({x,y,z}) = {x}. Then this choice function is strongly rationalizable,
but is not maximal rationalizable. The relation ⪰that strongly rationalizes c
is given by x ⪰y, y ⪰x, x ⪰z, z ⪰x, x ⪰x, y ⪰y, and z ⪰z. Note that ⪰is not
complete.
To see that c is not maximal rationalizable, suppose that ⪰maximally
rationalizes c. Then because c({x,y,z}) = {x}, it must be that either x ≻y or
z ≻y. But x ≻y is impossible, as c({x,y}) = {x,y}. On the other hand, suppose
that z ≻y. Then it follows that x ≻z, or else z ∈c({x,y,z}). But this contradicts
z ∈c({x,z}).
2.1.3
Quasitransitivity
The previous analysis makes use of the property of transitivity of a preference
relation. In contrast, a quasitransitive relation presents a challenge. In partic-
ular, a version of Theorem 2.6 for strong rationalization by a quasitransitive
relation is not available (see also Theorem 11.2). But one can easily prove a
similar result in one direction:
Proposition 2.14
If ⟨⪰c,≻c⟩is quasi-acyclic, then there is a complete and
quasitransitive relation strongly rationalizing c.
Proof. Suppose ⟨⪰c,≻c⟩is quasi-acyclic. By Lemma 1.7, there is a complete
and quasitransitive relation ⪰for which ⪰c ⊆⪰and ≻c ⊆≻. By Theorem 2.4,
⪰strongly rationalizes c.
But quasi-acyclicity of ⟨⪰c,≻c⟩is not necessary for strong rationalization
by a quasitransitive relation, as shown by our next example.
2 ⪰′ is then termed the canonical conjugate of ≻(Kim and Richter, 1986). This object appears
often in revealed preference theory.


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2.2 Satisﬁcing
23
Example 2.15
Suppose X = {x,y,z}, and let  = {{x,y},{y,z},{x,z},{x,y,z}}.
Deﬁne c({x,y}) = {x,y}, c({y,z}) = {y,z}, c({x,z}) = {x}, and c({x,y,z}) = {x,y}.
Then c is strongly rationalizable by the relation ⪰given by
⪰= {(x,y),(y,x),(y,z),(z,y),(x,z),(x,x),(y,y),(z,z)};
that is, all pairs are indifferent except for x and z. The relation ⪰is
quasitransitive. However, z ⪰c y and y ≻c z so that ⟨⪰c,≻c⟩is not quasi-acyclic.
Proposition 2.7 presents a characterization of quasitransitive strong ratio-
nalization in the case when c is singleton-valued. In general, however, it is
difﬁcult to ﬁnd a simple characterization of quasitransitive rationalizability.
The reasoning is simple. When we have a preference relation ⪰strongly
rationalizing c, we know that if x ∈c(B) and y ∈B, but y ̸∈c(B), then x ≻y. The
same cannot be said to hold for quasitransitive relations. In fact, suppose that
⪰is a complete relation on a ﬁnite set X for which ≻is acyclic. A relatively
easy proposition states that this ⪰is a preference relation iff for all ﬁnite B, if
y ∈B but y ̸∈c(B), then for all x ∈c(B), x ≻y. On the other hand, it can be
shown that ⪰is quasitransitive iff for all ﬁnite sets B, if y ∈B and y ̸∈c(B),
then there exists x ∈c(B), x ≻y.
2.2
SATISFICING
The notion of strong rationalization can be adapted to the study of a theory
of “bounded rationality,” the theory of satisﬁcing behavior. The theory, which
was ﬁrst proposed by Herbert Simon, claims that an individual always makes
choices which are “good enough,” though not necessarily optimal. In our
context, the assumption is that if an individual chooses an alternative, then
she also chooses anything which is at least as good.
We will say that a choice function c is satisﬁcing rationalizable by ⪰if for
all B ∈ and all x,y ∈B, x ∈c(B) and y ⪰x imply y ∈c(B).
The following two results characterize various notions of satisﬁcing
rationalizability. They are essentially extension results, whereby instead of
searching for ⪰for which ⪰c ⊆⪰and ≻c ⊆≻, we simply require ≻c ⊆≻.
Theorem 2.16
The following statements are equivalent:
I) c is satisﬁcing rationalizable by a strict preference relation.
II) c satisﬁcing rationalizable by a preference relation.
III) ≻c is acyclic.
Proof. First, note that if c is satisﬁcing rationalizable by a strict preference
relation, then it is obviously satisﬁcing rationalizable by a preference relation.
Second, we prove that (II) implies (III). So suppose that c is satisﬁcing
rationalizable by a preference relation ⪰, but suppose that there is a cycle
x1 ≻c x2 ... ≻c xK ≻c x1. We claim that xi ≻c xi+1 implies that xi ≻xi+1. Note
that as xi ≻c xi+1, there exists B ∈ for which {xi,xi+1} ⊆B, xi ∈c(B), and


--- Page 10 ---
24
Classical Abstract Choice Theory
xi+1 ̸∈c(B). Because ⪰satisﬁcing rationalizes c, it follows that xi ≻xi+1;
as otherwise, we must have xi+1 ∈c(B). This demonstrates that there is a ≻
cycle, contradicting the fact that ⪰is a preference relation. This shows that (II)
implies (III).
Finally, we show that (III) implies (I). So suppose that ≻c admits no
cycles. We can consider the transitive closure of the relation {(x,y) : x ≻c y or
x = y}; denote it by ⪰′. By acyclicity of ≻c, ⪰′ is a partial order. By Szpilrajn’s
Theorem (Theorem 1.4), it admits an extension to a strict preference relation
⪰. We claim that ⪰satisﬁcing rationalizes c. To see this, suppose that B ∈,
x ∈c(B), and y ⪰x. Then it must be the case that y ∈c(B), as otherwise, we
would have x ≻c y, which would imply x ≻y, a contradiction.
We say a choice function satisﬁes the weakened weak axiom of revealed
preference if ≻c is asymmetric. The following theorem states that the
weakened weak axiom is necessary and sufﬁcient for the existence of a
complete binary relation ⪰for which ≻c ⊆≻.
Theorem 2.17
A choice function is satisﬁcing rationalizable by a complete
binary relation iff it satisﬁes the weakened weak axiom of revealed preference.
Proof. First, suppose that c is satisiﬁcing rationalizable by a complete binary
relation ⪰, and suppose by way of contradiction that there exist x and y for
which x ≻c y and y ≻c x. Then there exist B,B′ ∈ for which {x,y} ⊆B ∩B′,
x ∈c(B), y ̸∈c(B), y ∈c(B′), x ̸∈c(B′). As ⪰satisﬁcing rationalizes c, we
therefore conclude that x ≻y and y ≻x, a contradiction.
On the other hand, suppose that c satisﬁes the weakened weak axiom of
revealed preference. We deﬁne x ⪰y when y ≻c x is false. Note that by
the weakened weak axiom of revealed preference, ⪰is complete. Further, ⪰
satisﬁcing rationalizes c. To see this, suppose that {x,y} ∈B for some B ∈,
and that x ∈c(B) and y ⪰x. Then if y ̸∈c(B), we know that x ≻c y. This implies
that y ≻c x is false (by the weakened weak axiom), so that x ⪰y. And since
x ≻c y, we know that y ⪰x is false (by deﬁnition of ⪰). We conclude that x ≻y,
a contradiction.
Note that there is an obvious parallel between Theorem 2.6 and Theo-
rem 2.16 on the one hand, and between Theorem 2.8 and Theorem 2.17 on
the other. However, there is no counterpart of Theorem 2.9 here. For example,
a choice function c deﬁned on all ﬁnite subsets of {x,y,z} by c({x,y}) = {x},
c({y,z}) = {y}, c({x,z}) = {z}, c({x,y,z}) = {x,y,z} satisﬁes the weakened
weak axiom of revealed preference, but there is no transitive relation ⪰that
satisﬁcing rationalizes c.
2.3
WEAK RATIONALIZATION
The notion of strong rationalization is based on the assumption that c(B) could
capture all the choices that an agent might potentially make from a set B. In


--- Page 11 ---
2.3 Weak rationalization
25
general, though, in an empirical environment, we have no reason to expect that
we will be able to observe such an object. What we usually observe is a choice
that an agent makes; but this does not preclude the possibility of other choices
being reasonable for that agent. The notion of weak rationalization is intended
to capture this idea.
The following theorem is an analogue to Theorem 2.5. However, it does
not require the hypothesis that ⪰be a preference relation. Thus, in general,
the property of being weakly rationalizable by a relation ⪰is completely
characterized by the revealed preference pair.
Theorem 2.18
A binary relation ⪰weakly rationalizes c iff the revealed
preference pair ⟨⪰c,≻c⟩satisﬁes ⪰c ⊆⪰.
Proof. Suppose c is weakly rationalizable by some ⪰. Note that if x ⪰c y, then
by deﬁnition, x ⪰y. So ⪰c ⊆⪰.
On the other hand, suppose that the revealed preference pair ⟨⪰c,≻c⟩
satisﬁes ⪰c ⊆⪰. Then we claim that ⪰weakly rationalizes c. We need to show
that for all B ∈, x ∈c(B) implies x ⪰y for all y ∈B. So suppose that x ∈c(B).
Then by deﬁnition, x ⪰c y. Consequently, x ⪰y.
Weak rationalization is trivial if we place no restrictions on the order which
can weakly rationalize a choice function. This is because complete indifference
weakly rationalizes anything. To this end, we will study rationalization by
preference relations which have monotonicity properties. Monotonicity will
be a discipline imposed on the revealed preference exercise.
Let ⟨≥,>⟩be an acyclic order pair. Say that a binary relation ⪰is monotonic
with respect to ⟨≥,>⟩if ⟨⪰,≻⟩is an order pair extension of ⟨≥,>⟩. The
meaning of ⟨≥,>⟩is that ≥and > reﬂect some observable characteristics of
the alternatives under consideration, and that we can require the unobservable
rationalizing relations to somehow conform to ≥and >.
The order pair ⟨≥,>⟩suggests a structure on budget sets. Say that B ⊆X is
comprehensive with respect to order pair ⟨≥,>⟩if whenever x ∈B and x ≥y,
y ∈B. The notion of a comprehensive budget is natural when budgets are
deﬁned to be a set of objects that are “affordable.” Suppose that B is deﬁned by
some notion of what the agent can purchase, like a budget deﬁned from prices
and income (see Chapter 3). Then, if x ∈B because the cost of purchasing x
is below the agent’s income, and x ≥y implies that the cost of purchasing y
cannot exceed that of purchasing x, then of course we have that y ∈B.
For a ﬁxed ⟨≥,>⟩, deﬁne the order pair ⟨⪰R,≻R⟩by x ⪰R y if x ⪰c y and
deﬁne x ≻R y if there exists B ∈ and z ∈B where {x,y,z} ⊆B, x ∈c(B) and
z > y.
The following theorem shows that the order pair ⟨⪰R,≻R⟩is the proper tool
for studying rationalization by a preference relation which is monotonic with
respect to ⟨≥,>⟩. We say that a choice function satisﬁes the generalized axiom
of revealed preference if the order pair ⟨⪰R,≻R⟩is acyclic. Theorem 2.19 is
relevant to the issues we shall study in Chapter 3.


--- Page 12 ---
26
Classical Abstract Choice Theory
Theorem 2.19
Suppose that the acyclic order pair ⟨≥,>⟩satisﬁes x > y ≥z
implies x > z, and that all B ∈ are comprehensive. Then there exists a
preference relation which is monotonic with respect to order pair ⟨≥,>⟩and
which weakly rationalizes c iff ⟨⪰R,≻R⟩satisﬁes the generalized axiom of
revealed preference. In addition, if there is a countable set Y = {y1,y2,...}
such that for all x,z satisfying x > z, there is k such that x > yk > z, then the
preference relation can be chosen to have a utility representation.
Proof. Suppose preference relation ⪰is monotonic with respect to ⟨≥,>⟩and
weakly rationalizes c. Suppose by way of contradiction that ⟨⪰R,≻R⟩is not an
acyclic order pair. If it is not acyclic, there is x1,...,xL for which x1 ⪰R x2 ... ⪰R
xL and xL ≻R x1. Since xL ≻R x1, there is z for which xL ⪰R z and z > x1. Since
⪰weakly rationalizes c, we have x1 ⪰... ⪰xL ⪰z, and by monotonicity, we
have z ≻x1, a contradiction to the fact that ⪰is a preference relation.
Conversely, suppose that ⟨⪰R,≻R⟩is an acyclic order pair. We shall
demonstrate that ⟨⪰R ∪≥,>⟩is also an acyclic order pair.3 It will therefore
follow by Theorem 1.5 that there is a preference relation ⪰for which ⪰R ⊆⪰,
≥⊆⪰, and >⊆≻. As a consequence, ⪰will be monotonic with respect to
⟨≥,>⟩, and by Theorem 2.18, ⪰will weakly rationalize c (since ⪰R = ⪰c). So,
suppose for a contradiction that ⟨⪰R ∪≥,>⟩is not an acyclic order pair. The
key observations here are that for all x,y,z ∈X:
• x ⪰R y ≥z implies x ⪰R z
• x ⪰R y > z implies x ≻R z
• x ≻R y ≥z implies x ≻R z
Indeed, the ﬁrst implication follows because if x ⪰R y, there exists B ∈ for
which {x,y} ⊆B and x ∈c(B). But if y ∈B, by comprehensivity, z ∈B as well, so
that x ⪰R z. The second implication follows by deﬁnition. The third implication
follows since, if x ≻R y, then there exist B ∈ and w ∈B for which x ∈c(B)
and w > y. As B is comprehensive and y ≥z, z ∈B. Further, as w > y ≥z, we
have by assumption that w > z. Consequently x ≻R z.
Since ⟨⪰R ∪≥,>⟩is not acyclic, there is a ⟨⪰R ∪≥,>⟩-cycle. For simplicity,
let Q =

⪰R ∪≥

. Let x1 Q ... Q xL > x1 be a ⟨⪰R ∪≥,>⟩-cycle, where L ≥2.
By the preceding observations, we can, without loss of generality, assume that
this cycle takes the form
x1 ≥x2 ≥... ≥xK ⪰R ... ⪰R xL > x1
(2.1)
by converting all relations of the form xi ⪰R xi+1 ≥xi+2 to xi ⪰R xi+2.
If K ̸= L, then we have xL−1 ⪰R xL > x1. Here, the second observation
implies that xL−1 ≻R x1. But x1 ≥... ≥xK, so by repeatedly applying the third
observation, we would have xL−1 ≻R xK. But then xK ⪰R ... ⪰R xL−1 ≻R xK,
contradicting acyclicity of ⟨⪰R,≻R⟩.
3 For future reference, note that this also implies that ⟨⪰R ∪≥,≻R ∪>⟩is an acyclic order pair.


--- Page 13 ---
2.3 Weak rationalization
27
On the other hand, if K = L, then the cycle in equation (2.1) is a ⟨≥,>⟩
cycle, contradicting the acyclicity of ⟨≥,>⟩. We have therefore established
that ⟨⪰R ∪≥,>⟩is acyclic.
To see the statement about the utility representation, let S =

⪰R ∪≥

, and
let T =

≻R ∪>

. Deﬁne the relation U by x U y if there are x1,...,xk with
x = x1 V ... V xk = y, where each V ∈{S,T}, and at least one instance coincides
with T. Note that U is transitive. Deﬁne u(x) = 
{k:xUyk} 2−k.
First, we claim that u represents ⪰which weakly rationalizes c. Thus,
suppose that x ∈c(B), and that y ∈B. Now, if y U yk, then we have x ⪰R y U yk,
whereby x U yk, so that u(x) ≥u(y). Second, we claim that ⪰is also monotonic
with respect to ⟨≥,>⟩. Thus, suppose x ≥y. Again, if y U yk, then by deﬁnition
x ≥y U yk, so that x U yk. Hence u(x) ≥u(y). Now, suppose that x > y. Then
there is yk for which x > yk > y. We claim that y U yk is false (obviously x U yk).
So, suppose by way of contradiction that y U yk holds, so that y U yk > y.
This implies the existence of a ⟨⪰R ∪≥,>⟩cycle, which we previously
demonstrated to be impossible.
If one is willing to sacriﬁce the transitivity of indifference, nearly any choice
function becomes weakly rationalizable by a monotonic binary relation. As we
shall see, the relevant property of c is that for all x, x ≻R x is false. In fact,
any choice function satisfying this property can be rationalized by the relation
deﬁned as x ⪰y if y > x is false. Let us denote this relation as >−1.
Proposition 2.20
Suppose ⟨≥,>⟩is an acyclic order pair for which x > y ≥
z implies x > z. Then >−1 is complete, quasitransitive, and monotonic with
respect to ⟨≥,>⟩.
Proof. To show that >−1 is complete, suppose that it is not. Then there are
x,y for which x > y and y > x. But this contradicts the acyclicity of order
pair ⟨≥,>⟩(as >⊆≥). Further, >−1 is quasitransitive. Suppose x >−1 y and
y >−1 z, but that y >−1 x and z >−1 y are false. It follows that x > y and y > z
must be true. As a consequence of our assumption that x > y ≥z implies x > z,
this implies that x > z, which implies by acyclicity that x >−1 z, and of course
z >−1 x is false. It is clear by acyclicity that >−1 is monotonic with respect to
⟨≥,>⟩.
Theorem 2.21
Suppose that the acyclic order pair ⟨≥,>⟩satisﬁes x > y ≥z
implies x > z, and that all B ∈ are comprehensive. Then the following
statements are equivalent:
I) There exists a complete binary relation which is monotonic with
respect to order pair ⟨≥,>⟩and which weakly rationalizes c.
II) For all x, x ≻R x is false.
III) >−1 weakly rationalizes c.
Proof. First, we show that (I) implies (II). So suppose that there is a complete
order monotonic with respect to order pair ⟨≥,>⟩and weakly rationalizing c.


--- Page 14 ---
28
Classical Abstract Choice Theory
We claim that for all x, x ≻R x is false. Suppose by way of contradiction that
there exists x for which x ≻R x. Then by deﬁnition there exists B ∈ and y ∈B
for which x ∈c(B) and y > x. But since y > x and ⪰is monotonic with respect
to order pair ⟨≥,>⟩, it follows that y ≻x. This implies that x ̸∈c(B), as ⪰
weakly rationalizes c, a contradiction.
To see that (II) implies (III), we need to show that >−1 weakly rationalizes
c; suppose that x ∈c(B). Suppose by way of contradiction that there is y ∈B
such that y >−1 x and x >−1 y is false. Then it follows that y > x. But since
x ∈c(B), y ∈B, and y > x, it follows that x ≻R x, a contradiction.
Finally, that (III) implies (I) is obvious.
The theorem demonstrates that the only empirical content of quasitransitive
choice is that for all B, if x ∈c(B), then x lies on the “boundary” of B.
The equivalence of (I) and (III) establishes that, amongst binary relations
monotonic with respect to ⟨≥,>⟩, the assumption of quasitransitivity adds
no additional empirical content to the hypothesis of weak rationalization by a
complete binary relation. This stands in strong contrast to weak rationalization
by a preference relation.
2.4
SUBRATIONALIZABILITY
A dual notion to the notion of weak rationalizability is what we will call
subrationalizability. We will say that a choice function c is subrationalizable
by ⪰if for all B ∈,
∅̸= {x ∈B : x ⪰y for all y ∈B} ⊆c(B).
Thus, a subrationalizable choice function admits all dominant elements of
⪰, but may admit other alternatives. We might be interested in such a condition
in an environment where an individual potentially makes mistakes.
Fishburn introduces the following notions. For a ﬁnite collection ′ ⊆,
we deﬁne
C(′) =

x ∈

B∈′
B : For all B ∈′,x ∈B ⇒x ∈c(B)

.
That is, C(′) is the set of all alternatives that are always chosen from ′
when they are available. We will say that choice function c satisﬁes partial
congruence if for all ′ ⊆ where 0 < |′| < +∞, C(′) ̸= ∅.
As its name suggests, partial congruence is weaker than congruence.
Proposition 2.22
If c satisﬁes congruence, then it satisﬁes partial congru-
ence.
Proof. Suppose c violates partial congruence, and let ′ ⊆ for which
0 < |′| < +∞and C(′) = ∅. Now, for each x ∈
B∈′ c(B), there is
y ∈
B∈′ c(B) for which y ≻c x. Pick an arbitrary x ∈
B∈′ c(B), and label


--- Page 15 ---
2.4 Subrationalizability
29
this x1. For each xi, ﬁnd xi+1 ∈
B∈′ c(B) for which xi+1 ≻c xi. Since |′| < ∞,
eventually there will be i > j for which there is B ∈′ so that {xi,xj} ⊆c(B).
Observe that xi ≻c ... ≻c xj, and xj ⪰c xi, contradicting congruence.
The following result characterizes subrationalizability by preference rela-
tions.
Theorem 2.23
Suppose a choice function has the property that |c(B)| < +∞
for all B ∈. Then the following are equivalent:
I) c is subrationalizable by a preference relation.
II) c is subrationalizable by a strict preference relation.
III) c satisﬁes partial congruence.
The requirement that |c(B)| < +∞for all B ∈ is necessary to
complete a critical induction step in the proof. The general characterization
of subrationalizable choice functions remains open. The condition of partial
congruence may be problematic from a falsiﬁability perspective. The reason
is that if 
E∈′ E is an inﬁnite set, the statement that there exists x ∈C(′)
is existential. This means that it postulates the existence of a certain object. If
our process of scientiﬁc observation consists of observing chosen elements of
choice sets one by one, and if we suppose that we can only observe ﬁnite data,
we can never falsify the hypothesis that such an element exists.
Proof. To see that (I) implies (II), suppose that c is subrationalizable by a
preference relation. One simply needs to “break ties.” To see this, suppose that
⪰is a preference relation subrationalizing c. Let ≥be a well-ordering of X.4
We deﬁne ⪰′ by x ⪰′ y if x ≻y or x ∼y and y ≥x. First, for every B ∈, there
exists a ⪰′-maximal element. To see this, consider the collection of ⪰-maximal
elements, and pick the ≥-minimal element (this necessarily exists because ≥is
a well-ordering). Then this element is clearly ⪰′-maximal. And further, every
⪰′-maximal element of B is also ⪰-maximal. So ⪰′ subrationalizes c, and is a
strict preference relation.
To see that (II) implies (III), suppose c is subrationalizable by a strict
preference relation ⪰. We need to show that it satisﬁes partial congruence.
To this end, let ′ ⊆ be a ﬁnite subcollection of budgets. Then for each
B ∈′, there exists a unique maximal element x(B) ∈B according to ⪰. By
letting x∗be the maximal element of {x(B) : B ∈′} (this exists as ′ is a ﬁnite
collection), we see that x∗∈C(′).
To see that (III) implies (I), suppose that c satisﬁes partial congruence. We
add to  all singleton and binary sets which are not originally present, and
extend c to these sets by c({x,y}) = {x,y}. This new choice function is also
clearly partially congruent, and we will show that it is subrationalizable (this
4 A well-ordering of a set X is a strict preference relation for which every nonempty subset of X
has a minimal element. Existence of well-orderings is guaranteed by the axiom of choice.


--- Page 16 ---
30
Classical Abstract Choice Theory
will establish that c is subrationalizable). So, without loss, we can assume that
 includes all binary sets.
Now, let E be the collection of ﬁnite subsets of . For a partial order ⪰and
′ ∈E, we deﬁne C(′,⪰) = {x ∈
E∈′ E : ∄y ∈
E∈′ E,y ≻x}. That is,
C(′,⪰) are the ⪰-maximal elements of 
E∈′ E.
We consider the collection R of all partial orders ⪰on X such that for all
′ ∈E, C(′,⪰) ∩C(′) ̸= ∅. We know that R is nonempty, as the relation
 = {(x,x) : x ∈X} satisﬁes  ∈R. We will show that R contains at least one
strict preference relation ⪰; this will be enough to complete the proof. To see
why, note that if ′ ∈E is ′ = {B}, C(′,⪰) is just the ⪰-maximal element
of B. And C(′) = c(B), so that we conclude that the ⪰-maximal element of B
is an element of c(B).
So, we order R by set inclusion. For any chain {⪰λ}λ∈ ⊆R, we claim that
⪰′=

λ∈ ⪰λ

∈R. We will thus be able to use Zorn’s Lemma to establish
the existence of a maximal element. First, it is easy to verify that ⪰′ is a partial
order; we will skip the details. We need to show that ⪰′ satisﬁes C(′,⪰′) ∩
C(′) ̸= ∅for all ′ ∈E. For a contradiction, suppose that there exists some
′ ∈E for which C(′,⪰′) ∩C(′) = ∅. In particular, this implies that for
every x ∈C(′), there exists y ∈
B∈′ B for which y ≻′ x. Now, note that as
c(E) is ﬁnite for each E ∈, this necessarily implies that C(′) is also ﬁnite
(since ′ is ﬁnite). This in particular implies that there exists ⪰λ such that for
each x ∈C(′), there exists y ∈
B∈′ B for which y ≻λ x. But this contradicts
the fact that ⪰λ∈R. We conclude by Zorn’s Lemma that there exists a maximal
⪰∗∈R. Our only task now is to show that ⪰∗is complete.
Now, suppose conversely that ⪰∗is not complete. Then there exist x′,y′ ∈X
for which x′ and y′ remain unranked according to ⪰∗. Recall from Chapter 1
that T denotes transitive closure. We consider two possible extensions of ⪰∗:
⪰1=

⪰∗∪{(x′,y′)}
T, and ⪰2=

⪰∗∪{(y′,x′)}
T. The two differ in how they
rank x′ and y′. It is easy to verify that x ⪰1 y if x ⪰∗y or x ⪰∗x′ and y′ ⪰∗y,
and that x ⪰2 y if x ⪰∗y or x ⪰∗y′ and x′ ⪰∗y.
Because ⪰∗is maximal and each of ⪰1 and ⪰2 are partial orders
(antisymmetry can be proved by supposing it to be false and establishing a
contradiction that x′ and y′ were ranked according to ⪰∗), we conclude that
there exist 1,2 ∈E for which C(1) ∩C(1,⪰1) = ∅and C(2) ∩C(2,
⪰2) = ∅.
This implies that
w ∈C(1) ∩C(1,⪰∗) ⇒∃x ∈

E∈1
E such that x ⪰∗x′ and y′ ⪰∗w,
(2.2)
and that
z ∈C(2) ∩C(2,⪰∗) ⇒∃x ∈

E∈2
E such that x ⪰∗y′ and x′ ⪰∗z.
(2.3)
Finally, let ∗= 1 ∪2 ∪{{x′,y′}}. Since  includes all singleton and
binary sets, ∗∈E, so that C(∗) ∩C(∗,⪰∗) ̸= ∅. So pick b ∈C(∗) ∩


--- Page 17 ---
2.5 Experimental elicitation of choice
31
C(∗,⪰∗). We claim that b ∈{x′,y′}. Observe that
C(∗) ∩C(∗,⪰∗) ⊆

C(1) ∩C(1,⪰∗)

∪

C(2) ∩C(2,⪰∗)

∪{x′,y′}.
So, suppose that b ∈C(1) ∩C(1,⪰∗). By (2.2), we conclude that b =
y′—otherwise, (2.2) tells us that y′ ≻∗b (recall ⪰∗is a partial order), so that
b ̸∈C(∗) ∩C(∗,⪰∗). Analogously, by (2.3), if b ∈C(2) ∩C(2,⪰∗), we
conclude that b = x′.
This establishes that b ∈{x′,y′}. So, suppose without loss of generality that
b = x′. Since C(1)∩C(1,⪰∗) ̸= ∅, we can conclude by (2.2) that there exists
z ∈
E∈1 E such that z ⪰∗x′. But unless we have z = x′, it would follow that
x′ ̸∈C(∗,⪰∗), which we know to be false. So we must conclude that x′ = z ∈

E∈1 E. But by deﬁnition of C, this implies that x′ ∈C(1), as x′ ∈C(∗).
Finally, x′ ∈C(∗,⪰∗) implies x′ ∈C(1,⪰1), as z ≻1 x′ implies z ≻∗x′.
So we have shown that C(1)∩C(1,⪰1) ̸= ∅, a contradiction. Similarly, if
we assume that b = y′, we arrive at the conclusion that C(1)∩C(2,⪰2) ̸= ∅.
As either possibility arrives at a contradiction, we have established that ⪰∗is
complete, and hence a strict preference relation.
2.5
EXPERIMENTAL ELICITATION OF CHOICE
Until now, we have abstained from assigning any concrete interpretation to the
notion of a choice function. However, for choice to be an empirical concept, it
must be observable. There are conceptual and practical issues that arise when
we operationalize the idea that choice should be observable.
Imagine a choice-theoretic experiment, the goal of which is to study a
particular subject’s choice function. In the experiment, the subject may be
presented with multiple budgets. If the goal of the experiment is to understand
what the individual would choose from each of the budgets, we run into a
basic problem. Imagine an individual presented with choices from the two
budgets: {hat,left shoe}, {jacket,right shoe}. Suppose that, when presented
with the budget {hat,left shoe}, the choice would be the hat, and when
presented with the choice {jacket,right shoe}, the choice would be the jacket.
If presented with both choices simultaneously, it is possible that the subject
might choose the pair of shoes. The two shoes are complements. The presence
of complementarities distorts choice.
There is a classical solution to this problem. The idea is to ask the subject
to announce which choice she would make from each budget, and then to
randomly select a budget, paying the subject the announced choice from that
budget. Suppose  is ﬁnite. Consider a ﬁnite set of states of the world , to
be realized in the future, and associate each B ∈ with some state ωB ∈. If
the subject is asked to report a single-valued choice function, and announces
c, the subject is paid the random variable paying off the single element of c(B)
in state ωB, which we can also call c with a slight abuse of notation. This
mechanism is referred to as the random decision selection mechanism.


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32
Classical Abstract Choice Theory
Note that under the random decision selection mechanism, the subject’s
ultimate payoff is an element of X, rather than of X. Thus, what matters
for her choices is her preference over X, rather than her preference over
X. We would expect there to be some relation between preferences on X and
preferences on X. We can say a preference ⪰∗over X is monotonic with
respect to a preference ⪰over X if whenever f,g ∈X satisfy f(ω) ⪰g(ω) for
all ω ∈, then f ⪰∗g, with a strict preference if in addition there is ω ∈ for
which f(ω) ≻g(ω).5 Monotonicity is usually understood as the claim that, if
the subject is made better off ex-post, regardless of which state obtains, then
the subject is made better off ex-ante.6
The punchline is that if ⪰is a complete and transitive relation over X,
and ⪰∗over X is monotonic with respect to ⪰, then a single-valued choice
function c ⪰∗dominates all elements of X iff for all B ∈, if c(B) = {y},
then y ⪰x for all x ∈B. Thus, if we are willing to assume monotonicity, then
the random decision selection mechanism is a good way of eliciting choice.
Importantly, monotonicity requires no form of separability (such as Savage’s
P2 “sure thing” principle), or an expected utility hypothesis.
2.6
CHAPTER REFERENCES
Probably one of the ﬁrst works to mention abstract choice theory is Arrow
(1951). The theory is mostly a generalization of the classical demand context,
described in the next section. Uzawa (1956) is one of the ﬁrst papers to study
the revealed preference approach in an abstract environment, followed shortly
thereafter by Arrow (1959), to whom Theorem 2.9 is due. Corollary 2.11
and Theorem 2.12 are due to Sen (1971), while condition α ﬁrst appears
in Chernoff (1954). Theorem 2.6 is ﬁrst established in full generality in
the independent works of Richter (1966) and Hansson (1968). The main
contribution in those papers was to describe restrictions on budget sets; prior
to this, most works in abstract choice assumed that every ﬁnite set could be a
potential budget set.
The term “congruence” is due to Richter. Theorem 2.3 on rationaliz-
ability is due to Richter (1971). A related result, characterizing maximal
rationalizability, is provided by Bossert, Sprumont, and Suzumura (2005).
Theorem 2.8 on the class of choice functions satisfying the weak axiom of
revealed preference is from Wilson (1970), see also Mariotti (2008). Wilson
notes the connection between choice functions satisfying the weak axiom and
those induced as von Neumann–Morgenstern stable sets. See also Plott (1974).
The characterizations of satisﬁcing rationalizability by preference relation
5 This assumes no state is null, so that we believe each state might possibly occur; see Chapter 8.
6 This statement is compelling so long as the ex-post notion of “better off” coincides with the
ex-ante notion. That is, in evaluating ex-post outcomes, the ex-ante preference ⪰over certain
prospects is applied. There are classical examples (Diamond, 1967 or Machina, 1989) where
we would not expect this relationship to hold.


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2.6 Chapter references
33
and strict preference relations appear in Aleskerov, Bouyssou, and Monjardet
(2007) and Tyson (2008). The notion of the weakened weak axiom of revealed
preference is due to Ehlers and Sprumont (2008), Theorem 2.17 appears there,
and the concept in that characterization appears in Wilson (1970) as the notion
of a “Q cut.” Theorem 2.23 is due to Fishburn (1976).
Theorem 2.19 is related to Afriat’s Theorem, discussed formally in the next
chapter. A result along the lines of Theorem 2.19 appears in Quah, Nishimura,
and Ok (2013).
The theory of satisﬁcing behavior in Section 2.2 was proposed by Simon
(1955). The observations on the distinction between rationalizability and
dominant rationalizability are due to Suzumura (1976a). Kim (1987) studies
generalized transitivity concepts which we have not discussed, all of which
turn out to be empirically equivalent to the standard concept. Bossert and
Suzumura (2010) is a detailed work devoted to studying the empirical content
of many generalized choice models, some of which we describe here.
A recent literature has used choice as a primitive to study various
“behavioral” theories. Manzini and Mariotti (2007) consider choice by the
successive application of different binary relations. Masatlioglu, Nakajima,
and Ozbay (2012) study attention, and formalize the notion that a decision
maker may only consider a subset of the available alternatives. Ok, Ortoleva,
and Riella (2014) develop a model of reference-dependent choice. Green and
Hojman (2007), de Clippel and Eliaz (2012), Cherepanov, Feddersen, and
Sandroni (2013), and Ambrus and Rozen (2014) describe models of agents
with multiple motivations. Finally, we should mention the paper by de Clippel
and Rozen (2012), which studies some of these behavioral developments under
the assumption of limited observability that is very relevant to the notion of
empirical content.
In a similar spirit, Green and Hojman (2007), Bernheim and Rangel (2007),
Chambers and Hayashi (2012), and Bernheim and Rangel (2009) use choice
theory to propose welfare criteria that remain valid when standard revealed
preference axioms fail.
The random decision selection mechanism is originally due to Allais
(1953), and discussed also by Savage (1954) and his “hot man” example. The
idea is also used in the famous elicitation mechanism of Becker, DeGroot,
and Marschak (1964). The framework described here is taken from Azrieli,
Chambers, and Healy (2012).
