Choice, Consideration Sets and Attribute Filters∗
Mert Kimya
†
Koç University, Department of Economics, Rumeli Feneri Yolu, Sarıyer 34450, Istanbul,
Turkey
August 26, 2016
Abstract
It is well known that decision makers do not always consider all of the available
alternatives when making a choice. When the alternatives have attributes, these
attributes provide a natural way to form the consideration set. I assume a procedure in which the decision maker uses the relative ranking of the alternatives on
each attribute to reduce the size of the choice set. I provide a characterization of
the procedure and illustrate how to identify the underlying preference and consideration set. The model explains certain choice anomalies such as the attraction and
compromise effects.
JEL classification: D01; D03; D11
Keywords: Bounded Rationality; Choice; Consideration Set; Revealed Preference
1
Introduction
Suppose you are searching online for a hotel in a popular destination. Once you enter the
dates and the city you will realize that there are too many alternatives to choose from.
In which case you will probably use the filters the website conveniently provides for you.
A typical website will have filters on price, star rating, review rating and distance. Most
likely you will use these filters to remove the hotels that rank poorly on some of these
dimensions. Say hotels with a review rating less than 6 out of 10, with a price more than
150 dollars and distance to a point of interest more than 5 miles. Having removed some
∗
I am grateful to Mark Dean for guidance, support and encouragement. I would also like to thank
Geoffroy de Clippel and two anonymous referees for helpful comments and suggestions.
†
E-mail Address: mert [email protected]
1
of the available alternatives this way, you would then select your preferred option from
those that remain.
It is well known that decision makers do not always consider all of the available
alternatives when making a choice. Although there are many models that characterize
how the consideration sets might be constructed, one thing that has been overlooked is
in many situations alternatives have observable attributes. When the alternatives have
attributes it seems natural that the decision maker (DM) uses these attributes to form
the consideration set. In this paper, I formalize and characterize such a choice procedure.
I model a DM who has a stable preference over a set of alternatives. The alternatives
have a set of attributes and the DM has an ordering of the alternatives on each of these
attributes. The ordering indicates the DM’s preference on each attribute. In the above
example, an attribute is price and the DM ranks cheaper hotels higher than expensive
ones on this attribute. Another attribute is distance and the DM ranks hotels with a
shorter distance to the point of interest higher than the hotels with a longer distance.
Our DM sets a threshold on each attribute ranking and does not consider any alternative
that stays below that threshold. She maximizes her preference on the set of remaining
alternatives.
The thresholds are not determined arbitrarily, but instead they depend on the relative ranking of alternatives in the sense that inclusion of an alternative above (below)
the threshold cannot move the threshold down (up) and it can only go up (down) to
that alternative.1 This restriction ensures that if an alternative stays above (below) the
threshold in some attribute and a new set is obtained by adding alternatives worse (better) than this alternative, then this alternative will again stay above (below) the threshold
in this new set.
I call the choice procedure that satisfy these conditions choice through attribute filters
(CAF). The model has several intuitive interpretations:
• As in the hotel choice example, the set might contain too many elements to consider,
in which case the DM might use the objectively defined attributes to reduce the
size of the set.
• The DM might be using a ‘rule of thumb’ to eliminate certain alternatives based
on their attribute rankings even when the size of the set is small. For instance
the DM might have a fixed cutoff (see Example 5 in Section 2.2) or she might be
using satisficing on one of the attribute rankings (see Example 6 in section 2.2) or
she might eliminate non-dominant alternatives in a set containing asymmetrically
1
Proposition 2 in Appendix A shows that if we do not put any restriction on how the thresholds are
determined then the model has little empirical content.
2
dominated alternatives (see Sections 3.2 and 4). Further examples of this kind is
available in Section 2.2.
CAF does not provide testable predictions on standard choice data: every observed
choice function is rationalizable as CAF (see Proposition 1 in Appendix A). I consider
a richer dataset, which I call ‘multicriteria choice data’. This data includes not only
the final choice but also the attributes and the rankings of the DM on these attributes.
Data that includes observable attributes has been used before both in theoretical and
experimental papers (see de Clippel and Eliaz (2012), Gabaix et al. (2006), Huber et
al. (1982), Simonson (1989), Tversky and Simonson (1993)). Multicriteria choice data is
easy to collect in many settings, especially in controlled laboratory environments, where
the alternatives can be defined with a set of attributes.2
I characterize the conditions under which multicriteria choice data is consistent with
CAF. This is done through an understanding of what revealed consideration and revealed
preference means for the model. The choice of one alternative over another does not
indicate preference as the DM may have eliminated the unchosen alternative. However,
if we can conclude that the unchosen alternative is considered (revealed consideration),
this would imply a preference in favor of the chosen alternative (revealed preference).
For example, suppose the choice of the DM is consistent with CAF and she chooses
x from some set S. This would trivially imply that x is considered in S. Now assume
that as we go from set S to set T , the relative position of x improves in every attribute,
but y is chosen in set T . As x is not eliminated by any attribute in set S and as its
relative position in every attribute improves as we move from S to T , x should not be
eliminated by any attribute in set T . Hence, x is revealed to be considered in set T
(revealed consideration). Furthermore, as y is chosen in T and as x is considered in T
this would imply that y is revealed to be preferred to x (revealed preference).
The necessary and sufficient condition for multicriteria choice data to be consistent
with CAF is the acyclicity of the revealed preference. Section 3 provides the necessary
and sufficient conditions for the data to be consistent with CAF. It also fully characterizes
what we can learn about the DM’s preference (revealed preference) and the consideration
set (revealed consideration) if the choice is CAF.
The CAF model is consistent with a number of intuitive choice procedures and it
is able to explain certain consistently observed choice anomalies such as the attraction
and the compromise effects. While explaining these effects, I show that CAF is also
inconsistent with the ‘opposites’ of each of these effects.
2
It is important to note that multicriteria choice data and CAF does not assume all of the available
attributes and the rankings on these attributes is observable. It only requires that those attributes that
are used for the construction of the consideration set are observable.
3
Furthermore, not only that CAF is inconsistent with the opposites of these effects
while consistent with the effects themselves, it is shown that an independent model that
only allows for choice anomalies in the spirit of these two effects is completely characterized by a simple class of CAF. This is shown in Section 4, where I take the axiomatic
approach and directly put restrictions on what kind of choice anomalies are allowed in
the data. It turns out that a model that only allows choice anomalies in the spirit of the
attraction and compromise effects is completely characterized by a simple class of CAF
in which there is only a single undominated alternative in each consideration set.
I start with the formal description of the model in Section 2, which also includes
a number of choice procedures that are examples of CAF. In Section 3, I provide the
necessary and sufficient conditions for the data to be consistent with CAF and show
what can be deduced about the preference and the consideration set of the DM if her
choice is consistent with CAF. Section 4 studies an axiomatic model that only allows
for choice anomalies in the spirit of attraction and compromise effects. This section
also shows how this model is related to CAF. Section 5 includes the literature review,
it also shows the independence of CAF from several influential models in the literature.
Appendix A contains additional auxiliary results and Appendix B contains the proofs.
2
Model
2.1
Multicriteria Choice Data
To study CAF I use multicriteria choice data. This data includes not only the choice but
also the attributes and the rankings of the alternatives on these attributes.
Let Z ⊂ Rk++ be a finite set of alternatives, Z denotes the set of all nonempty subsets
of Z. Each l ∈ {1, 2, ..., k} denotes an attribute and zl denotes the value of alternative z
on attribute l.3
For convenience I assume no ties in attribute rankings, i.e. for any z, z 0 ∈ Z with
z 6= z 0 and l ∈ {1, 2, ..., k}, we have zl 6= zl0 .
A choice function is a mapping c : Z → Z such that c(S) ∈ S for all S ∈ Z. It is
interpreted as the choice of the DM from each set. Multicriteria choice data includes Z
and c.
The assumption of no ties in attribute rankings and the restriction to choice functions
is only made for convenience. In Appendix A, I show that the main results are still
valid with minor modifications even if we relax these assumptions by looking at choice
correspondences and ties in attribute values.
3
The model is ordinal. Hence, only the order induced matters.
4
2.2
Choice Through Attribute Filters
The DM has a consideration set, a mapping Γ : Z → Z, where Γ(S) ⊆ S for each S ∈ Z.
Γ(S) denotes the set of alternatives she considers. She has a complete, transitive, strict
and monotone preference on Z.4 Once she forms the consideration set, she is assumed
to maximize on this consideration set. It is easy to see that without any restriction on
Γ any observed choice would be rationalizable with this procedure.5
The existence of attributes and rankings over these attributes provide a natural way
to form the consideration sets. I will consider the case where for each set S the DM has
a threshold on each attribute ranking and eliminates everything below this threshold.
Furthermore, the thresholds weakly move up (down) when we add an alternative above
(below) the threshold. And the extent of this movement is limited to the alternative we
include.
This last condition ensures that the elimination depends on the relative ranking of the
alternatives in the sense that if an alternative x is (not) eliminated by an attribute i in set
S. And if set T can be obtained from set S by adding an alternative that ranks (worse)
better than x in attribute i then x should (not) be eliminated in set T by attribute i.
The following definition formalizes these restrictions. Let Ei (S) denote the set of
eliminated alternatives because of attribute i. The consideration set will be given by
Γ(S) = S \ (∪i=1,..,k Ei (S)). Let > 0 be a small enough number such that |zl − zl0 | > for all z, z 0 ∈ Z with z 6= z 0 and for all l ∈ {1, 2, .., k}.
Definition 1. Γ is an attribute filter if there exists a threshold tS ∈ Rk for each S ⊆ Z
and a mapping Ei : Z → Z for each i ∈ {1, .., k}, where Ei (S) ⊆ S, Γ(S) = S \
(∪i=1,2,..,k Ei (S)), Γ(S) 6= ∅ such that
(A) Ei (S) = {z ∈ S|zi ≤ tSi }
(B) If F = S ∪ x where x ∈ Z and xi ≥ tSi then xi + ≥ tFi ≥ tSi
(C) If F = S ∪ x where x ∈ Z and xi ≤ tSi then xi − ≤ tFi ≤ tSi
Condition (A) states that anything that stays below the threshold is eliminated in
each attribute. Condition (B) ((C)) states that if an alternative is added above (below)
the threshold in attribute i, then the threshold can only move up (down). However, this
movement is limited by the included alternative; that the threshold can move up (down)
4
A preference is monotone if whenever zl > zl0 for all l ∈ {1, ..., k} we have z z 0 . This assumption
is made only because it is reasonable, Remark 2 in Appendix A explains how the results would change
if we do not make this assumption.
5
We can claim that Γ(S) = c(S) for all S.
5
enough to eliminate (save) this alternative, but not so much that it eliminates (saves)
the alternative ranking above (below) this alternative. The reason we include is that
the condition does not dictate whether the included alternative should or should not be
eliminated. The implication of dropping the latter two conditions will be discussed in
Appendix A.
The following examples illustrate the wide range of procedures that lead to an attribute filter. I will show how to set these up as attribute filters in Appendix B.
Example 1. (Rational Choice) Eliminate nothing
Example 2. (Top n) Eliminate nothing if the set contains less than n elements. If the
set contains more than n elements then consider the best n elements according to a single
attribute ranking. For example, consider the n cheapest alternatives.
Example 3. (Eliminate Worst n) Choose an attribute i. If the set contains less than
(or equal to) n elements then consider only the best element according to attribute i. If
the set contains more than n elements then eliminate the worst n elements in attribute
i. For example, consider only the cheapest if the set contains less than 3 elements and
eliminate the most expensive 3 elements if the set contains more than 3 elements.
Example 4. (Eliminate Worst n on each attribute) The DM eliminates the worst
n alternatives in each attribute ranking. If everything is eliminated then she eliminates
worst n − 1 instead, if still everything is eliminated she eliminates the worst n − 2 and
so on.
Example 5. (Fixed Cutoff ) The DM has a fixed cutoff a∗i for each attribute i and for
every set S she eliminates the alternatives that stay below the cutoff at each attribute
ranking. The DM also has a fixed ordering of attributes and if everything is eliminated
then she removes the cutoff of the attributes in that order until the set contains at least
one element.
Example 6. (Satisficing (Simon (1955))) The DM has stable preference and a threshold a∗ ∈ Z, she chooses an attribute i. She considers the best element in attribute i.
If this element is not preferred to a∗ then she also considers the second best element
in attribute i. She continues doing this until she finds an alternative that is preferred
to a∗ . If there is nothing preferred to a∗ then she considers everything. For example,
she considers the cheapest alternative. If it is better than a∗ then she doesn’t consider
anything else. If not, then she considers the second cheapest alternative and so on.
At this point, it is important to note that attribute filters are independent of other
prominent consideration sets used in the literature. In Section 5 this is formally shown
with a comparison to attention filters (Masatlioglu et al. (2012)) and consideration filters
(Lleras et al. (2010)).
6
Now, we are ready to define CAF.
Definition 2. Choice Through Attribute Filters (CAF)
A choice function c is a choice through attribute filters (CAF) if there exists a strict,
complete, transitive and monotone preference relation over Z and an attribute filter Γ
such that c(S) is the -best element in Γ(S) for every S ∈ Z.
3
Identification and Representation
What are the necessary and sufficient conditions for c to be CAF? And if c is CAF then
what can we confidently conclude about the preference and the consideration set of the
DM? This section aims to answer these questions. I will follow Masatlioglu et al. (2012)
in defining the revealed preference and consideration:
Definition 3. Suppose c is CAF and there are m different preference, elimination sets
and consideration sets representing c, {(Γ1 , {Ei1 }i={1,..,k} , 1 ), (Γ2 , {Ei2 }i={1,..,k} , 2 ), ...,
(Γm , {Eim }i={1,..,k} , m )}. Note that this is an exhaustive list of all possible representations
consistent with CAF. Then
• x is revealed to be preferred to y iff x j y for all j = 1, 2, ...., m
• x is revealed to be considered in S iff x ∈ Γj (S) for all j = 1, ..., m
• x is revealed to be non-eliminated by attribute i if x ∈
/ Eij (S) for all j = 1, ..., m.
This is a conservative definition as it requires for all representations to agree to conclude something about the preference or the consideration set of the DM. The following
condition will be helpful in our quest to understand these.
Condition 1. (RNi ) y ∈ T satisfies RNi in set T if there exists z ∈ T and a set S
including y and z such that c(S) = z, yi ≥ zi 6 and
• If x ∈ T and xi ≥ yi , then x ∈ S.
• If x ∈ S and zi ≥ xi , then x ∈ T .
Assume that c is CAF and y satisfies RNi in set T . Then z is chosen in S, so we know
that z is not eliminated by attribute i in S. Consider the set F obtained from set S by
removing everything that is not in T and that ranks better than z attribute i. Observe
that as we go from set S to set F we are just removing alternatives that rank better than
6
Since there are no ties in attribute rankings, this means that either y = z or yi > zi
7
z in attribute i. Therefore, by (B) and (C) in Definition 1, z is not eliminated in set F .7
Since yi ≥ zi , by (A) in Definition 1, y is not eliminated in set F .
But, as we go from set F to set T , we are only adding alternatives that rank below
y in attribute i, together with (B) and (C) in Definition 1 this implies that y is not
eliminated by attribute i in set T .8
This shows that if c is CAF and if y satisfies RNi in set T , then y is revealed to be noneliminated in set T by attribute i. Furthermore, if y satisfies RNi for each i ∈ {1, 2, ..., k}
then y is revealed to be considered in set T .
But if y is revealed to be considered in set T and if c(T ) 6= y then c(T ) is revealed to
be preferred to y. Also by monotonicity, for any two alternatives, x is preferred to y if x
dominates y. Lets define the relation P with these observations.
Let x D y if x dominates y, i.e. xl > yl for all l ∈ {1, 2, ..., k}.
Definition 4. xP y if x >D y or if there exists a set T containing y such that c(T ) = x,
x 6= y and y satisfies RNi in set T for each attribute i.
What conditions on P would ensure that the choice is consistent with CAF? The
following theorem shows that acyclicity is necessary and sufficient.
Theorem 1. c is CAF iff P is acyclic.
Now that we know which data is consistent with CAF, the question is if the choice is
consistent with CAF then what can we conclude about the preferences and the consideration sets of the DM? The next section provides a definitive answer to this question.
3.1
Identification
The discussion above has shown that if xP y then x is revealed to be preferred to y. Note
that since the underlying preference is transitive we can also conclude that if xP y and
yP z then x is revealed to be preferred to z. Which means that if we denote by PR the
transitive closure of P , then x is revealed to be preferred to y if xPR y. It turns out that
PR is actually all the revealed preference we can get.
Furthermore we have seen that if x satisfies RNi in set S then x is revealed to be
non-eliminated by attribute i in set S. And if x satisfies RNi for each attribute i, then x is
7
Suppose z is eliminated in set F by attribute i. As we move from set F to set S we are adding
alternatives that rank better than z in attribute i. But, by (B) and (C) the movement of the threshold
is limited to the alternative added to the set, so none of these alternatives will be able to drive the
threshold below z and hence z is eliminated in set S. A contradiction to c(S) = z.
8
As we move from set F to set T we are only adding alternatives that rank below y in attribute i,
since by (B) and (C) the movement of the threshold is limited to the alternative we add, none of these
alternatives can drive the threshold above y.
8
Attribute 2
x
y
z
Attribute 1
c(xy) = c(xz) = x, c(yz) = z and c(xyz) = z
Figure 1: Data corresponding to Example 7
revealed to be considered in set S. It turns out that this is all the revealed non-elimination
and consideration we can get. The following theorem summarizes the results.
Theorem 2. Suppose c is CAF then
• x is revealed to be preferred to y iff xPR y.
• x is revealed to be non-eliminated in attribute i in set S iff x satisfies RNi in set S.
• x is revealed to be considered in set S iff x satisfies RNi in set S for all i ∈ {1, ..., k}.
The proof of the theorem is in Appendix B. The theorem above provides a definitive
answer to what we can confidently say about the consideration set and the preference of
the DM by looking at her choice if the choice is CAF. The example below demonstrates
revealed consideration and preference.
Example 7. Suppose that Z = {x, y, z}, there are two attributes and the attribute rankings are given in Figure 1. Suppose the DM has the following (unobservable) preference,
x z y and uses the following choice procedure. She always considers two alternatives
and she does this by eliminating the worst |S| − 2 alternatives in attribute 1 if |S| ≥ 3.
Then she maximizes her preference in the corresponding consideration set. The data
corresponding to this example is in Figure 1.
It is easy to see that this choice is consistent with CAF. Table 1 lists the revealed
preference, revealed non-elimination, revealed consideration and the actual consideration
sets. Note that Ni (S) corresponds to the revealed non-elimination by attribute i.
Lets look at the set {x, z}. As the choice is x, we know that x trivially satisfies
RN1 and RN2 . As z ranks higher than x in attribute 1 this implies that z also satisfies
RN1 . Finally, as z is chosen in {x, y, z} and as we go from {x, y, z} to {x, z} we are
removing an alternative above z in attribute 2, z also satisfies RN2 in set {x, z}. This
9
S
{x, y}
{x, z}
{y,z}
{x, y, z}
xz
Revealed Preference
Revealed N1 (S)
{x, y}
{x, z}
{z}
{z}
Revealed N2 (S)
{x}
{x, z}
{y, z}
{x, y, z}
Revealed Consideration
{x}
{x, z}
{z}
{z}
Actual Γ(S)
{x, y}
{x, z}
{y, z}
{y, z}
Table 1: Revealed Preference and Consideration for Example 7
is all the revealed non-elimination we can get in this set. Revealed consideration follows
by Theorem 2. Furthermore as x is chosen and z satisfies RN1 and RN2 , by Definition
4 we have xP z and by Theorem 2, x is revealed to be preferred to z. It turns out that
this is the only revealed preference we can get in this example. A similar exercise can be
repeated to get every cell of the table above.
Note that for each set the set of alternatives that are revealed to be considered is a
subset of the actual consideration set. This will always be the case as revealed consideration includes only those alternatives that are included in the consideration set of every
possible representation. Furthermore, in this example the revealed preference is incomplete, which implies that there are multiple preferences that can rationalize the choice.
For example, the actual consideration set and the preference of the DM would rationalize
the choice, but also the following would: x y z, E2 (S) = {f ∈ S|f2 ≤ 0} = ∅ for
all S, E1 (S) = {f ∈ S|f1 ≤ 0} = ∅ for S ∈ {{x, z}, {x, y}}, E1 (S) = {f ∈ S|f1 ≤ y1 }
for S ∈ {{x, y, z}, {y, z}} , leading to Γ(xyz) = Γ(yz) = {z}, Γ(xy) = {x, y} and
Γ(xz) = {x, z}.
An example that is not consistent with CAF is available in the next section.
3.2
Choice Anomalies and CAF
The attraction and compromise effects are two of the most consistently observed violations
of WARP. The attraction effect has first been demonstrated by Huber et al. (1982), it
refers to the observation that inclusion of an asymmetrically dominated alternative may
lead to a shift in choice in favor of the dominant alternative. Compromise effect has first
been demonstrated by Simonson (1989), it refers to the observation that inclusion of an
extreme alternative may lead to a shift in choice in favor of an intermediate alternative.
A typical data that corresponds to these effects can be found in Figure 2.
CAF is able to explain both of these as resulting from the same choice procedure.
First, lets look at the data corresponding to the attraction effect. This data can be
10
Attribute 2
Attribute 2
x
x
y
y
z
z
Attribute 1
c(xy) = c(xz) = x, c(yz) = y, c(xyz) = y
Attribute 1
c(xy) = c(xz) = x, c(yz) = z, c(xyz) = y
Figure 2: Data corresponding to the attraction effect (on the left) and the compromise
effect (on the right)
rationalized with the following preference, x y z and the following elimination sets,
Ei (S) = ∅ for i = 1, 2 if |S| < 3, E1 (xyz) = x and E2 (xyz) = z. The elimination sets
satisfy the restrictions and c maximizes the preference in the corresponding consideration
set. The compromise effect can be rationalized with x z y and with the same
elimination sets.
The idea is simple, in the binary choice between x and y nothing is eliminated, so
the DM maximizes her preference in this set. In the attraction effect, when we include
z, it makes x look relatively poor on attribute 1, which leads x to be eliminated and y is
chosen. In the compromise effect, when faced with all three alternatives, she eliminates
the extremes, which leads to the choice of the intermediate one.
It is no use for a model to explain such anomalies if it does not put reasonable
restrictions on what is consistent with the model. We will now see that neither the
opposite of attraction nor the opposite of compromise can be explained with CAF. So,
assume that c(xy) = y and c(xyz) = x for both of the set of alternatives given in Figure
2.
The following argument shows that this is not compatible with CAF. First note that
y satisfies RN1 and RN2 in the set {x, y} trivially, as it is chosen. As we go from {x, y} to
{x, y, z} we are adding an alternative that is worse than y in attribute 2. So, y satisfies
RN2 in {x, y, z}. As x is chosen in {x, y, z} and as x ranks lower than y in attribute
1, y also satisfies RN1 in {x, y, z}. Therefore y is revealed to be considered in {x, y, z}
implying xP y.
As c(xyz) = x we know that x satisfies RN1 in {x, y, z}, but as we go to {x, y} we are
removing an alternative that ranks better than x in attribute 1. So, x satisfies RN1 in
{x, y}. As x ranks higher than y in attribute 2 and y is chosen in {x, y}, x also satisfies
11
RN2 in {x, y}. Hence, x is revealed to be considered in {x, y}. As c(xy) = y this implies
yP x. But then we have xP yP x.
Attraction or Compromise9
4
The previous section has shown that CAF provides an explanation for the attraction and
compromise effects. Given this feature of the model, one might be interested in how these
effects generalize to more than three alternatives and whether a class of attribute filters
can be characterized with such generalizations.
To answer these questions, in this section I will take an axiomatic approach and come
up with an independent model that only allows for choice anomalies in the spirit of the
attraction and compromise effects. I will show that this model is completely characterized
by a particularly simple class of CAF.
4.1
The Model
Throughout this section I will assume that there are only two attributes, i.e. Z ⊂ R2++ .
Axiom 1. A choice function c satisfies Axiom 1 if whenever c(S) = x and c(S ∪ z) = y,
where y 6= x and y 6= z we have either
1. y D z and x 6D z or
2. xi > yi > zi and zj > yj > xj , where i 6= j, i, j ∈ {1, 2}
The idea is that if choice satisfies Axiom 1 then any choice change will be either in
favor of a dominant alternative when an asymmetrically dominated alternative is added
to the set (1) or in favor of an intermediate alternative (2). Hence every choice anomaly
is either in the spirit of the attraction effect or the compromise effect.
Axiom 2. A choice function c satisfies Axiom 2 if whenever c(S) = x, c(S ∪ z) = y,
where y D z , x 6D z and if there exists z 0 ∈ S with z 0 D z then either
• y D z 0 or
• xi > yi > zi0 and zj0 > yj > xj , where i 6= j, i, j ∈ {1, 2}
Axiom 2 is a consistency condition on the ‘attraction effect’. It requires that if z leads
to the ‘attraction effect’ in favor of y, but if there is another alternative z 0 that dominates
z then it should be the case that either y dominates z 0 or y benefits from being in between
9
I am grateful to an anonymous referee for suggesting this section.
12
z 0 and the previously chosen alternative x. In other words, y either benefits from being
dominant or from being intermediate. Note that this Axiom has no bite when there are
only 3 alternatives.
Definition 5. A choice function c is AoC if it is monotonic10 and satisfies Axiom 1 and
Axiom 2.
Unlike CAF, AoC takes the axiomatic approach and directly puts restrictions on the
kind of choice anomalies allowed in the data. It mentions neither the preferences nor the
consideration sets and it is far from clear whether it is related to CAF. Surprisingly, it
turns out that AoC is equivalent to a particularly simple class of CAF.
For any set S ⊆ Z, let S nd ⊆ S denote the set of undominated alternatives in S, i.e.
S nd = {z ∈ S| 6 ∃z 0 ∈ S with z 0 D z}. CAF with total elimination (CAFTE) is CAF
when Γ(S)nd includes a single alternative:
Definition 6. (CAF T E) c is CAFTE if c is CAF and |Γ(S)nd | = 1 for all S ⊆ Z.
If choice is CAFTE then choice is CAF and every consideration set includes a single
undominated alternative. An example of data that is CAFTE is the data corresponding
to the attraction and compromise effects (see Figure 2). Indeed, the compromise effect
can be rationalized with Ei (S) = {z 0 ∈ S|zi0 ≤ c(S)i −} for i ∈ {1, 2} and all S ⊆ Z. The
attraction effect can be rationalized with Ei (S) = {z 0 ∈ S|zi0 ≤ c(S)i − } for i ∈ {1, 2}
and for S 6= {x, y, z}, E1 (xyz) = {z 0 ∈ S|z10 ≤ z1 −} and E2 (xyz) = {z 0 ∈ S|z20 ≤ y2 −}.
These elimination sets lead to an attribute filter in which Γ(S) = c(S) for all S ⊆ Z.
And hence the choice is CAFTE.
Furthermore, it is immediate that the data corresponding to the attraction and compromise effects is AoC, as all the choice changes are either (1) or (2) from Axiom 1.
For an example of data that is CAF, but not CAFTE see the data in Figure 1. We
have already seen that this is CAF. However, it cannot be rationalized as CAFTE, as
the revealed consideration set for the set {x, z} is {x, z} itself and contains multiple
undominated alternatives.
It is also easy to see that the data in Figure 1 is not AoC, as when we go from {x, z}
to {x, y, z}, y leads to a choice change that is neither consistent with the attraction nor
the compromise effects (so, neither 1) nor 2) in Axiom 1).
It turns out that what we see in these two examples hold in general. In particular,
CAFTE is equivalent to AoC.
Theorem 3. c is AoC iff c is CAFTE.
10
c is monotonic if whenever x D y for x, y ∈ S, we have c(S) 6= y.
13
This result further establishes the strong connection between the attraction and compromise effects and CAF. It shows that not only CAF allows for these effects, while
not allowing for their opposites, but a model that only allows for the attraction and
compromise effects is completely characterized by a simple class of CAF.
5
Related Literature
There is evidence that the decision makers do not consider all of the available alternatives
when making a choice. This phenomena has been extensively studied in the marketing
literature, see Hauser and Wernerfelt (1990), Roberts and Lattin (1991) and Wright and
Barbour (1977) for examples.
There are several models where choice is made as a result of a two-stage procedure,
first stage of which can be interpreted as the formation of the consideration set. Manzini
and Mariotti’s (2007) rational shortlist method considers sequential maximization of several rationales. In Cherepanov et al.’s (2013) rationalization, the DM chooses the best
alternative among the ones she can rationalize. In Manzini and Mariotti (2012) the DM
categorizes the alternatives, and given a set she maximizes her preference among the ones
in the ‘winning categories’. These models are closely connected to each other, as their
characterization involves similar axioms. In particular, a condition called weak WARP
is necessary for the choice to be consistent with any of these models. In Section 5.2, we
will see that CAF might violate weak WARP.
Masatlioglu et al. (2012) and Lleras et al. (2010) are more closely connected to
this work. Just like this paper, they assume that the DM forms a consideration set and
maximizes her stable preference on this consideration set. The model considered here
and these differ on the properties of the consideration sets involved. Masatlioglu et al.
(2012) consider attention filters, which require that if an alternative is not considered in
a given set then removal of this alternative doesn’t affect the consideration set. Whereas
Lleras et al. (2010) consider consideration filters, which require that if an alternative is
considered in a set then it should be considered in every subset of this set including the
alternative. In Section 5.2, I show that CAF is independent of both of these models.
Caplin and Dean (2011), Caplin et al. (2011) and Masatlioglu and Nakajima (2013)
look at the cases in which the consideration set is formed as a result of search through
the set of alternatives. Both of these models look at data that is richer than choice data,
in particular Caplin and Dean (2011) and Caplin et al. (2011) consider ‘choice process
data’, which includes how the choice changes through the process of search. The data
Masatlioglu and Nakajima (2013) considers includes the starting point of search. Both
the motivation and the empirical implications of these models are quite different from
14
CAF.
De Clippel and Eliaz (2012) model choice as the cooperative solution to a bargaining
problem among the different selves of the individual. In their model, each attribute
ranking is interpreted as the preference of one self. Each self assigns each alternative a
score equal to the number of alternatives in the lower contour set of that alternative and
the solution selects the alternative whose minimum score is highest. Hence, they do not
model the choice as a two step procedure and there is no underlying single preference
ordering that represents the preference of the DM. As with CAF, their model can also
explain both the attraction and compromise effects.
Another model that is able to explain both the attraction and compromise effects is
Natenzon (2015), where the author takes a fundamentally different approach and shows
that these effects can arise from Bayesian updating.
Ok et al. (2015) is able to explain the attraction effect, but not the compromise
effect. In their explanation, in the attraction effect the dominated alternative acts as
a reference for the agent and makes the dominant alternative more attractive. One
important distinction between their model and mine is that they use standard choice
data, whereas I use data that includes observable attributes. The approaches should be
seen as complementary as the former has the advantage of identifying the attraction effect
and the attributes when they are subjective, while the latter can make firmer predictions
when the attributes are objective (for instance by ruling out the opposites of these effects).
Aside from these Lombardi (2009) has a model that explains the attraction effect. In
this model, the DM first eliminates the dominated alternatives according to a fixed, but
possibly incomplete relation. And from the remaining alternatives, she eliminates the
ones whose lower counter set is strictly contained in another option’s.
Finally, Gerasimou (2012) also has a model that incorporates the attraction effect.
In this model the DM again eliminates the dominated alternatives according to a partial
order, but from the remaining set she chooses the ones that are at least preferred to some
other alternative.
Mandler et al. (2012) study a choice procedure in which the DM sequentially goes
through a checklist of properties, where at each stage she eliminates the alternatives
that do not have the property. They show that this procedure has close connections to
utility maximization. Unlike in this paper, the properties are unobservable and there is
no underlying preference that the DM maximizes.
Tversky (1972) considers a probabilistic choice model in which the alternatives are
characterized by a set of aspects, at each stage an aspect is selected and all the alternatives
that do not have this aspect are eliminated. This continues until all, but one of the
alternatives are eliminated. Unlike this model, this is a probabilistic model and the DM
15
does not have a stable preference. Furthermore, unlike Tversky (1972), the data I consider
here allows for the movement of a threshold relative to the set under consideration.
Manzini and Mariotti (2014) consider a stochastic model of consideration set formation in which the DM considers each alternative with a certain unobservable probability
and then maximizes her preference among the alternatives she considers. In their model
the elimination is done stochastically given set-independent probabilities, whereas in CAF
elimination is deterministic and depends on the choice set.
Aside from these there are applied theory papers that study the effects of limited
consideration on markets. Eliaz and Spiegler (2011a) provides one of the earliest applications, where the firms can use costly marketing devices to influence the set of alternatives
considered by the consumer. In a similar vein, Eliaz and Spiegler (2011b) consider a
model in which products are used to influence what the consumer pays attention to.
5.1
Consideration Sets
In this section, I will compare the attribute filters to the consideration sets studied in the
literature. I will make the comparison to attention filters (Masatlioglu et al. (2012)) and
consideration filters (Lleras et al. (2010)).
We say that a consideration set is an attention filter if when an alternative is not
considered then the removal of this alternative does not affect the consideration set
(Masatlioglu et al. (2012)). Formally Γ is an attention filter if for any S, Γ(S) = Γ(S \ x)
whenever x ∈
/ Γ(S). The corresponding choice procedure is called choice with limited
attention (CLA).
We say that a consideration set is a consideration filter if when an alternative is
considered in set S then it will be considered in every subset of S that includes this
alternative (Lleras et al. (2010)). Formally, Γ is a consideration filter if x ∈ S ⊆ T and
x ∈ Γ(T ) then we have that x ∈ Γ(S). The corresponding choice procedure is called
choice with limited consideration (CLC).
Remark 1. Attribute filters are independent of attention filters and consideration filters.
To show the remark, consider Example 3 in Section 2.2 above. Suppose that the DM
eliminates the n most expensive alternatives. This is not an attention filter as when
the nth most expensive alternative is removed from the set an alternative that was not
originally eliminated will be eliminated in the set we obtain. The same argument also
shows that it is not a consideration filter either, as an alternative is considered in a set,
but not considered in one of its subsets. Example 8 below is an example of an attention
and consideration filter that is not an attribute filter.
16
Attribute 2
t
z
y
x
Attribute 1
Figure 3: Attribute rankings for Example 9
Example 8. (Top n on each attribute) The DM considers the best n alternatives
on each attribute ranking. It is easy to see that this is both an attention filter and a
consideration filter. To see that it is not an attribute filter consider the following example.
Z = {a, b, c}, there are two attributes K = {1, 2}, the rankings are a1 > b1 > c1 and
c2 > b2 > a2 . Suppose the DM considers the top alternative on each ranking, then
Γ(abc) = {a, c}. If this is an attribute filter then b is eliminated either by attribute 1
or 2. But if b is eliminated by attribute 1 then by condition (A) c is also eliminated.
Similarly, if b is eliminated by attribute 2 then by condition (A) a is also eliminated. So,
this cannot be an attribute filter. And this proves Remark 1.
5.2
Comparison to Other Models
We have seen that attribute filters are independent of both attention and consideration
filters, but the question of whether the models are independent still remains.
Here, I will show that they are independent. The examples also show independence
of CAF from Rationalization (Cherepanov et al. (2013)) and Categorize then Choose
(Manzini and Mariotti (2012)) .11
Example 9. Suppose that Z = {x, y, z, t}, there are two attributes and the attribute
rankings are given in Figure 3. Suppose the DM has the following preference, x y t z and uses the following choice procedure. In a binary set she maximizes her
preference, but given a set containing three or more alternatives she first eliminates the
worst in each attribute ranking then she maximizes her preference in the remaining set.
This choice is consistent with CAF. But it is neither a CLA nor a CLC.
11
Note that the data I consider is richer than the choice data, which all the other models mentioned
here consider.
17
To see that it is not a CLA observe that c(xyz) = y and c(yx) = x, but this implies
that if the choice is a CLA then the DM paid attention to z in {x, y, z} implying y z.
Similarly, c(yzt) = z and c(zt) = t implies z y, a contradiction.
To see that this is not a CLC, observe that c(xyzt) = y and c(yzt) = z. But if the
choice is a CLC then since y is considered in {x, y, z, t}, y is also considered in {y, z, t},
so z y. Similarly, c(yzt) = z and c(yz) = y implies that y z, a contradiction.
This example shows that CAF is not a special case of CLA and CLC. Another widely
used axiom, which is a necessary condition in several models discussed in the Literature
Review is Weak WARP.
Definition 7. Weak WARP
c satisfies weak WARP if x 6= y, {x, y} ⊆ S ⊆ T , c(xy) = c(T ) = x, then c(S) 6= y.
For c to be consistent with the Rational Shortlist Method (Manzini and Mariotti
(2007)), Rationalization (Cherepanov et al. (2013)) and Categorize then Choose (Manzini
and Mariotti (2012)) it has to satisfy Weak WARP. The above example also shows that
c might be consistent with CAF, but violate Weak WARP. To see this, observe that in
the above example we have c(xyzt) = y, c(zy) = y, but c(yzt) = z. So, CAF is also not
a special case of any of these models.
Finally, Weak WARP has no bite when |Z| = 3. CLA and CLC also do not put any
restrictions on c when |Z| = 3. But then, all of these models are consistent with the
‘opposites’ of the attraction and compromise effects studied in Section 3.2, while CAF is
not consistent with these. Hence with Example 9, this observation proves that CAF is
independent from CLA,CLC, Rationalization and Categorize then Choose.
6
Conclusion
It is well known that the decision makers do not always consider all of the alternatives
when they are making a choice. In this paper I introduced choice through attribute
filters, which postulates that attribute rankings can provide a natural way to form the
consideration sets. I have characterized CAF and shown what can be inferred about the
preference and the consideration set of the DM if the choice is consistent with CAF.
Appendix A. Additional Results
Unobservable Attributes
Here I will show the model has no empirical content if the attributes are unobservable.
The definition below assumes that attributes and attribute rankings are unobservable.
18
So let Z be any finite set of alternatives.12
Definition 8. A choice function c is a choice through attribute filters with unobservable
attributes if there exists an integer k > 0, a function f : Z → Rk++ such that (c, f (Z)) is
CAF.
Proposition 1. Any c is a choice through attribute filters with unobservable attributes.
Proof. Take any c. I will show the result by constructing |S| − 1 attributes for each set S.
For any set S and x ∈ {S \c(S)} construct an attribute ranking with the property that the
elements in S rank strictly higher than the elements in Z \ S and x is the alternative that
ranks the lowest among the alternatives in S. In set S the threshold on this attribute will
be at x and in any other set the threshold will be below the lowest ranking alternative
on this attribute. So, x and only x will be eliminated using this attribute ranking in set
S. And nothing will be eliminated by this attribute in any set T 6= S. We can do this
for every alternative x ∈ S \ c(S) and for every set S ∈ Z.
It is easy to see that this leads to an attribute filter and Γ(S) = c(S) for all S ⊆ Z.
Hence, any linear order will rationalize the choice.13
Pre-Attribute Filters
As mentioned in the Introduction a model with Condition (A) alone has little empirical
content as any monotone c will be rationalizable with such a model. Here I will show
this. Note that I am assuming that the attributes and the rankings on these attributes
are observable, as otherwise there is nothing to show by Proposition 1.
Definition 9. Pre-Attribute Filter
Γ is an pre-attribute filter if there exists a threshold tS ∈ Rk for each S ⊆ Z and a
mapping Ei : Z → Z for each i ∈ {1, .., k}, where Ei (S) ⊆ S, Γ(S) = S \ (∪i∈K Ei (S)),
Γ(S) 6= ∅ such that
(A) Ei (S) = {z ∈ S|zi ≤ tSi }
If the choice is rationalizable with a pre-attribute filter I will call it choice through
pre-attribute filters (CPF).
12
In the paper we have taken Z ⊂ Rk . This specification incorporates the attributes and the rankings
on these attributes. Here, Z is any finite set of alternatives, so the attributes are unobservable.
13
Note that monotonicity has no bite, as there are no dominated alternatives.
19
Definition 10. Choice Through Pre-attribute Filters (CPF)
c is a CPF if there exists a strict, complete, transitive and monotone preference relation over Z and a pre-attribute filter Γ such that c(S) is the -best element in Γ(S)
for every S ∈ Z.
Proposition 2. Any monotone c is a CPF.
Proof. Take any monotone c. Let be any complete, strict, monotone and transitive
preference relation. For every set S, let tS = (c(S)1 − , c(S)2 − , ..., c(S)k − ), i.e.
Ei (S) = {z ∈ S|c(S)i − ≥ zi }. The resulting consideration set is a pre-attribute
filter, furthermore everything in S that does not dominate c(S) is eliminated. Since c is
monotone, c(S) maximizes in Γ(S).
Remark 2. This brings the question of how monotonicity affects the main results in the
paper, indeed the main results in the paper is robust in this sense. If we do not assume
that the underlying preferences are monotonic and if we drop the condition that xP y if
x >D y, then all of the main results (Theorems 1 and 2) will still go through with exactly
the same proofs.
Remark 3. Finally, if we do not require the monotonicity of the underlying preferences,
then for this particular result we will have that any c will be a pre-attribute filter. The
elimination sets in the proof will stay the same, but the preferences will have the peculiar
property that that x y whenever y >D x.
Allowing for Indifference
In this section I will allow for indifference in the preference of the DM and in the attribute
rankings. We will see that results go through with slight modification. As these are
changes in the fundamentals I will quickly go through the data and choice procedure
again.
Z ⊂ Rk++ is a finite set of alternatives, Z denotes the set of all nonempty subsets of
Z. Now it can be the case that zi = xi for some z, x ∈ Z with z 6= x and some attribute
i.
A choice correspondence is a mapping c : Z → Z such that c(S) ⊆ S for all S ∈ Z. It
is interpreted as the choice of the DM from each set. It will be assumed that Z and c is
observable.
Definition 11. Γ is an attribute filter if there exists a threshold tS ∈ Rk for each S ⊆ Z
and a mapping Ei : Z → Z for each i ∈ {1, .., k}, where Ei (S) ⊆ S, Γ(S) = S \
(∪i∈K Ei (S)), Γ(S) 6= ∅ such that
20
(A) Ei (S) = {z ∈ S|zi ≤ tSi }
(B) If F = S ∪ x where x ∈ Z and xi ≥ tSi then xi + ≥ tFi ≥ tSi
(C) If F = S ∪ x where x ∈ Z and xi ≤ tSi then xi − ≤ tFi ≤ tSi
We say that x D y if xi ≥ yi for all i ∈ {1, 2, ..., k}. A preference relation is
monotone if whenever x D y then x y.
Here is the definition of CAF.
Definition 12. Choice Through Attribute Filters (CAF)
A choice correspondence c is a choice through attribute filters (CAF) if there exists a
complete, transitive, monotone and reflexive preference relation over Z and an attribute
filter Γ such that c(S) = {x ∈ Γ(S)|x y for every y ∈ Γ(S)}
RNi needs a very slight modification to reflect the changes.
Condition 2. (RNi ) y ∈ T satisfies RNi for attribute i in set T if there exists z ∈ T
and a set S including y and z such that z ∈ c(S), yi ≥ zi and
• If x ∈ T and xi ≥ yi , then x ∈ S.
• If x ∈ S and zi ≥ xi , then x ∈ T .
Exactly the same argument following the original definition of RNi shows here that
whenever choice is CAF and x satisfies RNi in set S then x is revealed to be non-eliminated
by attribute i.
If c is CAF and if y is revealed to be non-eliminated by attribute i in set T for all
attributes i then y is revealed to be considered in set T . Which means that if x ∈ c(T )
then x is revealed to be preferred to y. Furthermore if y ∈
/ c(T ) then x is revealed to be
strictly preferred to y. Lets define the relations P and S with these observations and by
incorporating monotonicity. Let x D y if x D y, but y 6D x.
Definition 13. xSy if x D y or if there exists a set T such that y ∈
/ c(T ), x ∈ c(T )
and y satisfies RNi in set T for all i ∈ {1, ..., k}. xRy if x D y or if there exists a set
T , x ∈ c(T ) and y satisfies RNi in set T for all i ∈ {1, ..., k}.
Note that S is the asymmetric part of R. We say that R satisfies only weak cycles
(OWC) if given any x1 , x2 , ...., xn ∈ Z whenever x1 Rx2 R...xn Rx1 , for no j = 1, ..., n − 1
can it be the case that xj Sxj+1 nor can it be the case that xn Sx1 . As the underlying
preference is transitive OWC is a necessary condition for c to be CAF. The following
theorem states that it is also sufficient.
21
Theorem 4. c is CAF iff R satisfies OWC.
Proof. Necessity already shown, for sufficiency observe that as R satisfies OWC, there
exists a complete preorder extension of R. Let R be one such extension, note that R
is transitive, reflexive, complete and monotone. The rest of the proof follows the same
lines as the proof of Theorem 1, so I will not be providing it here.
Appendix B. Proofs
Attribute Filters for Examples 1 to 6
Here I will formally set the attribute filters of the examples discussed in Section 2.
Example 1. (Rational Choice) For each S, tS = (0, 0, ..., 0), i.e. Ei (S) = {z ∈ Z|zi ≤
0} = ∅ for all i ∈ {1, ..., k} and S ∈ Z.
Example 2. (Top n) Suppose the DM considers the top n alternatives in attribute
1. Let z n+1 (S) be the (n + 1)th best alternative in set S according to attribute 1 if
|S| ≥ n + 1, otherwise let z n+1 (S) = 0. Then tS = (z1n+1 (S), 0, ..., 0).
Example 3. (Eliminate Worst n) Suppose the DM eliminates the worst n alternatives
in attribute 1. Let ẑ n (S) be the (n)th worst alternative in set S according to attribute
1 if the set contains more than n alternatives, otherwise it is the second best alternative
according to attribute 1. Then for each S, we have tS = (ẑ1n (S), 0, 0, ..., 0)
Example 4. (Eliminate Worst n on each attribute) Let zil (S) be the value of
the lth worst alternative in attribute i in set S if l > 0, otherwise zil (S) = 0. Let
tS (l) = (z1l (S), z2l (S), ...., zkl (S)). Let l∗ be the largest integer l smaller or equal to n such
that the consideration set induced by tS (l) is nonempty. Then tS = tS (l∗ ).
Example 5. (Fixed Cutoff ) Without loss of generality let the ordering of the attributes
be {k, k − 1, ..., 1}. Let tS (l) be such that tSi (l) = a∗i for i < l and tSi (l) = 0 for i ≥ l. Let
l∗ be the largest integer l smaller than or equal to k + 1 for which the consideration set
induced by tS (l) is nonempty. Then tS = tS (l∗ ).
Example 6. (Satisficing (Simon (1955))) Assume that the DM searches through the
attribute ranking of attribute 1. Let N1 (S) = {x ∈ S| ∃y ∈ Z such that y1 > x1 and y a∗ }.
Let n1 (S) be the element of N1 (S) with the highest value of attribute 1 if N1 (S) 6= ∅,
otherwise n1 (S) = 0. Then tS = (n1 (S), 0, 0, ..., 0).
Proofs
Lemma 1. Take any T ⊆ Z and x ∈ T
22
(i) Suppose y ∈ T , yi > xi and x satisfies RNi in set T . Then y satisfies RNi in set T .
(ii) If x satisfies RNi in set T and if yi < xi then x satisfies RNi in set {T ∪ y}.
(iii) If x does not satisfy RNi in set T and if yi > xi then x does not satisfy RNi in set
{T ∪ y}
Proof. Suppose x satisfies RNi in set T 0 . Then by definition, there exists z ∈ T 0 and a
set S including x and z such that c(S) = z, xi ≥ zi and
• If y 0 ∈ T 0 and yi0 ≥ xi then y 0 ∈ S, and
• If y 0 ∈ S and zi ≥ yi0 then y 0 ∈ T 0
For part (i), take T 0 = T . Since yi > xi , by the first bullet y ∈ S and furthermore any
y 0 ∈ T such that yi0 ≥ yi also satisfies yi0 ≥ xi and hence by the first bullet y 0 ∈ S. But
then y also satisfies RNi in set T . For part (ii), take T 0 = T . In set {T ∪ y}, the first
bullet point is satisfied as no alternative is added above x, furthermore the second bullet
point is satisfied whether zi ≥ yi or not, hence the result follows. For part (iii), take
T 0 = {T ∪ y} and towards a contradiction assume that x satisfies RNi in this set. But
then since as we go from T 0 to T an alternative above x is removed, we have that x also
satisfies RNi in set T , a contradiction.
Proof of Theorem 1. Necessity already shown, here we will show sufficiency. Suppose
that P is acyclic.
Step 1: Constructing the preference
Since P is acyclic, there exists a linear order extension of P . Let R be a linear
order extension of P . Then R is a strict, transitive, complete and monotone preference
relation.
Step 2: Forming the Elimination Sets and the Consideration Set
For any set S, we will include everything that does not satisfy RNi in set S in Ei (S).
Let N RNi (S) ⊆ S be the set of alternatives that does not satisfy RNi in set S.For
any set S let tSi = 0 if N RNi (S) = ∅, otherwise let tSi = max{z∈N RNi (S)} zi
So, for any set S we have Γ(S) = S \ ∪i∈K Ei (S), where for each i ∈ {1, ..., k} ,
Ei (S) = {z ∈ S|zi ≤ tSi }
By Lemma 1 (i), Ei (S) eliminates an alternative x iff x does not satisfy RNi in set S.
Step 3: Show that c(S) maximizes R in Γ(S)
c(S) satisfies RNi in every attribute i in set S, hence c(S) ∈ Γ(S). Now, we need to
show that c(S) R maximizes Γ(S).
Towards a contradiction, suppose there exists y ∈ S such that y R c(S) and y ∈
/
Ei (S) for all i ∈ {1, ..., k}. But then y satisfies RNi in set S by every attribute. By the
definition of P we have c(S)P y implying c(S) R y, a contradiction.
23
Step 4: Show that Γ is an attribute filter, i.e. elimination sets satisfy (A), (B) and (C)
in Definition 1.
(A) follows from Lemma 1(i).
To show (B), assume that yi > tSi and F = {S ∪ y}. Towards a contradiction assume
tFi < tSi or tFi > yi . In the former, there exists z ∈ F such that z satisfies RNi in set
F and z does not satisfy RNi in set S and yi > zi . This is a contradiction to Lemma
1(iii). In the latter, there exists z ∈ S that satisfies RNi in S with zi > yi , but z does
not satisfy RNi in set F , which is a contradiction to (ii).
A symmetric argument shows (C). Hence, it is indeed an attribute filter.
Step 5: The result
We have established that Γ is an attribute filter and c(S), R -maximizes Γ(S). Therefore (Γ, R ) rationalizes the choice.
Proof of Theorem 2. We already showed that if xPR y then x is revealed to be preferred
to y. For the other way, suppose that it is not the case that xPR y, then there exists an
extension of PR , say R , that ranks y over x. The proof of Theorem 1 shows that there
exists an attribute filter ΓR such that (ΓR , R ) rationalizes the choice as CAF.
We have also shown that if x satisfies RNi then x is revealed to be non-eliminated by
attribute i. For the other way suppose that x does not satisfy RNi in set S, again the
proof of Theorem 1 shows that there exists elimination sets that rationalizes the choice
and includes x in Ei (S). Revealed consideration follows the same argument.
Lemma 2. If c is CAFTE then c is AoC.
Proof. Suppose c is CAFTE. Take any S ⊆ Z, suppose c(S) = x 6= c(S ∪ z) = y 6= z.
Since c is CAF, it is monotone, so without loss of generality assume x2 > y2 and y1 > x1 .
Trivially, we have y ∈
/ E2 (S ∪ z) and since c is CAFTE, we have y ∈ E2 (S). But then
by (B) and (C) in Definition 1, z2 < y2 . Similarly x ∈
/ E1 (S) and since c is CAFTE,
we have x ∈ E1 (S ∪ z). But then by (B) and (C) in Definition 1, z1 > x1 . These two
observations show that Axiom 1 is satisfied.
We need to show Axiom 2. So, suppose y D z and there exists z 0 ∈ S with z 0 D z.
Assume that y 6D z 0 , we will show that y is in between z 0 and x. First suppose z20 > y2 ,
but then since c is CAFTE we have z 0 ∈ E1 (S ∪ z). Also since x ∈
/ E1 (S), by (A),
z0 ∈
/ E1 (S).14 By (B) and (C) this implies that z1 > z10 , a contradiction. So, y2 > z20 .
Since y 6D z 0 , we also have z10 > y1 . But then, z10 > y1 > x1 and x2 > y2 > z20 .
14
This is because z10 > x1 . And that is because by Axiom 1 (which is shown to hold above) x 6D z
(so z1 > x1 ), but z 0 D z.
24
Lemma 3. Suppose c is AoC. If c(S) = x and T ⊆ Z is obtained from S by removing
alternatives that rank better than x in attribute i, then xi ≥ c(T )i .
Proof. Towards a contradiction suppose c(T )i > xi . As we go from T to S we are adding
alternatives that rank better than x in attribute i. Lets assume that one by one we start
adding these alternatives. Let z be the first alternative that leads to a choice change and
F be the resulting set. Then either c(F ) = z or the choice change is due to 1) in Axiom
1 or 2) in Axiom 1, but in all of these cases c(F )i > xi , this is because both zi > xi and
c(T )i > xi . But then we can continue adding alternatives until we reach the set S and
by the same argument the choice will never change in favor of x. A contradiction.
Proof of Theorem 3. By Lemma 2 above, if c is CAFTE then c is AoC. Here, we
will show the other way. So, suppose that c is AoC. Let be any monotone preference
relation and let the elimination sets be defined as in the proof of Theorem 1. From
the proof of Theorem 1, we already know that these lead to an attribute filter. So, all
we need to show is that for any S ⊆ Z, the resulting consideration set Γ(S) satisfies
|Γ(S)nd | = 1.15
Towards a contradiction, suppose not. So, there exists T ⊆ Z with c(T ) = x, but
there exists y ∈ Γ(T ), y 6= x that is not dominated by x. Since the choice is AoC, x is
also not dominated by y, so without loss of generality assume x2 > y2 and y1 > x1 . Then
y satisfies RN2 in set T . Then, by the definition of RN2 , there exists z ∈ T and a set S
including y and z such that c(S) = z, y2 ≥ z2 and
• If t ∈ T and t2 ≥ y2 , then t ∈ S.
• If t ∈ S and z2 ≥ t2 , then t ∈ T .
Let F be the set obtained from S by removing everything that is not in T and that
ranks above z in attribute 2. By Lemma 3, z2 ≥ c(F )2 . Furthermore since c(F ) cannot
be dominated, we have c(F )1 ≥ y1 .
Set T is obtained from F by adding alternatives that rank worse than y in attribute
2. Lets divide this set into two: (T \ F )D , which includes all the alternatives in (T \ F )
that are dominated by x and (T \ F )U , which includes the rest.
Note that everything in (T \ F )D is also dominated by y. Lets start adding the
alternatives in (T \ F )D to set F in any order , let z1 be the first alternative that leads
to a choice change and let Z1 be the corresponding set. As z1 is dominated, the choice
cannot be z1 . If the choice change is due to 1) in Axiom 1, then c(Z1 )1 ≥ y1 . This is
because of Axiom 2, as both x and y dominate z1 , but y is in between c(F ) and x, x or
15
Note that if this holds, then maximization of the preference trivially holds.
25
anything that ranks worse than y in attribute 1 cannot be chosen. Finally, if the choice
change is due to 2) in Axiom 1, then c(F )1 > c(Z1 )1 > z1 and z2 > c(Z1 )2 > c(F )2 . Note
that as c(Z1 ) cannot be dominated by y, we also have c(Z1 )1 > y1 . So, in either case
c(Z1 )1 ≥ y1 .
Now, we can continue adding the remaining alternatives in (T \ F )D in any order,
whenever there is a choice change, the same argument shows that the chosen alternative
ranks at least as good as y in attribute 1, but then c(F ∪ (T \ F )D )1 ≥ y1 .
Note that everything in (T \ F )U ranks better than x in attribute 1, so by Lemma 3,
c(T \ (T \ F )U )1 ≤ x1 .
But then we have c(T \ (T \ F )U )1 ≤ x1 < y1 ≤ c(F ∪ (T \ F )D )1 = c(T \ (T \ F )U )1 .
The last inequality is because (T \ (T \ F )U ) = (F ∪ (T \ F )D ). A contradiction.
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