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CHAPTER 13
Revealed Preference and Model Theory
In this book, we have studied the concept of empirical content in disparate
environments. To conclude our study, we wish to suggest that there is a
unifying theme behind these exercises. The idea of the empirical content of
a theory as the set of all falsiﬁable predictions of the theory is generally
applicable, and subject to formal study.
A theory can make predictions which are non-falsiﬁable. A case in point
is the theory of representation by a utility function. Recall Theorem 1.1. The
theorem implies that if a preference relation ⪰over Rn
+ possesses a utility
representation, then there is a countable set Z ⊆Rn
+ such that for all x,y ∈X
for which x ≻y, there exists z ∈Z for which x ⪰z ⪰y. This implication of
the theory of utility is not falsiﬁable. To demonstrate that the theory has been
falsiﬁed, one would need to establish the non-existence of such a set Z. Doing
so involves checking, one-by-one, every possible countable subset Z of Rn
+, a
task which can never be completed.
A ﬁrst and basic issue in understanding empirical content has to do with
universal vs. existential axiomatizations. The idea was already introduced in
Chapters 9 and 12, where we saw the removal of existential quantiﬁers as a
source of testable implications. The issue of universal and existential axioms
goes back to Popper (1959), who thought that a theory with a universal
description is falsiﬁable, while an existential theory is not.
Popper offers the example of the theory that claims “all swans are white.”
This theory is universal, in the sense that it states a property of all swans, or
“universally quantiﬁes over swans.” It is easy to see that, in principle, such a
theory can be falsiﬁed by ﬁnding a single swan that is not white. Contrast with
Popper’s example of an existential theory: that “there exists a black swan.”
The existential theory cannot be falsiﬁed. Falsifying the theory would involve
collecting all possible swans and verifying that each one is not black. We could
only do this if we could somehow be sure to have exhaustively checked all the
swans in the universe.
Universality is clearly important for falsiﬁability, but there is a second
component that is particularly relevant for the subject of this book. Popper’s


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Revealed Preference and Model Theory
187
idea captures the notion of empirical content and falsiﬁability very well in
many environments. However, economic theories and data are often burdened
by partial observability. Implicit in Popper’s examples is the idea that when we
observe a swan, we observe its color. And indeed, in the swans example, this is
entirely natural. With economic choice data, on the other hand, it is perfectly
reasonable to observe objects but not all their properties. It is common, for
example, to observe a pair of alternatives and not be able to observe which of
the pair an individual chooses, or would choose. Think of choice from a budget
B. When x ∈B is chosen this reveals something about the pairs (x,y) for y ∈B,
but nothing about the pairs (z,y) for z ̸= x. This issue has been important in
many of the results we have established. It lies behind the non-testability of
concave utility in Afriat’s Theorem, for example.1
Thus, in contrast to the theories that Popper had in mind, economic theories
have the feature that we may be able to observe objects without observing the
properties these objects enjoy. The phenomenon of partial observability will
be very important for our discussion.
The considerations of universality and partial observability will be reﬂected
in the kinds of axioms that capture the empirical content of a theory. Recall
GARP, which states that the revealed preference pair ⟨⪰R,≻R⟩is acyclic.
Acyclicity of a revealed preference pair is a universal theory, but it is more.
Using the universal quantiﬁer ∀and the negation symbol ¬, we can write
acyclicity succinctly as for all n,
∀x1...∀xn¬
1n−1
6
i=1
(xi ⪰R xi+1) ∧(xn ≻R x1)
2
.
(13.1)
The structure of Equation (13.1) explains its falsiﬁability: it begins with
universal quantiﬁcation, followed by a negation, followed by a conjunction ∧
(meaning “and”), followed by basic properties about observables. Speciﬁcally,
such “basic” properties amount to observable relations among observables.
This mathematical equation is therefore a Universal Negation of Conjunction
of Atomic Formulas: it is an UNCAF formula.
Negating atomic formulas
means that one rules out that a particular (observable) relation holds among
observable entities or quantities.
Contrast GARP with the hypothesis that ⪰R is a preference relation (a weak
order) and ≻R is its strict part. This hypothesis has a universal axiomatization,
but not of the UNCAF form. We shall see that this distinction matters. The
hypothesis that ⪰R is a weak order and ≻R is its strict part has the following
axiomatization:
I) ∀x∀y(x ⪰R y) ∨(y ⪰R x)
II) ∀x∀y∀z(x ⪰R y) ∧(y ⪰R z) →(x ⪰R z)
III) ∀x∀y(x ≻R y) ↔(x ⪰R y) ∧¬(y ⪰R x).
1 If one could observe comparisons between all pairs of alternatives, then concavity would
clearly be testable.


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Revealed Preference and Model Theory
Some claims made by this theory are falsiﬁable. For example if we observe
that x ≻R y, y ≻R z, and z ≻R x (a violation of transitivity), then obviously
we have falsiﬁed the theory. But not all claims made by the theory are
falsiﬁable. A pair x,y could be observed without observing any relation
between them for example: this is common in the consumption datasets
analyzed in Chapters 3–5. Therefore, in the presence of partial observability,
completeness cannot be falsiﬁed.2 Interestingly, GARP is usually understood
as forming the empirical content of axioms (I), (II), and (III). We demonstrate
below the formal sense in which this is true.
13.1
A MODEL FOR OBSERVABLES, DATA, AND THEORIES
There are three important concepts we need to explain. The ﬁrst is the primitive
of the model: the things we can observe are the primitive, and these things will
be speciﬁed through a language. The second is what we mean by a dataset:
datasets are ﬁnite and consist of partial observations. The third is our notion of
a theory: a theory is a formal way of hypothesizing that certain relationships
hold between objects of interest.
A ﬁrst-order language L is given by a ﬁnite set of relation symbols and, for
each relation symbol R, a positive integer nR, the arity of R.
Let L be a language. An L-dataset D is given by:
I) A ﬁnite non-empty set D (the domain of D).
II) An n-ary relation RD over D for every n-ary relation symbol R of L.
Each element (x∗
1,...,x∗
nR) ∈RD is intended to represent the observation that
(x∗
1,...,x∗
n) stand in relation R. The notion of a dataset is intended to capture
the idea that there can only be a ﬁnite number of observations.
As an example, consider the language L that has two binary relation
symbols, ⪰and ≻. We mean the former to signify revealed weak preference
observations, and the latter to signify revealed strict preference observations.
Consider a dataset D1, with domain D1 = {a,b,c}, and where we observe
that a is revealed weakly preferred to b, and b to c, but we do not observe
any strict comparisons. In symbols, D1 = (D1,⪰D1,≻D1); ⪰D1 is the relation
given by a ⪰D1 b and b ⪰D1 c, while ≻D1 is empty. The example illustrates
partial observability: We might theorize that a ⪰D1 b implies either b ⪰D1 a or
a ≻D1 b, but often data will not contain this kind of information. In fact partial
observability is very prominent in economics, for example in the consumption
datasets used by the papers described in Chapter 5.
2 In an environment of full observability (meaning, not partial) and strong rationalization,
Eliaz and Ok (2006) investigate choice functions based on maximization of a relation that
retains transitivity, but which need not be complete. In such a context, they show that
completeness adds real content. They also show how indifference can often be distinguished
from incompleteness in such an environment.


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13.1 A model for observables, data, and theories
189
We will now deﬁne a notion of theory. We are interested in investigating a
notion of empirical content, so from this perspective, we imagine that all of
the relevant aspects of a theory can be captured by describing the relations
among potential observables that we hypothesize hold. To this end, deﬁne an
L-structure M to consist of the following objects:
I) A nonempty set M (the domain of M).
II) An n-ary relation RM over M for every n-ary relation symbol R of L.
A structure forms a hypothesis about the relations which we might expect to
observe, but we never expect to see the entire structure. Rather, we imagine
that a dataset is consistent with a structure when all of the observations are
members of the structure.
With this in mind, we say that an L-structure M rationalizes an L-dataset
D if the following conditions are satisﬁed:
I) D ⊆M, where D and M are the domains of D and M.
II) RD ⊆RM.
The deﬁnition of rationalization requires that RD ⊆RM rather than that RD is
the restriction of RM to D. The idea is again simply that we do not imagine
that all existing relations are necessarily observed. This is the nature of partial
observability: observing only a weak preference for coffee over tea does not
refute the possibility that coffee is strictly preferred to tea.
We say that two structures are isomorphic if we can relabel the objects
across the two structures so that all relations are maintained. Let M and N
be L-structures with domains M and N respectively. Formally, an isomorphism
from M to N is a bijective map η : M →N that preserves the interpretations
of all symbols of L: (a1,...,anR) ∈RM iff (η(a1),...,η(anR)) ∈RN for every
relation symbol R of L and a1,...,anR ∈M.
Informally, two structures are isomorphic if there is no way to use our
language to distinguish between them.
Finally, we deﬁne a theory to be a class of structures. Formally, an L-theory
T is a class of structures that is closed under isomorphism. A dataset D is
T-rationalizable if there is a structure in T that rationalizes D. Otherwise, D
falsiﬁes T.
As an example, consider again the language with two binary symbols, ⪰
and ≻. We have the theory Two consisting of the class of triples (M,⪰M,≻M)
such that ⪰M is a preference relation on M (a weak order), and ≻M is the strict
preference derived from ⪰M. The theory Two can be thought of as the theory of
rational choice. Note that Two rationalizes the dataset D1 described above: D1
has observed objects D1 = {a,b,c}, where a is revealed weakly preferred to b,
and b to c. The data D1 is Two-rationalizable because it can be rationalized, for
example, by the set M of all letters in the English alphabet, with ⪰M being the
lexicographic order on M. Of course, there are many other structures in Two
that could rationalize D1.


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Revealed Preference and Model Theory
We emphasized that D1 is silent on some aspects of the relationship between
a, b, and c; these aspects are not observed, but we do not view this partial
observability as a conﬂict with Two. Some “theoretically true” relations are
simply not observed in the dataset. The structure of all the letters in the English
alphabet is a possible rationalization of D1 in Two. In this structure a is strictly
preferred to c, a property that has not been observed in D1.
In contrast with data D1, consider the dataset D2; where D2 = {a,b,c} (the
same as D1), but where we observe no weak comparisons, and instead observe
that
a ≻D2 b ≻D2 c ≻D2 a.
No structure in Two could rationalize D2 because the “theoretical” strict
preference ≻M in such a structure would need to exhibit the cyclic comparisons
a ≻M b ≻M c ≻M a. This is impossible for a weak order.
Another example of a theory is the theory of utility maximization, denoted
Tu, which is the set of triples (M,⪰M,≻M) in Two such that there is u : M →R
with x ⪰M y iff u(x) ≥u(y). Note that
Tu ⊊Two.
The theory of utility maximization is more stringent than Two. But Tu can
also rationalize D1 (but not D2). In fact, it is easy to see that any dataset that is
Two-rationalizable is also Tu-rationalizable, even though utility maximization
is more stringent than rational choice. Put differently, one can weaken Tu to
Two without observable consequences. When that happens we shall say that
the two theories have the same empirical content.
13.1.1
Empirical content
We want the empirical content of a theory to capture all of the observable
consequences of that theory, but no other consequences. To this end, we want
to weaken the theory as much as possible without changing the observable
consequences of the theory, removing all non-observable consequences. For
example, we can obtain a new theory by adding structures to Tu (thus
weakening Tu) without changing the datasets that falsify the new theory. The
empirical content of Tu is the most one can weaken Tu in this fashion.
Hence we deﬁne the empirical content of a theory T, denoted ec(T), to be
the class of all structures M that do not rationalize any dataset that falsiﬁes
T. The main result of this chapter is that the empirical content of a theory is
captured by the UNCAF axioms that are true in that theory.
Given a language L, we can write formulas using the symbols in L. In
addition to the relation symbols speciﬁed by L, we shall use standard logical
symbols: the quantiﬁers “exists” (∃) and “for all” (∀); “not” (¬); the logical
connectives “and” (∧) and “or” (∨); a countable set of variable symbols
x,y,z,u,v,w,...; parentheses “(” and “)”; and equality and inequality symbols
“=” and “̸=”.


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13.1 A model for observables, data, and theories
191
Strings of symbols are put together to form axioms. Rules for the formation
of axioms can be found, for example, in Marker (2002).
We are primarily concerned with a special class of axioms, the UNCAF
axioms. These are deﬁned as follows. First we must deﬁne the notion of an
atomic formula.
An atomic formula ϕ of a language L consists of either
I) t1 = t2 or t1 ̸= t2, where t1,t2 are variable symbols or
II) R(t1,...,tnR) where R is a relation symbol of L and t1,...,tnR are
variable symbols.
Atomic formulas are closely related to the notion of observation discussed
above. Let us consider again the language with two binary relation symbols, ⪰
and ≻. In this example, all atomic formulas use at most two variable symbols.
For example, the string x ⪰y is an atomic formula, as is x ≻y. These strings
are unquantiﬁed and as such do not yet form axioms. But one can imagine that
an observation might consist of some pair a and b for which a is observed to
stand in relation ⪰to b.
We will form axioms out of the atomic formulas. The axioms are intended
to be statements precluding the existence of certain ﬁnite collections of
observations from holding. To this end, let L be a language. A universal
negation of a conjunction of atomic formulas (UNCAF) axiom is a string of
the form
∀v1∀v2 ...∀vn¬(ϕ1 ∧ϕ2 ··· ∧ϕm)
where ϕ1,ϕ2,...,ϕm are atomic formulas with variables from v1,...,vn.
As an example, consider again the language that has two binary relation
symbols, ⪰and ≻. An example of an UNCAF axiom in this language is:
∀x∀y¬(x ⪰y ∧y ≻x);
which we might call the weak axiom of revealed preference (WARP).
Similarly, GARP can be seen to be an UNCAF axiom. In a different language,
congruence (recall Chapter 2) is UNCAF.
Let  be a set of UNCAF axioms of L. Let T () consist of the structures
for which all axioms in  are true; thus, T () is a theory. If T = T () for
some set  of axioms, we say that  is an UNCAF axiomatization of T.
Given a theory T, let uncaf(T) be the set of UNCAF axioms that are true in
all members of T. The following result establishes that the empirical content
of a theory always has an UNCAF axiomatization. This is true whether or not
the theory itself does. Moreover, one such UNCAF axiomatization consists of
the UNCAF axioms true for every structure in the theory.
Theorem 13.1
For every theory T, ec(T) is the theory axiomatized by the
UNCAF axioms that are true in T: ec(T) = T (uncaf(T)).
Thus, in the presence of partial observability, UNCAF axioms, not universal
ones, properly describe the empirical content of the model. In the context of


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Revealed Preference and Model Theory
the examples we have been using, Theorem 13.1 presents the formal sense in
which GARP is the empirical content of completeness and transitivity.
Theories that coincide with their empirical content are in some sense special.
They cannot be weakened in any way without adding falsifying datasets. We
shall see in 13.2 an example of a theory with T = ec(T).
13.1.2
Relative theories
In economics, a researcher often wants to take certain hypotheses as being
given. For example, economic theorists often view continuity axioms as a
technical assumption. By itself, continuity has no empirical content. Hence,
they are not interested in testing for this property, though it is often useful
for providing a representation. Even though continuity by itself usually has no
empirical content, the axiom may have empirical content when imposed with
other axioms. So what we really care about are the empirical implications of a
preference in the presence of the hypothesized continuity.
Moreover, there are often obvious constraints imposed on us by the structure
of the model. We can imagine an individual with preferences over bundles
of coffee and tea. Bundles of coffee and tea are elements in a linear space.
It would not be interesting to “test” the axioms for a vector space. One can
then talk about the empirical content of the economically meaningful theory,
relative to the theory of linear spaces.
Consider two theories, T and T′, where T ⊆T′. We can deﬁne the empirical
content of T relative to T′, written ecT′(T), as the class of all structures M ∈T′
that do not rationalize any dataset that falsiﬁes T, i.e.,
ecT′(T) = ec(T) ∩T′.
(13.2)
The following is an immediate consequence of Theorem 13.1.
Corollary 13.2
For any theories T and T′ such that T ⊆T′, ecT′(T) =
T (uncaf(T)) ∩T′.
We say that a collection of UNCAF axioms  is an UNCAF axiomatization
of T relative to T′ if T = T () ∩T′. Corollary 13.2 implies that the empirical
content of T relative to T′ admits an UNCAF axiomatization relative to T′.
13.2
APPLICATION: STATUS QUO PREFERENCES
As mentioned above, theories that satisfy T = ec(T) are particularly interest-
ing. As an application, we discuss a recent theory of choice in the presence of
status quo due to Masatlioglu and Ok. There is a sense in which their theory
makes no non-falsiﬁable claims.
Let L be a language including the binary relation ∈and the ternary relations
c, ˜c. The latter two are meant to express “chosen” and “not chosen.” We deﬁne
the theory of choice with status quo Tcsq to be the class of structures (M,∈M,


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13.2 Application: Status quo preferences
193
cM, ˜cM) whereby there is some set X, a collection  ⊆2X\{∅}×X satisfying
(E,b) ∈ implies b ∈E, and a nonempty valued function c∗:  →2M\{∅}
for which c∗(E,b) ⊆E for all (E,b) ∈ for which the following are satisﬁed:
I) M = X ∪
II) ∈M is the usual set theoretic relation
III) cM(a,b,E) if and only if a ∈c∗(E,b)
IV) ˜cM(a,b,E) if and only if a ̸∈c∗(E,b);
as well as all structures isomorphic to these. The idea here is that each budget
set E possesses a status quo b. We observe the choices made (and not made)
from budget sets.
The theory of status quo rationalizable choice is the subtheory Tsq ⊆Tcsq
whereby there exists a function Q : X →2X\{∅} such that for all x ∈X,
x ∈Q(x), and a complete and transitive binary relation ⪰for which the
corresponding function c∗can be expressed as c∗(E,b) = argmax⪰Q(b) ∩E.
The idea is that the status quo alternative determines a “reference set” from
which the agent will choose.3
Theorem 13.3
ecTcsq(Tsq) = Tcsq.
Proof. We provide an explicit syntactic characterization. The formula
(x ∈E) ∧(y ∈E) ∧c(x,d,E)
is abbreviated x ⪰(E)d y, and the formula
(x ∈E) ∧(y ∈E) ∧c(x,d,E) ∧˜c(y,d,E)
is abbreviated x ≻(E)d y. Note that each of these formulas is a conjunction of
atomic formulas.
Similarly, we deﬁne, for a structure (M,∈M,cM, ˜cM) ∈Tcsq, the ternary
relations x ⪰M
d
y to mean that there is E for which x,y ∈E and x ∈c∗(d,E),
and x ≻M
d
y means there is E for which x,y ∈E and x ∈c∗(d,E) and y ̸∈c∗(d,E)
(here, we suppress the dependence of the relation on E as it is not needed).
Now, a cycle for a structure M ∈Tcsq is a collection x1,...,xn, y1,...,yn,
and z1,...,zn all in M such that xi+1 ̸= yi and xi ⪰M
zi xi+1 with at least one strict
part, and for each zi, xi+1 ⪰M
zi yi or xi+1 = zi.
Note that the collection of axioms which rule out all cycles is UNCAF, as
each formula x ⪰(E)d y is a conjunction of atomic formulas, as is the formula
x ≻(E)d y. In other words, the UNCAF list of axioms given by
∀x1 ...∀zn∀E1 ...∀En∀F1 ...∀Fn¬
1 n6
i=1
(xiRi(Ei)zixi+1) ∧(Qi(xi+1,yi,zi,Fi))
2
,
3 Masatlioglu and Ok (2014) also allow there to be no status quo, which they represent with
a status quo “alternative” of ♦. We could accommodate this easily by introducing a constant
symbol for ♦, and none of the results would change. However, the analysis would become
notationally much more burdensome.


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Revealed Preference and Model Theory
where each Ri is either ≻or ⪰and at least one is ≻, and each Qi(xi+1,yi,zi,Fi)
is either the expression xi+1 ⪰(Fi)zi yi or the expression xi+1 = zi, is the
appropriate collection of sentences.
Finally, given M ∈Tcsq, we have its associated ternary relations ⪰M and
≻M. We claim that M ∈Tsq if and only if it has no cycles.
To this end, suppose that M ∈Tsq. Then M is isomorphic to a triple
(X,,c∗). There is a binary relation R and a function Q as stated in the
deﬁnition. Suppose for a contradiction that there is a cycle. Each xi+1 ∈Q(zi),
since xi+1 ⪰M
zi
yi (implying it is chosen at some point when zi is the status
quo) or xi+1 = zi, which again implies that xi+1 ∈Q(zi). Therefore, xi ⪰M
zi xi+1
implies xi R xi+1 and xi ≻M
zi xi+1 implies xi R xi+1 but not xi+1 R xi, contradicting
transitivity.
Conversely, suppose M ∈Tcsq and that there are no cycles. Without loss
of generality, we may assume that M is speciﬁed by a triple (X,,c∗). We
deﬁne Q(d) = {y : ∃E ∈ such that y ∈c∗(d,E)}. Deﬁne ⪰d = {(x,y) ∈⪰d:
y ∈Q(d)} and ≻d = {(x,y) ∈≻d: y ∈Q(d)}. Finally, deﬁne ⪰= 
d∈X ⪰d and
≻= 
d∈X ⪰d. Then because there are no cycles, there exist no x1,...,xn for
which x1 ⪰...⪰xn, where at least one ⪰is ≻. By Theorem 1.5, there is a
complete and transitive R for which x⪰y implies x R y and x≻y implies x R y.
The pair Q,R then status quo rationalizes c∗.
13.3
CHOICE THEORY AND EMPIRICAL CONTENT
As mentioned earlier in the chapter, theories that coincide with their empirical
content are in some sense special. We shall here consider the structure of such
theories relative to a theory of choice.
The theory of rationalizable choice functions enjoys a very interesting
property. Recall Chapter 2 and, in particular, Theorem 2.6. The theorem
characterizes the empirical content of a preference relation by the congruence
axiom. Congruence is a collection of ﬁrst-order axioms precluding the exis-
tence of certain types of revealed preference cycles. Importantly, congruence
can be described in a very parsimonious language: the language is able to
express the relations ∈, ̸∈, as well as the properties of being chosen or not.
This means that the data the economist must possess in order to falsify the
model are quite simple.
In this section, we demonstrate a very simple result claiming that most
choice theories do not share this property. As motivation, consider the
following simple example.
Example 13.4
An economist asks whether a given individual maximizes a
preference relation (a weak order). She observes three budget sets, A, B, and
D, and three potential choices, x, y, and z. She sees that each of x and y are
feasible in A, y and z are feasible in B, and x and z are feasible in D. She also
sees that x is chosen from A, y is chosen from B, and z is chosen from D, while x


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13.3 Choice theory and empirical content
195
is never chosen from D. These data clearly refute the hypothesis of preference
maximization.
To establish this refutation, the economist did not see every feasible option
from each of the three budget sets or any “global set” on which the choice
function might potentially be deﬁned, nor did she need to see what the
individual would choose from an unrelated budget E.
Consider two languages. The ﬁrst, L = {∈,̸∈,c, ˜c}, includes four binary
predicates. The predicates ∈and ̸∈are to be understood in their usual way,
while c(a,B) means a is chosen from B, and ˜c(a,B) means a is not chosen
from B. The second language is more expressive: L′ = {∈,̸∈,c, ˜c,⊆,̸⊆}. The
binary relations ⊆and ̸⊆are given their usual interpretation.
A choice function consists of a triple (X,,c), where X is a set,  ⊆2X\{∅},
and c :  →2X\{∅}, so that for all E ∈, c(E) ⊆E. Each choice function is
naturally identiﬁed with a structure in either language.
In either language, L or L′, a choice theory T is a class of choice functions
and all structures isomorphic to an element of this class. The choice theory
which consists of all choice functions is written Tchoice in language L and T′
choice
in language L′, respectively.
Say that the L-theory T is rich if for all (X,,c) in T, there is z ̸∈X and
a choice function (X ∪{z},′,c′) ∈T, where  = {E ∩X : E ∈′} and for
all E ∈′, c′(E) = c(E ∩X). Intuitively, a rich choice theory is one with the
property that, for any of its structures, we can add an alternative that is never
chosen to the domain. Say that a choice theory satisﬁes condition α if all choice
functions in the theory satisfy condition α.
When a theory satisﬁes ec(T) ∩Tchoice = T, then we say that it makes no
non-falsiﬁable claims relative to the theory of choice. We are now in a position
to show that, under some conditions, the property of not making non-falsiﬁable
claims can imply WARP.
Theorem 13.5
Suppose that T is rich, satisﬁes condition α, and that ec(T)∩
Tchoice = T. Then every (X,,c) ∈T satisﬁes WARP. Further, the class of choice
functions satisfying WARP is the maximal T which is rich, satisﬁes condition
α, and ec(T) ∩Tchoice = T.
Proof. Suppose by way of contradiction that (X,,c) ∈T violates WARP.
Thus, there are x,y ∈X and E,F ∈ for which x,y ∈E ∩F, x ∈c(E), y ∈c(F),
and x ̸∈c(F). Appeal to richness and consider z ̸∈X. Let ∗consist of all
budgets of , with the exception that z has been added to E. The choice
function c∗then coincides with c, and c∗(E ∪{z}) = c(E).
Now consider the following choice function: X′ = {x,y,z}, ′ = {E′ =
{x,y,z},F′ = {x,y}}, and c′({x,y,z}) = c(E) ∩{x,y,z}, c′({x,y}) = {y}. This
choice function is clearly not a member of T as it violates condition α: x ∈
c′({x,y,z}), but x ̸∈c′({x,y}). However, every dataset consistent with (X′,′,c′)
can be rationalized by (X ∪{z},∗,c∗). This is a contradiction.


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196
Revealed Preference and Model Theory
Clearly, the class of choice functions satisfying WARP has the desired prop-
erty (WARP is an UNCAF axiom, so the result follows from Corollary 13.2).
A special example of choice theories are choice theories rationalizable by
some binary relation. Let R be a class of complete binary relations. We
deﬁne the theory of R-rationalizable choice, TR, to be the class of choice
functions (X,,c) for which there exists R∈R such that for all E ∈,
c(E) = {x ∈E : x R y for all y ∈E}.
Let the relation ⪰∗on Z+ ∪{ω} be deﬁned so that for all n,m ∈Z+, we have
n ⪰∗m if and only if n ≥m, and otherwise, ω ∼∗n for all n ∈Z+.
A simple corollary follows.
Corollary 13.6
For the following R, ec(TR) ∩Tchoice ̸= TR:
• R is the set of all complete binary relations.
• R is the set of all quasitransitive and complete binary relations.
• R is the set of all complete binary relations for which there exists a
pair of linear orders ⪰1 and ⪰2 for which ≻=≻1 ∩≻2. These are the
set of 2-Pareto rationalizable choice functions.
Proposition 13.7
If ⪰∗∈R, then ec(TR) ∩T′
choice ̸= ec(TR).
Proof. Let X = Z+ ∪{ω} and let us consider the collection  which includes
all binary subsets, as well as the set X itself. The choice function is speciﬁed
by c(X) = {ω}, c({i,ω}) = {i,ω} for all i ∈Z+, and ﬁnally, c({i,j}) = {j} when
j > i. Then (X,,c) ∈TR, and has the feature that ω is uniquely chosen from
X, but it beats no i ∈Z+.
Consider the following choice function: (X′,′,c′), where X′ = {1,2,ω},
′ = {{1,2},{2,ω},{1,2,ω}}, and c′({1,2}) = {2}, c′({2,ω}) = {2,ω}, and
c′({1,2,ω}) = {ω}. Clearly (X′,′,c′) ̸∈TR as if it were, any R which
rationalizes c′ must have that 2 R 1 and 2 R ω, which would imply that
2 ∈c′({1,2,ω}), a contradiction.
However any dataset contained in (X′,′,c′) can be rationalized by (X,,c),
a contradiction.
Proposition 13.7 demonstrates that none of the choice theories mentioned in
Corollary 13.6 are equivalent to their empirical content, even when we can
express set containment.
13.4
CHAPTER REFERENCES
Theorem 13.1 and Corollary 13.2 are due to Chambers, Echenique, and
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relies on the axiom of choice, while Theorem 13.1 does not. The working paper
version of Chambers, Echenique, and Shmaya (2014) has results for languages


--- Page 12 ---
13.4 Chapter references
197
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Echenique, and Shmaya (2014).


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--- Page 30 ---
Index
act, 123
act-dependent probability representation, 126
Afriat inequalities, 40, 64, 84, 118, 123, 138,
139, 177
Afriat’s efﬁciency index, 72, 78
Afriat’s Theorem, 40, 139, 177, 178
aggregate excess demand function, 130
aggregation rule, 159
Allais paradox, 115, 127
allocation, 136
atomic formula, 191
axiom, 191
axiom of revealed stochastic preference, 96
axiomatization, 191
UNCAF, 191
balanced, 118
doubly, 122
bargaining
egalitarian, 150
Nash, 150
utilitarian, 150
Bayesian environment, 148
beliefs, 116
binary relation, 1
acyclic, 5
antisymmetric, 1
asymmetric part, 1
completeness, 1
convex, 4
extension of, 1
irreﬂexive, 1
linear order, 2, 159
partial order, 2
quasitransitive, 1, 17, 18, 22, 27
reﬂexive, 1
strict extension of, 1
strict part, 1
strictly convex, 4
symmetric, 1
symmetric part, 1
transitive, 1
weak order, 2
Birkhoff–von Neumann Theorem, 92
bistochastic matrix, 92
Block–Marschak polynomials, 97
blocking pair, 154
Bronars’ index, 75
canonical conjugate, 22
certainty inclusive, 126
chain, 2
choice function, 143
complementarity, 57, 58, 66
complete markets, 117
component, 2
comprehensive set, 25, 87
condition α, 20, 96, 144, 195
condition β, 20, 144
cone, 60, 110, 180
congruence, 18, 179, 191, 194
partial, 28
conic independence, 92
constant act, 124
constant returns to scale, 90
consumption dataset, 35, 60
convex cone, 110
convex hull, 109
core, 140
correspondence, 3
cost rationalization, 83
Cournot oligopoly, 177
critical cost efﬁciency index, 73, 81
cycle, 154


--- Page 31 ---
216
Index
cyclic monotonicity, 9, 41, 65
cylinder, 99
data envelopment analysis, 94
dataset
bargaining, 150
consumption, 35, 60
cross-country, 78
cross-section, 78
discrete, 58
economy-wide, 136
experimental, 79
input–output, 83
panel, 77
partial production, 91
partially observed, 51, 52
production, 87
time series, 79
voting record, 168
demand correspondence, 34
demand function, 35, 130
dictator game, 80
direct revealed preference, 37
strict, 37
direct revelation mechanism, 147
disagreement point, 149
discipline, 16
downward-sloping demand, 117
edge, 154
election, 172
empirical content, 190
endowment vector, 129
Engel curve, 75
envy-free, 91
Epstein test, 121
equivalence relation, 2
Euclidean norm, 3
Euclidean space, 2
excess demand function, 130
exchange economy, 129
existential, see existential axiom
existential axiom, 139, 175, 180, 184, 186
expected utility, 115, 116
extension lemma, 6
extreme point, 109, 169
falsiﬁcation, 176, 189
Farkas’ Lemma, see Theorem of the
Alternative
ﬁrst-order stochastic dominance, 121
game, 143
game form, 143
GARP, see generalized axiom of revealed
preference
generalized axiom of revealed preference, 26,
37, 46, 139, 176, 177, 187, 191
gradient, 3
graph, 154
gross complements, 66
Hal Varian, 55, 61, 73, 78
HARP, see homothetic axiom of revealed
preference
Herbert Simon, 23
homothetic, 60
homothetic axiom of revealed preference, 61
homothetic revealed preference pair, 61
ideal, 180
ideal point, 164
idempotence, 133
IIA, 103
Inada conditions, 150
inclusion–exclusion principle, 98
independence axiom, 109, 115
indicator function, 3
indicator vector, 3
indifference relation, 1
indirect revealed preference, 37
inner product, 2
interchangeable, 147
interior, 3
isomorphism, 189
join, 2, 144
joint choice function, 143
Karl Popper, 184, 186
L-dataset, 188
L-structure, 189
language, 188
lattice, 2, 144
linear order, 2, 159
logit model, 106
lottery, 114
lower bound, 2
lower contour set, 4
lower production set, 88
Luce independence of irrelevant alternatives,
103
Luce model, 104


--- Page 32 ---
Index
217
majority rule, 159
matching, 154
stable, 154
maximal element, 2
maxmin, 150
mechanism design, 147
meet, 2, 144
M¨obius inversion, 98
monetary act, 116
money pump index, 74
monotonic, 109
monotonicity, 9
multiplicative monoid, 181
n-ary relation, 2
N-congruence, 146
Nash bargaining, 150, 182
Nash equilibrium, 143
no arbitrage, 117
normal-form game, 143
null state, 124
numeraire, 65
objective probability, 114
observation, 188
order pair, 5
acyclic, 5, 179
asymmetric, 9
extension, 5, 17
quasi-acyclic, 8, 22
Pareto rationalizable, 145
partial observability, 187
partial order, 2
path, 10, 154
permutation matrix, 92
persistence, 140
persistence under contraction, 144
persistence under expansion, 144
polyhedral cone, 110
polynomial, 180
polynomial inequality, 138
Positivstellensatz, 181
preference
additive separability, 64
additively separable, 64
homothetic, 60
separable, 64
preference relation
objective expected utility, 115, 117
probabilistically sophisticated, 121
strict, 3, 96
subjective expected utility, 122, 124
preference proﬁle, 154
preference relation
continuous, 3
Euclidean, 164
locally nonsatiated, 3, 37, 51
monotonic, 3, 25, 124
monotonic with respect to order pair, 25
rational, 3
representation, 4
smooth utility representation, 4
strict, 144
strictly monotonic, 3
uniformly monotonic, 126
prior, 116
probabilistically sophisticated, 121
production function, 83
production set, 87
proﬁt function, 90
psychiatric patients, 79
quantiﬁer elimination, 138, 175
quasiconcave, 4
quasilinear utility, 65
random decision selection, 32, 79
random utility, 96
rate of violation of GARP, 78
rationalizable, 91, 189
additively separable, 64
convex, 168
egalitarian (maxmin), 150
Euclidean, 164
expected utility, 109, 115, 117
g, 150
majority rule, 159
n-unanimity, 160
Nash bargaining, 150
Pareto, 145, 159
probabilistic sophistication weakly, 121
production dataset, 87
quasilinear, 65
random utility, 96
satisﬁcing, 23
stable matching, 154
strongly, 15, 47
strongly Nash, 143
strongly pair, 168
subjective expected utility, 122
team, 145
unanimity rule, 159
utilitarian, 150, 162
Walras, 136


--- Page 33 ---
218
Index
rationalizable (cont.)
weakly, 16, 35
weakly Nash, 144
zero sum, 147
real closed ﬁeld, 138
regular, 109
regularity, 96, 109
relation symbols, 188
relative theory, 192
revealed preference
general budget sets, 46
order pair, 16, 36, 54, 115
order pair (strong), 48, 58
revealed preference graph, 53
rich, 195
risk aversion, 117
risk-neutral prices, 117
SARP, see strong axiom of revealed preference
semideﬁnite programming, 181
separating hyperplane theorem, 11
states of the world, 116
status quo, 192
stochastic frontier analysis, 94
strategy proﬁle, 143
strict preference, see preference relation
strictly quasiconcave, 4
strong rationalization, 15
strong axiom of revealed objective expected
utility, 118
strong axiom of revealed subjective expected
utility, 122
strong axiom of revealed preference, 48, 51,
58, 130, 176
strong rationalization, 47
structuralism, 197
subjective expected utility, 122
subjective probability, 121
submodular, 2, 57
subrationalizable, 28
substitutability, 58, 68
superdifferential, 12
supergradient, 11, 182
supermodular, 2, 57
Szpilrajn’s Theorem, 7
Tarski–Seidenberg Theorem, 138, 175
team rationalizable, 145
Theorem of the Alternative, 12, 42, 43, 47, 86,
87, 91, 102, 116, 148, 156, 167, 170, 171,
175
theory, 189
transitive closure, 5, 24, 37, 78, 146, 160
type space, 147
unanimity rule, 159
UNCAF, see universal negation of conjunction
of atomic formulas
uniform monotonicity, 126
unit vector, 3
universal, see universal axiom
universal axiom, 180, 184, 186
universal negation of conjunction of atomic
formulas, 187, 191
unordered, 1
upper bound, 2
greatest, 2
least, 2
upper contour set, 4
strict, 4
upper production set, 89
utilitarianism, 150
utility function, 4
separable, 64
smooth, 50
V-axiom, 17
Varian’s efﬁciency index, 73
vector, 2
veriﬁcation, 176
vertex, 154
Walras’ Law, 130
Walrasian equilibrium, 136
Walrasian equilibrium price, 130
WARP, see weak axiom of revealed preference
weak axiom of production, 88
weak axiom of proﬁt maximization, 88
weak axiom of revealed preference, 19, 35, 36,
38, 46, 60, 61, 67, 73, 74, 76, 137, 179,
191, 195
weak order, 2
weak rationalization, 16
demand, 35
in choice, 16
weakened weak axiom of revealed preference,
24
weakly rationalizable, 35
well-ordering, 29
Zorn’s Lemma, 2, 8, 30


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