WARP and Combinatorial Choice*
Samson Alva
University of Texas at San Antonio†
May 29, 2016
Abstract
I show that in combinatorial choice environments the Weak Axiom of Revealed Preference (WARP)
is equivalent to the Irrelevance of Rejected Items (IRI), a consistency condition whose importance for
familiar results in matching theory is demonstrated by Aygün and Sönmez (2013). Moreover,
WARP is necessary and sufficient for there to exist a complete and reflexive rationalization of
a combinatorial choice function. I also study the connection between IRI and other important
choice conditions in the classical choice framework, including independence of irrelevant alternatives,
the Chernoff property, and path independence. Each of these conditions is equivalent to IRI and to
WARP in single-valued combinatorial choice settings. I extend the definition of IRI for multivalued combinatorial choice and show that the equivalence with WARP continues to hold.
These results rely upon a faithful representation of a combinatorial model as a classical choice
model.
JEL Classification: C78, D01
Keywords: weak axiom of revealed preference; irrelevance of rejected contracts; matching; combinatorial choice; rationalizability; path independence
1
Introduction
In the classical abstract choice model,1 there is a set of mutually exclusive alternatives from which
a subset is chosen. When more than one alternative is chosen, the interpretation is that different
chosen alternatives are equally good options that are not simultaneously feasible. This model is an
abstraction of the consumer choice model, where alternatives are vectors of commodities, budgets
are generated by linear prices and a money-income constraint, and choice is modeled through
demand correspondences.
*
For helpful comments and suggestions, I thank Sean Horan and Bertan Turhan, and especially Vikram Manjunath
and Alex Teytelboym
†
e-mail: [email protected]; address: Department of Economics, UTSA, One UTSA Circle, San Antonio TX
78249; website: http://faculty.business.utsa.edu/salva/.
1
The abstract individual choice model is due to Uzawa (1956) and Arrow (1959).
1
On the other hand, in many markets with indivisible goods, an economic actor must choose
from an available set of items Y some subset C(Y ) to constitute a consumption bundle. In two-sided
matching, a firm has to assemble, from a given set of workers, a team of workers to achieve a particular task. In a spectrum auction, a telecommunications company seeks to assemble a profitable
package of bands of electromagnetic spectrum from an available set of band-price pairs. These
choice problems are combinatorial (Echenique, 2007) because the items are combinable into consumption alternatives rather than being mutually exclusive alternatives themselves.2
The matching with contracts framework of Hatfield and Milgrom (2005) provides a general setting for the study of many combinatorial economic problems. Contracts are the items of choice, and
can contain terms such as price, delivery date, etc. They follow the early matching literature in assuming each agent’s choice is determined by the maximization of a complete, transitive preference
relation. However, in this case, choice satisfies the Strong Axiom of Revealed Preference (SARP).
Aygün and Sönmez (2013) show that the results of Hatfield and Milgrom (2005) continue to hold
when choice is primitive and SARP is replaced with a weaker choice condition, the Irrelevance of
Rejected Items (IRI).3 In particular, when choice is determined by means other than maximization
of a transitive preference relation, as in the recent applications of the matching with contracts framework to market design, only IRI, rather than SARP, need be assumed to ensure that the cumulative
offer mechanism satisfies the desirable properties of strategy-proofness and stability.4
Nevertheless, like SARP, the IRI condition is a form of choice consistency.5 In this article, I
demonstrate that imposing IRI on a combinatorial choice model is equivalent to requiring that the
Weak Axiom of Revealed Preference (WARP) holds. Moreover, I show that a combinatorial choice
function can be rationalized as the maximization of a binary relation if IRI is satisfied.6 Given the
importance of IRI for the good properties of the cumulative offer mechanism, Theorem 1 shows
2
To my knowledge, Alkan (2002) is the first paper in matching to assume combinatorial choice functions as primitive
rather preferences. However, Blair (1988) also works directly with choice functions despite the superfluous assumption of
preferences, and in a many-to-many matching model with variable terms for a match. Roth (1984) assumes a preference
relation as a primitive, and works with the choice generated by maximization of this preference relation, while Kelso
and Crawford (1982) work with demand correspondences generated by utility maximization. Fleiner (2003) is also
notable for its choice-based approach.
3
An extensive study of the role of this condition in the matching with contracts models of Hatfield and Milgrom
(2005) and Hatfield and Kojima (2010) can be found in Aygün and Sönmez (2013) and Aygün and Sönmez (2012), who
call it the Irrelevance of Rejected Contracts. Since combinatorial choice environments may not be in the matching
with contracts framework, I avoid referring to contracts in the definition of this, otherwise equivalent, condition, and
use instead the generic term “items”.
4
Examples of the market design literature that design object choice functions in a matching with contracts framework to solve real-world allocation problems include the papers on US Army cadet-branch matching (Sönmez and
Switzer (2013), Sönmez (2013)).
5
Versions of this condition have appeared previously in the matching literature (Blair (1988), Alkan (2002), Alkan
and Gale (2003), Fleiner (2003)). See Section 3.6 for more details.
6
One such rationalizing relation is the Blair relation (Blair, 1988) on the set of possible bundles, which is equivalent
to the revealed preference relation under IRI.
2
that a designed choice function will satisfy IRI if and only if it can be rationalized as maximizing
a binary relation over bundles of items. The rationalizing relation need not be transitive, however,
and so SARP may not be satisfied.7 Thus, the equivalence of WARP with IRI demonstrates the
importance of this classical rationality axiom for the familiar theory of matching with contracts.8
Indifferences often arise in market design settings. In school choice (Abdulkadiroğlu and Sönmez, 2003), for instance, school districts frequently have coarse priorities that are resolved using
tie-breakers based on random numbers.9 I define a natural extension of IRI for multi-valued combinatorial choice rules on the basis of exogenous tie-breakers. I show that the equivalence result
between IRI and WARP holds for this extension. The key implication for a matching model with
combinatorial choice rules is that WARP must be satisfied for IRI to be guaranteed for every exogenous tie-breaker over bundles. Otherwise, there exists a tie-breaker that generates a tie-broken
combinatorial choice function failing IRI, and possibly leading to failure of the received theory of
stable matchings (Aygün and Sönmez, 2013). I also study two other possible extensions of IRI, and
demonstrate how they are unsatisfactory candidates; one is too strong and the other too weak.
To obtain the results of this paper, I define a representation of a combinatorial choice model as
a classical choice model, where the interpretation that items are combinable but bundles of items
are mutually exclusive alternatives is faithfully represented. In the representation, choice alternatives
are subsets of the items in the combinatorial model, and every budget set comprises exactly those
bundles that can be composed using some items from a given set of items. The representation serves
as a bridge to transport classical choice conditions into the combinatorial model.
I explore the implications for the combinatorial choice model of the following classical choice
conditions: independence of irrelevant alternatives; the Chernoff property (also known as Sen’s α
condition); Arrow’s axiom; and path independence. Each of these conditions has been important
in the study of rationalizability of classical choice rules. I show that for single-valued combinatorial
choice, all these conditions are equivalent to WARP, and hence to IRI.
The equivalence of path independence with IRI may seem surprising. In the matching literature, beginning with Alkan (2002), the combination of IRI with the property of substitutability of
combinatorial choice has also been called path independence.10 My results on path independence
do not contradict these authors. The differences arise from the way in which the path independence
condition is mapped into a combinatorial model. These studies use the forgetful representation of
a combinatorial choice model as a classical choice model, so called because the mutually exclusive
7
Arrow (1959) and Sen (1971) show that WARP and SARP are equivalent if the choice domain is rich enough to
include as budget sets every pair and every triple of bundles, a condition not satisfied by a combinatorial model.
8
In the tradition of revealed preference analysis, Echenique et al. (2013) study the theory of stable matchings for its
empirical content.
9
See Erdil and Ergin (2008) and Abdulkadiroğlu et al. (2009) for welfare issues created by using tie-breakers in
school choice.
10
See for instance Alkan and Gale (2003), Fleiner (2003), and Chambers and Yenmez (2014).
3
alternatives requirement of classical choice is “forgotten”. The forgetful representation generally
leads to different results from the faithful representation.
Related Literature
Chambers and Yenmez (2014) study combinatorial choice functions that
satisfy the combination of IRI and substitutability, and provide a new technique to analyze matching models. Under the forgetful representation, which they implicitly use, IRI and substitutability
is equivalent to path independence. They show that imposing a weakened version of WARP on this
representation helps characterize some path independent choice functions.11 My results in Section
4.4 examine the implications of WARP and of path independence working with the faithful representation of a combinatorial choice function, and so our results differ. Their paper shows how our
understanding of combinatorial choice functions and of matching can be advanced through the use
of the forgetful representation, and my analysis is complementary to theirs.
In the classical choice literature, a rationalization of choice is usually taken to be a binary relation
on the set of alternatives. Brandt and Harrenstein (2011) study conditions for rationalizability by a
binary relation on the power set of alternatives, an approach they call set-rationalizability. If one works
with the forgetful representation in reverse, starting with a classical choice model and defining from
it a single-valued combinatorial choice model, their α̂ condition and γ̂ condition (which are adaptations of Sen’s α and γ for sets of alternatives) correspond to IRI and substitutability, respectively.
Also, their set-rationalizability corresponds to rationalizability of the single-valued combinatorial
choice model. One of their main results is that the α̂ condition characterizes classical choice rules
that are set-rationalizable (their Theorem 2). By using the forgetful representation, it is clear that
this result is the same as the equivalence of IRI and rationalizability of single-valued combinatorial
choice, one of the results in Theorem 1. However, the interpretations of these results are different,
and there is no clear analog when combinatorial choice is multi-valued.
The rest of the paper is organized as follow. In Section 2, I illustrate the faithful representation
technique and the main result. In Section 3, I describe the classical choice model and the combinatorial choice model, as well as choice conditions and representations. In Section 4, I describe and
prove the results in four subsections, one each for single-valued combinatorial choice (Section 4.1),
multi-valued combinatorial choice (Section 4.2), general classical choice (Section 4.3), and choice
satisfying path independence (Section 4.4). I conclude in Section 5.
11
This weakened WARP is studied by Ehlers and Sprumont (2008).
4
2
From combinatorial to classical choice: an example
To illustrate how to represent faithfully a combinatorial choice problem as a classical choice
problem, and to show how IRI and WARP are related in the single-valued combinatorial choice
setting, consider the following example set in the matching with contracts framework (Hatfield and
Milgrom, 2005).
There is a firm that has three possible bilateral contracts X = {x, x′ , y} that it could sign, where
for each of these contracts the other party involved is a worker. Suppose contracts x and x′ involve
the same worker, wx , and contract y involves another worker wy .
Contracts are not mutually exclusive alternatives for the firm, because it might be feasible for
the firm to sign more than one contract, and so two contracts are not alternatives in the sense
of being mutually exclusive. Instead, a choice alternative for the firm is a subset Y of contracts
from X . Such a set of contracts is a bundle. The firm in this example has eight possible bundles it
could choose. It could pick: no contract (Z∅ = ∅); only one contract (Zx = {x} or Zx′ = {x′ }
or Zy = {y}); only two contracts (Zxx′ = {x, x′ } or Zxy = {x, y} or Zx′ y = {x′ , y}); all three
contracts (Zxx′ y = {x, x′ , y}).
Sometimes, certain bundles are not feasible, even though the items constituting the bundle can
be combined with other items to form feasible bundles. For example, if bilateral contracts are
comprehensive, then x and x′ are mutually exclusive bilateral contracts for the firm and worker
wx , so it is not feasible for the firm to pick bundles Zxx′ or Zxx′ y . In this paper, as is typical in the
matching literature, I assume that the empty bundle ∅ is always feasible.
In an instance of a choice problem for the firm, a subset of contracts Y ⊆ X , an opportunity set, is
available from which contracts can be used to construct bundles. Therefore, every bundle consisting
exclusively of contracts in Y is a choice alternative available given opportunity set Y . This set of
possible bundles is the budget set generated by Y . For example, the opportunity set {x} generates the
budget set with two possible bundles, Zx and Z∅ , the opportunity set {y} generates the budget set
with Zy and Z∅ as possible bundles, and the opportunity set ∅ generates the budget set with only Z∅
as a possible bundle. Notice that Z∅ is in every budget set generated by an opportunity set, and so
there is always a feasible alternative available. However, there are sets of bundles that cannot be a
budget set generated by an opportunity set. In particular, there is no opportunity set that generates
only bundles Zx and Zy as a budget set. If an opportunity set allows for bundles Zx and Zy , it will
also allow for Z∅ and, if feasible, Zxy .
If the firm’s choice is {x, y} when faced with opportunity set X , it has revealed a “preference”
for the bundle Zxy over bundles Zx , Zy , and Z∅ . Here the firm’s combinatorial choice is singlevalued, which simply means that exactly one bundle is chosen, and not that a bundle necessarily
consists of exactly one item. If the firm chooses both the bundle Zxy and the bundle Zx , then
5
its combinatorial choice from the opportunity set X is multi-valued. In this case, its choice is not
decisive, which might be interpreted as indifference between the two bundles, rather than the two
bundles being a jointly feasible consumption possibility.
Interestingly, when budget sets are generated combinatorially, the firm could never reveal a
preference for Zx over Zy , given that Zxy is revealed preferred to Zx and Zy , even if this firm chooses
bundles by maximizing the preference relation Zxy ≻ Zx ≻ Zy ≻ Z∅ . In fact, from combinatorial
choice data alone, one could not separately identify this preference from the preference Zxy ≻′
Zy ≻′ Zx ≻′ Z∅ .12
Now suppose instead the firm, when faced with the opportunity set {x, y}, chooses bundle Zx ,
but when faced with the opportunity set {x} chooses Z∅ . Such choice behavior seems inconsistent
with rational choice. Of what relevance is contract y in the decision of whether to choose contract x,
if the firm rejects y when x is available? The IRI condition requires that the choice from {x} be the
same as that from {x, y} since the only difference between the two opportunity sets {x, y} and {x}
is the removal of the contract y that is “rejected”, i.e. not chosen, from {x, y}. So, this firm’s choice
fails the IRI condition. By choosing bundle Zx from opportunity set {x, y}, the firm has revealed
a preference for Zx over Z∅ . But then by choosing Z∅ from {x}, it has revealed a preference for
Z∅ over Zx . This violates WARP, which forbids such 2-cycles in the revealed preference relation.
One of the main results of this paper is that simultaneous failure of IRI and WARP is general: a
combinatorial choice function C satisfies IRI if and only if it satisfies WARP.
3
Two choice models
I begin with some definitions concerning binary relations. Then, I describe the classical choice
model, followed by the combinatorial choice model. Next, I define the faithful and forgetful representations of a combinatorial choice model as a classical choice model, after which I introduce
various choice conditions for both classes of choice models.
3.1
Binary relations
Given a set A, let 2A denote the power set of A, i.e. 2A = {A′ : A′ ⊆ A}.
A binary relation R on a set A is a subset of the product space A × A. For any a, a′ ∈ A, let
a R a′ denote (a, a′ ) ∈ R. A binary relation R on a set A is reflexive if a R a for every a ∈ A. It is
complete if for every a, a′ ∈ A with a ̸= a′ , a R a′ or a′ R a. It is antisymmetric if for every a, a′ ∈ A,
if a R a′ and a′ R a then a = a′ . It is asymmetric if for any a, a′ ∈ A, a R a′ implies ¬(a′ R a). It
12
This is the basis for the results of invariance of the core across particular classes of preference profiles (Martínez et
al., 2008, 2012).
6
is transitive if for every a, a′ , a′′ ∈ A, a R a′ and a′ R a′′ implies a R a′′ . It is a partial order if it is
reflexive, transitive, and antisymmetric. It is a linear order if it is a complete partial order.
A binary relation R on A is an extension of a binary relation R′ on A if for every a, a′ ∈ A,
i) a R′ a′ implies a R a′ ii) a R′ a′ and ¬(a R′ a′ ) implies a R a′ and ¬(a R a′ ) The transitive
closure R̄ of a binary relation R on set A is the smallest transitive relation that contains R. Formally,
∩
R̄ = R′ ∈Rt R′ , where Rt is the set of all transitive relations on A.13
3.2
Classical choice model
Let X be a nonempty set. Elements of X are the mutually exclusive alternatives that are to be
considered for choice. Alternatives in X could be elements of a real vector space, as in consumer
choice theory, or subsets of contracts from a universal set of potential contracts, as in matching with
contracts. This additional structure is not relevant for the classical abstract choice model.
A budget set is a nonempty subset B ⊆ X . Let B ⊆ 2X be a choice domain, which is a
collection of budget sets.14
A choice rule is a correspondence c : B ⇒ X such that, for every B ∈ B , c(B) is nonempty
whenever B ̸= ∅ and c(B) ⊆ B , i.e. for every nonempty budget set B , at least one alternative
must be chosen and a chosen alternative must be available. A choice rule c is single-valued if,
for all nonempty B ∈ B , exactly one element of B is chosen: |c(B)| = 1. It is multi-valued
otherwise. When single-valued, the choice rule c can be naturally identified as a choice function
c : B → X . A choice function c̃ : B → X is a selection from a choice rule c if, for all nonempty
B ∈ B , c̃(B) ∈ c(B).
A classical choice model is a 3-tuple (X , B, c) consisting of a set of alternatives, a choice
domain, and a choice rule defined on this domain. For given X , a choice structure (B, c) is a
2-tuple consisting of a choice domain and a choice rule defined on this domain.
3.3
Combinatorial choice model
Let X be the universal set of items. A bundle is a subset of items Z ⊆ X . Let FX be the
collection of feasible subsets of X . Restrictions on how items can be combined to form bundles
can be modeled through the specification of the set of feasible bundles. The empty set as a bundle
is assumed to be feasible.
An opportunity set is a subset Y ⊆ X of items available for the construction of bundles. A
combinatorial choice rule is a correspondence C : 2X ⇒ 2X such that, for every opportunity
13
To see this is well-defined, note that the relation A × A is transitive, and that transitivity is preserved by the
intersection of relations.
14
Allowing for an empty budget set is useful to show how a classical choice model is a particular kind of representation
of a combinatorial choice model, and is harmless. See Section 3.4 for more on this.
7
set Y ⊆ X , C(Y ) ⊆ 2Y and C(Y ) ⊆ FX , where the latter condition ensures feasibility constraints
are respected. A combinatorial choice rule is single-valued if |C(Y )| = 1, and multi-valued otherwise.
Note that C being single-valued implies that for every Y ⊆ X , C(Y ) = {Z} for some Z ⊆ Y ∩FX .
So, it is possible that |C(Y )| = 1 and |Z| > 1, because the unique bundle chosen is composed of
multiple items. If single-valued, I refer to C as a combinatorial choice function and identify
C(Y ) with the unique bundle it contains.
A combinatorial choice model is a 3-tuple (X, FX , C) consisting of a set of items, a set of
feasible bundles, and a combinatorial choice rule.
3.4
Representations of a combinatorial choice model
Suppose (X, FX , C) is a combinatorial choice model.
Let (X , B, c) be the classical choice model defined by
1. X = FX ,
2. B = {2Y ∩ FX : Y ⊆ X},
3. c(2Y ∩ FX ) = C(Y ) for every Y ⊆ X .
The classical choice model (X , B, c) so defined is the faithful representation of (X, FX , C).
The faithful representation of C respects the interpretation that different elements of X , which are
now bundles from X , are mutually exclusive alternatives.15 The choice domain B generated in this
way is a combinatorial choice domain.
Instead, one could identify (X, FX , C) with the classical choice model (X , B, c) defined by
1. X = X ,
2. B = 2X ,
3. c(Y ) = C(Y ) for every Y ⊆ X .
In this case, (X , B, c) is the forgetful representation of (X, FX , C), since distinct alternatives
in X are not necessarily mutually exclusive. The forgetful representation of a combinatorial choice
model preserves the syntax, but alters the interpretation. It is important for this representation that
the empty budget set and empty choice be allowed in classical choice representation, the latter of
which is not common in the classical choice literature.16
Since the empty bundle, which is feasible, is in the power set of any subset of items Y ⊆ X , every budget set
B ∈ B contains at least one bundle.
16
See Echenique (2007) for further discussion on this point and for a few examples of classical choice papers allowing
for empty choice. Arguably, the possibility of not choosing in the classical setting should be modeled explicitly as a
labeled alternative, in which case one returns to the requirement that choice be nonempty.
15
8
3.5
Classical choice conditions and revealed preference relations
Fix X of a classical choice model (X , B, c).
Given a choice domain B and a binary relation R on X , I define the choice rule generated
by R, denoted cR , as follows: cR (B) = {Y ∈ B : for all Z ∈ B, Y R Z}, i.e. for every B ∈ B ,
cR (B) is the set of R-greatest elements amongst the bundles in the budget set B .17
A choice structure (B, c) is rationalizable if there exists a binary relation R on X such that
c = cR .
Define the revealed preference relation Rc of a choice structure (B, c) as follows: Z Rc Z ′
if and only if there exists B ∈ B such that Z ∈ c(B) and Z ′ ∈ B . I say that Z is revealed preferred
to Z ′ if Z Rc Z ′ . Define the revealed strict preference relation Rcs as follows: Z Rcs Z ′ if and
only if there exists B ∈ B such that Z ∈ c(B) and Z ′ ∈ B\c(B). I say that Z is revealed strictly
preferred to Z ′ if Z Rcs Z ′ .
A choice structure (B, c) satisfies the weak axiom of revealed preference (WARP)18 if
for any Z, Z ′ ∈ X , Z Rcs Z ′ implies ¬ (Z ′ Rc Z). A choice structure (B, c) satisfies the strong
s
axiom of revealed preference (SARP) if for any Z, Z ′ ∈ X , Z R̄c Z ′ implies ¬ (Z ′ Rc Z),
where R̄cs is the transitive closure of Rcs .
A choice structure (B, c) has the Chernoff property19 if the following holds: For any B, B ′ ∈
B and for any Z ∈ B , if B ⊆ B ′ and Z ∈ c(B′ ), then Z ∈ c(B).
A choice structure (B, c) satisfies independence of irrelevant alternatives (IIA)20 if the
following holds: For any B, B ′ ∈ B , if c(B ′ ) ⊆ B ⊆ B ′ , then c(B) = c(B ′ ).
A choice structure (B, c) satisfies Arrow’s axiom21 if the following holds: For all B, B ′ ∈ B ,
if B ⊆ B ′ and c(B ′ ) ∩ B ̸= ∅, then c(B) = c(B ′ ) ∩ B .
3.6
Combinatorial choice conditions and the Blair relation
Fix X and FX of a combinatorial choice model (X, FX , C).
A combinatorial choice function C satisfies irrelevance of rejected items (IRI) choice condition if the following holds: for any Y, Y ′ ⊆ X , if C(Y ) ⊆ Y ′ ⊆ Y , then C(Y ) = C(Y ′ ). Most
An alternative derivation of a choice rule from a binary relation R is the following: c̃R (B) = {Y ∈ B : for all Z ∈
B, ¬(Z R Y )}. Then c̃R (B) is the set of R-maximal elements in B . In this paper I do not study choice functions
generated in this manner. See Bossert et al. (2006) and Bossert and Suzumura (2010) for results concerning c̃R and for
the relation to cR .
18
This definition is due to Arrow (1959), and extends the original definition of Samuelson (1938, 1948) from budgets
sets of the consumer choice model to a general choice framework.
19
First introduced as Postulate 4 by Chernoff (1954), this is also known as “Sen’s α” property (Sen, 1971). The
condition was also studied in Uzawa (1956). It is called the “heritage property” in Aizerman and Malishevski (1981).
20
This condition is introduced by Nash (1950) as an axiom for a bargaining solution, and is Postulate 5* in Chernoff
(1954). It is closely related to the “outcast property” in the choice literature (Aizerman and Malishevski (1981).
21
Arrow (1959) defines this condition and shows that it is equivalent to WARP and to SARP when the choice domain
is finite-complete. A choice domain B is finite-complete if B = {B ⊆ X : |B| ∈ Z++ }.
17
9
recently defined in Aygün and Sönmez (2013), IRI appears repeatedly in the matching literature,
first in Blair (1988), and then in Alkan (2002) and Alkan and Gale (2003) as “consistency”.22
The IRI condition would seem a requisite of neoclassical rational choice. However, there are
at least two reasons why it may not hold. First, the presence of an object that is not chosen might
still matter because its presence indicates the manner in which the choice situation came about,
which might in turn affect choice behavior through an impact on valuations of objects. Luce and
Raiffa (1957, pg. 288) provide a famous example of the informational value of a menu. Second,
the presence of an ultimately unchosen object might affect the choice behavior because of reference
dependence, cognitive costs, or choice fatigue.23 Nevertheless, these explanations appeal to aspects
of the choice environment that ought to be explicitly modeled. If one is restricted to the primitives
at hand, IRI is an important requirement for rational choice in combinatorial settings.24
A combinatorial choice function C satisfies substitutability of items if the following holds:
for any Y, Y ′ ⊆ X and any y ∈ Y , if Y ⊆ Y ′ and y ∈ C(Y ′ ), then y ∈ C(Y ). Equivalently,
C(Y ′ ) ∩ Y ⊆ C(Y ). Substitutability as defined by here is due to Roth (1984), though Kelso and
Crawford (1982) introduced the analogous “gross substitutes” condition on demand functions in an
indivisible goods model with money.
Echenique (2007) points out the need to distinguish the interpretation of such formally identical
conditions when moving between combinatorial and classical choice models. Note that IIA in the
classical choice model is formally identical to IRI in the combinatorial choice model, and so equivalent to IRI under the forgetful representation of the latter. Analogously, the Chernoff property and
substitutability are equivalent under the forgetful representation. In Section 4.1 I show that, under
the faithful representation, IIA is equivalent to IRI with single-valued combinatorial choice, but the
Chernoff property and substitutability are not. In fact, it turns out that the Chernoff property is
equivalent to IRI.
The Blair relation25 RCB of a combinatorial choice function C is defined on FX as follows:
Z RCB Z ′ if and only if Z = C(Z ∪ Z ′ ), where Z, Z ′ ∈ FX . Note that RCB is an antisymmetric
relation since C is a function. It is straightforward to see that if a bundle is related to another by
22
The condition is also used by Fleiner (2003).
There is a growing literature on representing boundedly rational choice. For studies on reference dependence, see
for example Ok et al. (2015) and its references. For cognitive costs, see for example Ortoleva (2013) and its references.
For choice fatigue, see for example Iyengar and Lepper (2000) and Frick (2015) and their references.
24
In the context of matching with contracts, the combinatorial choice function of an agent is sometimes defined over
the set of all possible contracts in the economy, rather than just those involving this agent, even though the chosen sets
are restricted to contain only contracts involving the agent. If the IRI condition is imposed upon choice, this would rule
out the possibility that the presence of contracts of other agents influences the chosen set of the given agent, as might
be natural in a model with externalities (Pycia and Yenmez, 2015) or in a strategic model. In the case of externalities,
however, the true objects of choice are actually allocations for all agents, so with an appropriate modification one can
impose the IRI condition and still have the agent’s choice depend upon the contracts available to other agents.
25
This relation on single-valued combinatorial choice functions was introduced in the many-to-many matching
model of Blair (1988).
23
10
the Blair relation then it is also related to this other by the revealed preference relation.26 In the
context of classical many-to-one matching, Martínez et al. (2012) use the Blair relation to identify
the information in preferences that is relevant to determine the set of core matches. Echenique and
Oviedo (2006) also use the Blair relation to study a version of the core in many-to-many setting.
4
Results
I first discuss results for single-valued combinatorial choice, which is the usual choice model for
matching theory, with the main result being the equivalence of WARP and IRI. Next, I discuss extensions to the multi-valued combinatorial choice model, and obtain a similar equivalence of WARP
and IRI for an extension of IRI.27 I then furnish some results connecting various classical choice
conditions to WARP, and discuss their implications for the interpretation of these conditions in a
combinatorial choice setting. Finally, I discuss the classical choice condition of path independence
and its relationship to the combinatorial choice conditions of IRI and substitutability.
4.1
Single-valued combinatorial choice
Fix a combinatorial choice model (X, FX , C), and let (X , B, c) be its faithful representation.
The main theorem of this section is the equivalence of IRI, WARP, and rationalization by the
Blair relation whenever the combinatorial choice rule is single-valued, which is the typical choice
model in matching.
Theorem 1. If C is single-valued, then the following statements are equivalent:
1. C satisfies IRI.
2. c satisfies WARP.
3. The Blair relation RCB is equivalent to the revealed preference relation Rc .
4. The Blair relation RCB rationalizes (B, c).
5. There exists a complete and reflexive binary relation on FX that rationalizes (B, c).
6. There exists a binary relation on X that rationalizes (B, c).
Proof. [IRI implies WARP] Suppose that WARP is not satisfied, so that there exist Z, Z ′ ∈ X such
that Z Rcs Z ′ and Z ′ Rc Z . Then, there exist Y, Y ′ ⊆ X such that Z, Z ′ ⊆ Y ∩ Y ′ , C(Y ) = Z ,
and C(Y ′ ) = Z ′ . Now, C(Y ) = Z ⊆ Y ∩ Y ′ ⊆ Y so, by IRI, C(Y ∩ Y ′ ) = C(Y ) = Z . Also,
26
While the Blair relation bears some formal resemblance to the base relation (Bossert et al., 2006) of a classical choice
rule, it is not the same relation.
27
See Erdil and Kumano (2014) and Alva and Manjunath (2016) for recent papers that work directly with multivalued combinatorial choice rules.
11
C(Y ′ ) = Z ′ ⊆ Y ∩ Y ′ ⊆ Y ′ so, by IRI, C(Y ∩ Y ′ ) = C(Y ′ ) = Z ′ . But then I obtain Z = Z ′ ,
contradicting the hypothesis that Z Rcs Z ′ . Thus, WARP must hold if IRI is satisfied.
[WARP implies RCB = Rc ] It is clear that RCB ⊆ Rc even without WARP. Now, suppose Z Rc Z ′
for some Z, Z ′ ∈ X , i.e., there exists B ∈ B such that Z, Z ′ ∈ B , Z ∈ c(B). Since B is a
combinatorial choice domain, there exists Y ⊆ X such that B = 2Y ∩ FX , so Z ∪ Z ′ ⊆ Y . Since
c is single-valued, for every Z ′′ ∈ B where Z ′′ ̸= Z , Z Rcs Z ′′ . In particular, for every Z ′′ ∈ B ′ ,
′
where B ′ = (2Z∪Z ∩ FX ) ⊆ B , it is the case that Z Rcs Z ′′ . Then, by WARP, ¬(Z ′′ Rc Z). Thus,
since c is not empty-valued, c(B ′ ) = {Z}, i.e. C(Z ∪ Z ′ ) = Z , implying Z RCB Z ′ .
[RCB = Rc implies RCB -rationalizability] Let B ∈ B and Z ∈ c(B). Then, for all Z ′ ∈ B , Z Rc Z ′ .
B
Thus, by definition, Z ∈ cRc (B). Hence, c(B) ⊆ cRc (B) = cRC (B), where the latter equality
B
follows by assumption. Next, let Y ⊆ X and B ′ = 2Y ∩ FX . If Z, Z ′ ∈ cRC (B ′ ), then by definition
Z RCB Z ′ and Z ′ RCB Z . The definition of the Blair relation implies that Z = C(Z ∪ Z ′ ) and
B
Z ′ = C(Z ′ ∪ Z), so Z = Z ′ . Thus, c(B) ⊇ cRC (B). Thus, RCB rationalizes c.
[RCB -rationalizability implies complete, reflexive rationalizability] Define a binary relation R on X by the
follow conditions: a) for each Z ⊆ X , Z R Z , b) for all Z, Z ′ ⊆ X with Z ̸= Z ′ , Z RCB Z ′ implies
Z R Z ′ and ¬(Z ′ R Z), and c) for all Z, Z ′ ⊆ X , ¬(Z RCB Z ′ or Z RCB Z ′ ) implies Z R Z ′ and
Z ′ R Z . It is clear that R is complete and reflexive.
Let Y ⊆ X and B = 2Y ∩ FX . First, define Z = C(Y ). Since RCB rationalizes C , for all
Z ′ ⊆ Y , Z RCB Z ′ , and so, by Condition b) of the definition of R, Z R Z ′ . Then, Z ∈ cR (B), so
c(B) ⊆ cR (B).
Next, let Z, Z ′ ∈ cR (B). Then, Z R Z ′ and Z ′ R Z , which implies the hypothesis of Condition
a) or Condition c) holds. If Z ̸= Z ′ , it must be the hypothesis of Condition c) that holds, which
implies that C(Z ∪ Z ′ ) ̸∈ {Z, Z ′ }. Let Ẑ = C(Z ∪ Z ′ ). By RCB -rationalizability, Ẑ RCB Z , so by
Condition b) , Ẑ R Z and ¬(Z R Ẑ). But Ẑ ⊆ Y , so Z ∈ cR (B) implies Z R Ẑ , a contradiction.
Thus, Z = Z ′ , and so cR (B) is single-valued. Now, let Z ∈ cR (B) and suppose C(Y ) = Z ′ ̸= Z .
Then Z R Z ′ and Z ′ RcB Z . By Condition b) , Z ′ R Z and ¬(Z R Z ′ ), a contradiction. So
cR = c.
[complete, reflexive rationalizability implies rationalizability] This is immediate.
[rationalizability implies IRI] Let binary relation R rationalize c. Let Y ⊆ Y ′ ⊆ X , B = 2Y ∩ FX ,
′
and B ′ = 2Y ∩ FX . Let Z ′ ∈ c(B ′ ), i.e. Z ′ = C(Y ′ ). Then, rationalization by R implies Z ′ R Z
for all Z ∈ B ′ . Thus, if Z ′ ⊆ Y , then Z ′ R Z for all Z ∈ B , and so Z ′ ∈ c(B). Since c is
single-valued, c(B) = {Z ′ } = c(B ′ ), i.e. C(Y ) = C(Y ′ ).
It is known that WARP is not necessary for rationalizability of a choice rule when the choice
domain is arbitrary (Richter, 1966, 1971), even when the choice rule is single-valued. Theorem 1,
12
however, shows that WARP is necessary for a combinatorial choice domain.28
The notion of the Blair relation as a type of revealed preference relation is discussed in Alkan
(2002). Equivalence requires the assumption of IRI, as the following example demonstrates.
Example 1. Let (X, FX , C) be the combinatorial choice model defined by X = {x, y}, FX = 2X ,
and C(Y ) = Y if Y ⊆ X \ {y} and C(Y ) = ∅ if y ∈ Y . The combinatorial choice function
C satisfies substitutability but not IRI. Note that C(X) = ∅ implies ∅ Rc {x}, where c is the
faithful representation of C . However, ∅ ⊆ {x} and C({x}) = {x}, so {x} RCB ∅. Since RCB is
antisymmetric by definition, ¬(∅ RCB {x}).
When the combinatorial choice function C satisfies substitutability, some of the statements in
Theorem 1 can be strengthened. Most of the results in the following proposition are known in the
literature, but I state the equivalence here to relate them and to contrast them with Theorem 1.
Proposition 1. If C is single-valued and satisfies substitutability, then the following statements are equivalent:
1. C satisfies IRI.
2. The Blair relation RCB transitively rationalizes (B, c).
3. There exists a linear order on FX that rationalizes (B, c).
4. c satisfies SARP.
Proof. [IRI implies RCB -transitive-rationalizability] Alkan (2002) shows that RCB is transitive when C satisfies IRI and substitutability.29 By Theorem 1, IRI implies that RCB rationalizes (B, c).
[RCB -transitive-rationalizability implies linear order rationalizability] Since C is single-valued, RCB is antisymmetric. By the Szpilrajn extension theorem, there exists a linear order R on FX that extends
B
RCB . By the following claim, cR = cRC , and so by the assumption of rationalizability by RCB , cR = c.
Claim: Let R̃ be an antisymmetric relation on FX and R̃′ be an extension of R̃. Suppose cR̃
′
is nonempty-valued. Then, cR̃ is single-valued and cR̃ = cR̃ .
Proof of Claim: Let B ∈ B . First, nonempty-valued cR̃ and antisymmetry of R̃ implies that cR̃ is
single-valued, since Z, Z ′ ∈ cR̃ (B) implies both Z R̃ Z ′ and Z ′ R̃ Z , and so Z = Z ′ .
Let Z ∈ cR̃ and let Z ′ ∈ B .
′
By definition of cR̃ , Z R̃ Z ′ , so by definition of an extension, Z R̃ Z ′ . Since Z ′ is arbitrary,
′
′
Z ∈ cR̃ (B), so cR̃ (B) ⊆ cR̃ (B). Now suppose Z ′ ̸= Z , and note that Z is well-defined since cR̃
is nonempty-valued. From above Z R̃ Z ′ , so antisymmetry of R̃ implies ¬(Z ′ R̃ Z). But then the
′
requirement for an extension implies that ¬(Z ′ R̃ Z , and so Z ′ ̸∈ cR̃ (B).
28
Richter (1966, 1971) defines a choice condition, the V-Axiom, and proves it necessary and sufficient for rationalizability. A choice structure (B, c) satisfies the V-Axiom if and only if for all B ∈ B and for all Z ∈ B , Z Rc Z ′ for all
Z ′ ∈ B implies Z ∈ c(B).
B
29
In fact, Alkan (2002) shows that RC
is a join-semilattice on the range of C . See also Koshevoy (1999) and Johnson
and Dean (2001) for general analyses of classical choice rules on the finite, complete domain satisfying IIA and the
Chernoff property for analogous results.
13
′
Therefore, since B is arbitrary, cR̃ = cR̃ .
■
[linear order rationalizability implies SARP] See Richter (1966, 1971).
[SARP implies IRI] SARP is stronger than WARP, and so, by Theorem 1, implies IRI.30
Weaker versions of substitutability have been studied in the matching with contracts literature
(Hatfield and Kojima, 2010).31 However, Proposition 1 is tight in the sense that these weaker versions would not suffice. Aygün and Sönmez (2012) have examples satisfying IRI and weakened
substitutability that do not satisfy SARP.
I now consider the relationship between IRI, a combinatorial choice condition, and the classical
choice conditions of IIA, the Chernoff property, and Arrow’s axiom.
Given a single-valued combinatorial choice rule C , suppose, for example, the faithful representation c satisfies IIA. What restrictions are placed upon C ? It turns out that the exact restriction
placed is equivalent to IRI.
The reason for this equivalence, however, does not lie with similarity of the formal description
of IIA and IRI. The Chernoff property and substitutability also have similar formal descriptions.
As a consequence, under the forgetful representation, IIA and IRI are equivalent to each other, and
the Chernoff property and substitutability are equivalent to each other. However, under the faithful
representation, the Chernoff property does not imply and is not implied by substitutability. Instead,
the Chernoff property is equivalent to IRI under this representation, and so is equivalent to IIA.
Proposition 2. If C is a single-valued, then the following statements are equivalent:
1. c satisfies WARP.
2. C satisfies IRI.
3. c satisfies IIA.
4. c satisfies the Chernoff property.
5. c satisfies Arrow’s axiom.
Proof. [WARP implies IRI] This is true by Theorem 1.
[IRI implies IIA] Let B, B ′ ∈ B such that B ⊆ B ′ . Since B is a combinatorial choice domain,
′
there exist Y ⊆ Y ′ ⊆ X such that B = 2Y ∩ FX and B ′ = 2Y ∩ FX . Suppose that c(B ′ ) ⊆ B .
Since choice is single-valued, this implies C(Y ′ ) ⊆ Y . Then, by IRI, C(Y ) = C(Y ′ ), which means
c(B) = c(B ′ ).
′
[IIA implies WARP] Let Z, Z ′ ∈ X . Let B = 2Z∪Z ∩ FX ∈ B . Since B is a combinatorial
domain, for every B̃ ∈ B , if Z, Z ′ ∈ B̃ , then B ⊆ B̃ . Suppose Z Rcs Z ′ . Clearly, Z ̸= Z ′ . Then
30
31
Aygün and Sönmez (2013) show that IRI implies SARP given substitutability.
These are unilateral and bilateral substitutability.
14
there exists B ′ ∈ B such that Z, Z ′ ∈ B ′ , Z ∈ c(B ′ ), and Z ′ ̸∈ c(B ′ ). Since c is single-valued,
c(B ′ ) = {Z}. Then c(B ′ ) ⊆ B ⊆ B ′ , so by IIA, c(B) = c(B ′ ) = {Z}. Now suppose, for the sake
of contradiction, that Z ′ Rc Z , i.e., there exists B ′′ ∈ B such that Z, Z ′ ∈ B ′′ and Z ′ ∈ c(B ′′ ).
Since c is single-valued, c(B ′′ ) = {Z ′ } ⊆ B ⊆ B ′′ . But since Z ̸= Z ′ , c(B ′′ ) ̸= c(B), contradicting
IIA.
This establishes the equivalence of statements 1. to 3.
The equivalence of conditions 3. to 5. is established more generally for any single-valued classical choice model in Proposition 5 below. Therefore, all five conditions are equivalent.
Finally, suppose that combinatorial choice is not just single-valued but that the bundles chosen
contain exactly one item, i.e. unit-demand. If this unit-demand requirement is modeled by having
FX contain only singletons, then the forgetful and faithful representations yield the same classical
choice model. In essence, items are alternatives, and the combinatorial and classical choice model
are the same.
4.2
Multi-valued combinatorial choice
In this section, I explore extensions of IRI to multi-valued combinatorial choice and relate them
to WARP.
The first approach is to use tie-breakers to resolve indecisiveness of choice. Indifferences can
arise in the applications of matching theory to market design, and tie-breakers are often used to
produce a single-valued combinatorial choice function.32 The main result for combinatorial choice
is the equivalence of extended-IRI with tie-breakers and the WARP condition.
Let (X , B, c) be a classical choice model. A tie-breaker τ is a complete, transitive, and asymmetric binary relation on the set X . If c is a choice rule and τ a tie-breaker, then cτ is a selection
from c defined as follows: for every B ∈ B , Z ∈ cτ (B) if and only if Z ∈ c(B) and for every
Z ′ ∈ c(B), Z ′ ̸= Z implies Z τ Z ′ . In other words, cτ is a tie-broken choice function from c.
Similarly, given a combinatorial choice model (X, FX , C), a tie-breaker τ is a complete, transitive,
and asymmetric binary relation on the set 2X ∩ FX . Let C τ denote the tie-broken selection from
C according to tie-breaker τ , defined for all Y ⊆ X by C τ (Y ) = Z , where Z ∈ C(Y ), Z τ Z ′
for all Z ′ ∈ C(Y ) \ {Z}. Note that this definition of C τ is equivalent to identifying C τ with the
tie-broken selection cτ from the faithful representation of C .
Consider the following extension of the definition of IRI. Let (X, FX , C) be a combinatorial
choice model. Then C satisfies extended-IRI if and only if for every tie-breaker τ on 2X ∩ FX , C τ
satisfies IRI. This natural extension requires that the systematic approach of resolving indecisiveness
32
Erdil and Ergin (2008) and Abdulkadiroğlu et al. (2009) study school choice with tie-broken priorities structures.
See also Erdil and Kumano (2014).
15
through tie-breaking guarantees that the realized single-valued choice is well-behaved. Of course,
extended-IRI is equivalent to IRI when C is single-valued.
The main result of this subsection is the equivalence of extended-IRI with WARP, a partial
analogue to Theorem 1 for multi-valued combinatorial choice.
Theorem 2. Suppose C is a combinatorial choice rule. Then it satisfies extended-IRI if and only if it satisfies
WARP.
To establish the equivalence, I first show in Proposition 3 that for an arbitrary classical choice
structure, WARP is satisfied if and only if it is satisfied for every tie-broken selection. The proof of
Theorem 2 then follows from a combination of Theorem 1 and Proposition 3.
Proposition 3. Let (B, c) be a choice structure. Then c satisfies WARP if and only if for every tie-breaker τ
the tie-broken selection cτ from c satisfies WARP.33
Proof. [WARP for c implies WARP for all cτ ] Suppose c satisfies WARP. Let τ be a tie-breaker and let
cτ be the tie-broken selection from c. Let B ∈ B , Z = cτ (B), Z ′ ∈ B , and Z ′ ̸= Z . For cτ to
satisfy WARP, it is necessary that cτ (B ′ ) ̸= Z ′ for every B ′ ∈ B such that Z, Z ′ ∈ B ′ .
The first case is where Z ′ ∈ c(B). Then, by definition of cτ , it must be that Z τ Z ′ . Now,
consider any B ′ ∈ B such that Z, Z ′ ∈ B ′ . Since c satisfies WARP, Z ∈ c(B ′ ) if and only if
Z ′ ∈ c(B ′ ). Then, since Z τ Z ′ , cτ (B′ ) ̸= Z ′ .
The second case is where Z ′ ̸∈ c(B), so that Z Rcs Z ′ . Consider any B ′ ∈ B such that
Z, Z ′ ∈ B ′ . Since c satisfies WARP, ¬(Z ′ Rc Z), and so Z ′ ̸∈ c(B′ ), which immediately implies
cτ (B ′ ) ̸= Z ′ .
[WARP for all cτ implies WARP for c] Suppose for every tie-breaker τ the tie-broken selection cτ
from c satisfies WARP. Let B ∈ B , Z ∈ c(B), and Z ′ ∈ B \ c(B), so that Z Rcs Z ′ . For c to satisfy
WARP, it is necessary that Z ′ ̸∈ c(B ′ ) for every B ′ ∈ B such that Z, Z ′ ∈ B ′ , so that ¬(Z ′ Rc Z).
Consider B ′ ∈ B such that Z, Z ′ ∈ B ′ . Let τ be the tie-breaker where Z ′ τ Zτ Z ′′ for every
Z ′′ ∈ X \ {Z, Z ′ }. Then, cτ (B) = Z . Since cτ satisfies WARP, cτ (B ′ ) ̸= Z ′ . But since Z ′ is the
highest ranked bundle under τ , this implies that Z ′ ̸∈ c(B ′ ), completing the proof.
A second possible approach to extending IRI to multi-valued combinatorial choice rules is the
following stronger one: (B, c) satisfies strong extended-IRI if for every selection c̃ of c, c̃ satisfies
IRI. This is equivalent having IRI satisfied by every selection generated by budget-set-specific tiebreakers.
33
Ehlers and Sprumont (2008) study the implications choice behavior that satisfies a weaker revealed preference
axiom, the weakened weak axiom of revealed preference (WWARP), defined as follows: If Z, Z ′ ∈ X and there exists
B ∈ B such that Z ∈ c(B) and Z ′ ∈ B \ c(B), then there does not exist B ′ ∈ B such that Z ′ ∈ c(B ′ ) and
Z ∈ B ′ \ c(B′ ). Equivalently, Z Rcs Z ′ implies ¬(Z ′ Rcs Z). It is clear that WWARP is equivalent to WARP when the
choice rule is single-valued, and so any selection from a choice rule satisfies WARP if and only if it satisfies WWARP.
Thus, the analog of Proposition 3 for WWARP is not true.
16
The following two results are corollaries of Proposition 3 and Theorem 1. The take-away is that
any combinatorial choice rule satisfying strong extended-IRI also satisfies WARP, but the converse
need not be true. In particular, Example 2 shows there can be a selection from a combinatorial
choice rule where the selection fails WARP even though the rule satisfies it.
Corollary 1. Let (B, c) be a choice structure. If every selection c̃ of c satisfies WARP, then c satisfies WARP.
Proof. Since every selection of c satisfies WARP, so does the tie-broken selection cτ for every tiebreaker τ . Then the conclusion obtains from Proposition 3.
Corollary 2. Suppose C is a combinatorial choice rule. If it satisfies strong extended-IRI, it also satisfies WARP.
Proof. By strong extended-IRI, every selection from C satisfies IRI. Then by Theorem 1, every
selection from C satisfies WARP, and so by Corollary 1, the conclusion is obtained.
Example 2. Let X be any set of items, with |X| ≥ 3, and suppose every bundle from X is feasible. Let C be a combinatorial choice rule on X defined by C(Y ) = 2Y for every Y ⊆ X , i.e.
C chooses every feasible bundle when available. This combinatorial choice rule satisfies WARP
trivially. However, consider a selection C̃ that satisfies the following conditions: Fix three distinct
items x, y, z ∈ X . For each Y ⊆ X , C̃(Y ) = {x} if x ∈ Y and z ̸∈ Y , and C̃(Y ) = {y} if y ∈ Y
and z ∈ Y .
Note that C̃({x, y}) = {x} and C̃({x, y, z}) = {y}, by definition of C̃ . But then clearly C̃
violates IRI and so WARP.
Since IRI and IIA are equivalent for single-valued combinatorial choice structures, a third possible extension of IRI to the multi-valued setting is to identify IRI with IIA. Indeed, Echenique
(2007) defines IRI for the combinatorial choice model and calls it IIA. While IIA of the faithful representation of a combinatorial choice model need not have identical implications for combinatorial
choice as IRI of the combinatorial choice model, it turns out they are satisfied by the same set of
single-valued combinatorial choice functions, as shown in Proposition 2.
Unfortunately, as Example 3 below demonstrates, requiring IIA of the faithful representation of
a combinatorial choice model is too weak a requirement when choice is multi-valued. It is possible
that there does not exist any selection that satisfies IRI. Given the result in Proposition 3, the example
also demonstrates that there are multi-valued combinatorial choice rules that satisfy IIA but not
WARP.
Example 3. Let the set of items X = {x, y, z} and suppose B = {2Y : Y ⊆ X}. Suppose c is a
choice rule on B defined as in the following table:
17
B
∅
2
2{x}
2{y}
2{z}
c(B)
{∅}
{{x}}
{{y}}
{{z}}
B
{x,y}
2
2{x,z}
2{y,z}
2{x,y,z}
c(B)
{{x}}
{{x}}
{{y}, {z}}
{{y}, {z}}
The choice structure (B, c) satisfies IIA. For example, c(2{x,y,z} ) ⊆ 2{y,z} , and the requirement
that c(2{x,y,z} ) = c(2{y,z} ) is satisfied. However, WARP is not satisfied, since {x} Rcs {y} via budget
set 2{x,y} but {y} Rcs {x} via budget set 2{x,y,z} . Moreover, it can be verified that every selection
from c will violate IRI, i.e. there is no tie-breaking rule (exogenous or budget-specific) that will
produce a tie-broken selection satisfying IRI (and so WARP). Moreover, c is not rationalizable.
4.3
Arbitrary classical choice domains
Fix a nonempty set of alternatives X .
In Proposition 2 on combinatorial choice domains, it was shown that IIA is equivalent to WARP
with single-valued rules. It is the case that WARP implies IIA in general, even when combinatorial
choice domain restriction is removed and choice can be multi-valued.
Proposition 4. Suppose a choice structure (B, c) satisfies WARP. Then it satisfies IIA.
Proof. I begin with the following claim.
Claim: If Y Rc Z and Z Rc Y , then for every B ∈ B such that Y, Z ∈ B , Y ∈ c(B) if and
only if Z ∈ c(B).
Proof of Claim: By WARP, ¬ (Y Rcs Z) and ¬ (Z Rcs Y ). Given that Y ∈ c(B) and Z ∈ B ,
¬ (Y Rcs Z) implies Z ∈ c(B). Symmetrically, it can be established that Y ∈ c(B) if Z ∈ c(B) and
Y ∈ B , completing the proof.
■
′
′
′
′
′
Let B, B ∈ B , B ⊆ B and suppose c(B) ⊆ B . By definition, c(B ) ⊆ B . I need to show that
c(B ′ ) = c(B).
Let Y, Z ∈ c(B). By the claim and the hypothesis that c(B) ⊆ B ′ , Y ∈ c(B ′ ) if and only if
Z ∈ c(B′ ). Thus, if c(B) ∩ c(B ′ ) ̸= ∅, then c(B) ⊆ c(B ′ ).
Now, let Y, Z ∈ c(B ′ ). By definition and by hypothesis, c(B ′ ) ⊆ B ′ ⊆ B , so by the claim,
Y ∈ c(B) if and only if Z ∈ c(B). Thus, if c(B) ∩ c(B′ ) ̸= ∅, then c(B ′ ) ⊆ c(B).
Finally, suppose for the sake of contradiction that c(B) ∩ c(B ′ ) = ∅. Let Y ∈ c(B) and Z ∈
c(B ′ ), which are well-defined since the choice rule is non-empty valued. Notice that Z ∈ B \ c(B)
and Y ∈ B ′ \ c(B ′ ), so have Y Rcs Z and Z Rcs Y , which contradicts WARP.
18
For arbitrary choice domains, however, WARP is a more restrictive requirement than IIA even
when choice is single-valued.
Example 4. Consider B = {B, B ′ }, where B = {Z1 , Z2 , Z3 } and B ′ = {Z1 , Z2 , Z4 }, and for
i, j ∈ {1, 2, 3, 4}, Zi ∈ X for some set of mutually exclusive alternatives X and Zi = Zj if and
only if i = j . Suppose c(B) = {Z1 } and c(B ′ ) = {Z2 }. Then choice is single-valued, and WARP
is violated, but IIA is trivially satisfied. Moreover, c is rationalizable by the revealed preference
relation Rc .
Without a single-valued choice rule, IIA is a weaker requirement than WARP even when the
choice domain is combinatorial, as Example 3 makes clear. Therefore, the equivalence of IIA
and WARP in Proposition 2 relies upon both the assumptions of single-valued choice and of a
combinatorial choice domain.
Next, I show that the previously discussed classical choice conditions of IIA, the Chernoff property, and Arrow’s axiom are all equivalent when choice is single-valued.
Proposition 5. Suppose (B, c) is a single-valued choice structure. Then the following choice conditions are
equivalent:
1. c satisfies the Chernoff property.
2. c satisfies IIA.
3. c satisfies Arrow’s axiom.
Proof. Let B, B ′ ∈ B , B ⊆ B ′ .
First, suppose the Chernoff property holds, so c(B ′ ) ∩ B ⊆ c(B). Suppose c(B ′ ) ⊆ B . Then,
c(B ′ ) ⊆ c(B), and since c is single-valued (and non-empty valued), c(B ′ ) = c(B), so IIA is satisfied.
Instead, suppose c(B ′ ) ∩ B ̸= ∅. Since c is single-valued, c(B ′ ) ∩ B ̸= ∅ implies c(B ′ ) ⊆ B .
Suppose IIA holds. Then c(B) = c(B ′ ) = c(B ′ ) ∩ B , so Arrow’s axiom is satisfied.
Next, Arrow’s axiom is a strengthening of the Chernoff property, and so implies it (even without
single-valuedness of c).34
An immediate corollary to Propositions 4 and 5 and Example 4 is that WARP is a strictly stronger
requirement of single-valued choice than each of IIA, the Chernoff property, and Arrow’s axiom.
Corollary 3. Suppose (B, c) is a single-valued choice structure. Then it satisfies IIA, the Chernoff Property,
and Arrow’s axiom if it satisfies WARP. The converse need not be true.
34
See Arrow (1959) for a discussion.
19
4.4
On path independence
A choice domain is complete if B = 2X \ {∅}. The following condition was introduced by
Plott (1973), in a setting with a finite set of alternatives and a complete choice domain. A choice
structure (B, c) with a complete domain and finite set of alternatives X satisfies path independence
if the following holds: For every B, B ′ ∈ B , c(B ∪ B ′ ) = c(c(B) ∪ B ′ ).35
Aizerman and Malishevski (1981) show that, when B is complete, path independence is satisfied if and only if the Chernoff property and IIA are satisfied. Thus, the following corollary to
Proposition 5 is immediate.
Corollary 4. Let (X , B, c) be a classical choice model, and suppose X is finite, B is complete, and c is
single-valued. Then the following choice conditions are equivalent:
1. c satisfies path independence.
2. c satisfies IIA.
3. c satisfies the Chernoff property.
4. c satisfies Arrow’s axiom.
5. c satisfies WARP.
Because of the formal similarity between IIA and IRI and between the Chernoff property and
substitutability, a combinatorial choice function C satisfying IRI and substitutability is sometimes
said to be “path independent”, given the result of Aizerman and Malishevski (1981).36 However,
given a combinatorial choice model (X, FX , C), it is not clear how to apply path independence
condition to the faithful choice representation (B, c), because the choice domain B = {2Y ∩ FX :
Y ⊆ X} is not complete. Consider, for example, a combinatorial choice model with X = {x, y}.
Then in the faithful representation, Bx = 2{x} and By = 2{y} are two budget sets in B , but
Bx ∪ By = {∅, {x}, {y}} ̸∈ B , although Bx∪y = 2{x}∪{y} ∈ B .
In fact, for any combinatorial choice model, if B1 and B2 are budget sets in the choice domain of
the faithful representation, then B1 ∪ B2 ∈ B if and only if B1 ⊆ B2 or B2 ⊆ B1 . Since each budget
set is uniquely associated with an opportunity set, there must be Y1 , Y2 ⊆ X such that B1 = 2Y1 and
B2 = 2Y2 . So, path independence condition applied to the faithful representation can only have
implications on opportunity sets such that Y1 ⊆ Y2 or Y2 ⊆ Y1 .
Consider the following adaptation of the path independence condition to incomplete domains:
a choice structure (B, c) satisfies path independence if for every B, B ′ ∈ B such that B ∪ B ′ ∈ B ,
c(B ∪ B ′ ) = c(c(B) ∪ B ′ ) whenever c(B) ∪ B ′ ∈ B .
35
Plott (1973) actually provides two other definitions of path independence and shows that all three are equivalent
in his setting with a finite set of alternatives and a complete choice domain.
36
See, for example, Alkan (2002), Alkan and Gale (2003), Fleiner (2003), and Chambers and Yenmez (2014).
20
Proposition 6. Suppose (X, FX , C) is a single-valued combinatorial choice model, with faithful representation
of (X , B, c). The representation c satisfies path independence if and only if C satisfies IRI.
′
Proof. Let B, B ′ ∈ B , and let Y, Y ′ ⊆ X be such that B = 2Y and B ′ = 2Y . Then, B ∪ B ′ ∈ B
′
if and only 2Y ∪ 2Y = 2Z for some Z ⊆ X . This could only be if Y ⊆ Y ′ or Y ′ ⊆ Y . So, the
path independence condition applies only when considering opportunity sets ordered by inclusion.
Moreover, it is also necessary that c(B) ∪ B ′ ∈ B . Then, given that C is single-valued (and so
′
c(B) = {C(Y )}, the path independence condition applies only when C(Y ) ∈ 2Y = B ′ , so that
c(B) ∪ B ′ = B ′ ∈ B .
Therefore, path independence for the representation is equivalent to the following condition on
the combinatorial choice function: for Y, Y ′ ⊆ X with Y ⊆ Y ′ , C(Y ∪ Y ′ ) = C(C(Y ) ∪ Y ′ )
whenever C(Y ) ⊆ Y ′ , and C(Y ∪ Y ′ ) = C(Y ∪ C(Y ′ )) whenever C(Y ′ ) ⊆ Y . The first part
of this condition is trivially satisfied by every combinatorial choice function. Now, suppose that C
satisfies IRI. Then, since Y ⊆ Y ′ , if C(Y ′ ) ⊆ Y then C(Y ) = C(Y ′ ). But then C(Y ∪ Y ′ ) =
C(Y ′ ) = C(Y ) = C(Y ∪ C(Y ′ )), so obtain the second part of the condition. Thus, if C satisfies
IRI then its representation satisfies path independence.
To show the converse, let Y ⊆ Y ′ ⊆ X and suppose C(Y ′ ) ⊆ Y . Then C(Y ′ ) = C(Y ∪ Y ′ ) =
C(Y ∪ C(Y ′ )), where the latter equality follows from path independence given the assumption that
C(Y ′ ) ⊆ Y . But the same assumption yields the conclusion that C(Y ∪ C(Y ′ )) = C(Y ) and so
path independence implies IRI.
Therefore, path independence of the faithful representation does not imply IRI and substitutability, but only IRI. The following corollary of Theorem 1 and Proposition 2 is immediate.
Corollary 5. Suppose the combinatorial choice rule C is single-valued and c is its faithful representation. Then
c is path independent if and only if:
1. C satisfies IRI.
2. c satisfies WARP.
3. The Blair relation RCB rationalizes (B, c).
4. c satisfies the Chernoff property.
5. c satisfies Arrow’s axiom.
5
Conclusion
In this paper, I study the implications of well-known classical choice conditions for the combina-
torial choice model, when the latter model is represented so that the mutually exclusive alternatives
assumption of the former model is respected. In particular, I show that the important combinatorial choice consistency condition of Irrelevance of Rejected Items (Aygün and Sönmez, 2013) is
21
equivalent to the classical choice rationality condition of Weak Axiom of Revealed Preference. I
provide novel results for the combinatorial choice domain, by showing that many different classical
rationality conditions are identical for this domain. I also introduce a natural generalization of the
Irrelevance of Rejected Items condition for multi-valued combinatorial choice, and show that the
equivalence with the Weak Axiom continues to hold. Other implications of classical choice results
for the combinatorial choice domain await exploration.
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