Choice, Consideration Sets and Attribute Filters

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1 Choice, Consideration Sets and Attribute Filters Mert Kimya Brown University, Department of Economics, 64 Waterman St, Providence RI USA. October 6, 2015 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 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 and show that by using the information contained in the attributes we can make firmer I am grateful to Mark Dean for guidance, support and encouragement. I would also like to thank Geoffroy de Clippel for helpful comments and suggestions. Address: mert kimya@brown.edu 1

2 predictions and explain certain choice anomalies such as the attraction and compromise effects. 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 threshold is not determined arbitrarily, but instead it depends on the relative ranking of alternatives. An immediate way to incorporate this is the following. 1 Assume that to a set S that includes alternative x, we add alternatives that rank lower than x in attribute i and/or we remove alternatives that rank better than x in attribute i. In the new set, x is better than a larger set of alternatives on attribute i, while it is worse than a smaller set of alternatives. So, we can conclude that x ranks relatively better in attribute i in this new set compared to the original set. If x was not eliminated in the original set because of attribute i then it would be reasonable to require that x should not be eliminated because of attribute i in the set we obtain. To better understand this restriction consider the following example. A university admission committee is looking through a large applicant pool and to reduce the number of applicants to be considered they eliminate every applicant with a score lower than 70/100 on a standardized test. This means that an applicant, say Amy, with a score of 75/100 is not eliminated because of her test score. If we remove from the applicant pool students with a score higher than 75 and/or include students with a score lower than 75 then in the applicant pool we obtain Amy would be ranking relatively higher on the test score dimension. Hence, the condition dictates that Amy should not be eliminated in this new applicant pool because of her test score. I call the choice procedure that satisfy these conditions choice through attribute filters (CAF). 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 An advantage of using multicriteria choice data is that in many situations it allows us to make firmer predictions than the class of existing choice procedures that use standard choice data. For instance, many models from the class of existing choice procedures are completely silent on a domain that includes only three alternatives. These include 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 no empirical content. 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. 2

3 Cherepanov et al. (2013), Lleras et al. (2010), Manzini and Mariotti (2012) and Masatlioglu et al. (2012). 3 This is despite the fact that there are some consistently observed choice anomalies that only involve the existence of three alternatives. Prominent examples of these are the attraction and compromise effects. Furthermore, it is not possible for a model that uses standard choice data to explain these anomalies satisfactorily, as standard choice data does not make a distinction between these effects and the opposites of each of these effects. Take the attraction effect that refers to the tendency of a DM to shift her choice to the dominant alternative when an asymmetrically dominated alternative is added to the choice set (see Section 4 for details). Let the alternatives be x, x and y, where x dominates x. A typical data consistent with the attraction effect is c(x, y) = y and c(x, y, x ) = x. Any model that solely relies on standard choice data and that is consistent with the attraction effect will also be consistent with the following data: c(x, y) = x and c(x, y, x ) = y. The latter can be regarded as the opposite of the attraction effect and it has no empirical support. Any model that claims to explain the attraction effect should at least make a distinction between these two choice data. Hence out of necessity, multicriteria choice data or some other data that is enriched with the information on attributes should be utilitized if we are to understand and model such anomalies. 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 fully characterizes what we can learn about the DM s preference (revealed preference) and the consideration set (revealed consideration) if the choice is CAF. It also characterizes the conditions under which the data is consistent with CAF. The CAF model is consistent with a number of intuitive choice procedures and it is able to explain certain choice anomalies such as the attraction and the compromise effects. The model is independent of several related influential models in the literature, which include choice with limited attention (Masatlioglu et al. (2012)) and choice with limited consideration (Lleras et al. (2010)). I start with the Literature Review. This is followed by 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 show what can be deduced about the preference and the consideration set of the DM if her choice is consistent with CAF. Which is followed by the characterization of the choice procedure. In Section 4, I study the choice anomalies 3 Any observed choice is consistent with these when there are only three alternatives. 3

4 explained by CAF, in particular I look at the attraction and compromise effects which are both consistent with CAF. Section 5 concludes. Appendix A contains additional auxiliary results and Appendix B contains the proofs. 1.1 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 3, we 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 3, 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 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. 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 4

5 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 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. 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. Z is a finite set of alternatives, Z denotes the set of all nonempty subsets of Z. Let K = {1, 2,..., k} denote the set of attributes. For each i K, there exists a linear order i on Z that denotes the ranking of Z on attribute i. 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, c, K and i for i = 1,.., k. 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 complete preorders in attribute rankings. 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 and strict 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. And I will require that the thresholds are not arbitrary, but they depend on the relative rankings of the alternatives on each attribute. 4 In some settings it might be reasonable to assume that the preference is monotonic, i.e. x y whenever x i y for all i K. Remark 3 in Appendix A explains how this assumption would change the results. 5 We can claim that Γ(S) = c(s) for all S. 5

6 Specifically, I will require 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 alternatives that rank worse than x in attribute i or by removing alternatives that rank better than x or by doing both, then x should not be eliminated in set T by attribute i. The idea is that in set T the alternatives that are better than x in attribute i is a subset of the alternatives that are better than x in set S. And the alternatives that are worse than x in attribute i is a superset of the alternatives that are worse than x in set S. Hence, x ranks relatively better in attribute i in set T. As x is not eliminated in set S by attribute i and as its relative ranking is better in set T, it should not be eliminated in set T because of attribute i. The following definition formalizes these restrictions. Let E i (S) denote the set of eliminated alternatives because of attribute i. The consideration set will be given by Γ(S) = S \ ( i K E i (S)). For x S and attribute i K, let UCS(x, i, S) = {y S y i x} and LCS(x, i, S) = {y S x i y}. These are the (strict) upper and lower contour sets of x in set S according to i. Definition 1. Γ is an attribute filter if there exists a mapping E i : Z Z for each i K, where E i (S) S, Γ(S) = S \ ( i K E i (S)), Γ(S) and each E i satisfies the following for any set S (A) If x E i (S) and y S is such that x i y then y E i (S) (B) If x / E i (S) and set T containing x is such that UCS(x, i, S) UCS(x, i, T ) and LCS(x, i, S) LCS(x, i, T ) then x / E i (T ) Condition (A) states that if an alternative x is eliminated because of attribute i then anything that is worse than x in attribute i should also be eliminated. Condition (B) states that if x is not eliminated by attribute i in set S and if as we go from set S to set T the upper contour set of x in attribute i shrinks while the lower contour set expands then x should not be eliminated by attribute i in set T. The implication of dropping this condition will be discussed in Appendix A. An alternative, but equivalent way of thinking of an attribute filter is the following. For each i K, let u i : Z (0, 1) be a utility function corresponding to i, i.e. for x, y Z, u i (x) > u i (y) iff x i y. Then for each set S and attribute i we can find a threshold, a number u S i (0, 1), such that every alternative in S having a u i smaller than this threshold is eliminated by attribute i. Furthermore, if we add an alternative x to a set S that has a utility above the threshold in attribute i (u i (x) > u S i ), then the threshold can only move up, but cannot be higher than the utility of the first alternative ranking above x in set S. Similarly, if we add an alternative x to a set S that has a utility below the threshold in attribute i (u S i > u i (x)), then the threshold can only move down but cannot be lower than the utility of the first alternative ranking below x in set S according to i. The following lemma provides this alternative definition. First, lets extend each u i to include the following two imaginary alternatives, O and O, such that u i (O ) = 1 and u i (O ) = 0 for all i K. These alternatives will be used when there is nothing above or below the alternative we add to a set. For any x Z and S Z let x i (S) be the utility of the first alternative ranking above x in set S in attribute i (if no alternative in S ranks 6

7 above x then x i (S) = 1), i.e. x i (S) = min{u i (z) z {S O }, u i (z) > u i (x)}. Similarly let x i (S) = max{u i (z) z {S O }, u i (z) < u i (x)}. 6 Lemma 1. Γ is an attribute filter iff there exists a mapping E i : Z Z for each i K and u S i (0, 1) for each set S and attribute i such that Γ(S) = S \( i K E i (S)), Γ(S) and (I) E i (S) = {z S u S i u i (z)} (II) If T = S x where x Z and u i (x) u S i then x i (S) > u T i u S i. (III) If T = S x where x Z and u i (x) u S i then u S i u T i > x i (S). The proof of the lemma, as well as all other proofs can be found in Appendix B. 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 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 according to i. If this element is not preferred to a then she also considers the second best element according to 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 would be instructive to 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)). 6 This notation is needed as Condition (B) does not say whether the alternative we put to the set should or should not be eliminated. In particular, when we add an alternative x above the threshold then the threshold might stay below x (x not eliminated) or might go above x (x eliminated). That is why, instead of looking at x we are looking at the alternative above x, that the threshold will definitely stay below this alternative. 7

8 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). 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). Remark 1. Attribute filters are independent of attention filters and consideration filters. To show the remark, consider Example 3 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 7 below is an example of an attention and consideration filter that is not an attribute filter. Example 7. (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 a 1 b 1 c and c 2 b 2 a. 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. 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 and transitive 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 If c is CAF then what can we confidently conclude about the preference and the consideration set of the DM? 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 pairs of preference and attribute filter representing c, {(Γ 1, 1 ), (Γ 2, 2 ),..., (Γ m, 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 Γ j (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. I will start with the identification of the consideration set. 8

9 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 8 Definition 4. x T is revealed to be non-eliminated by attribute i in set T if there exists z T and a set S including x and z such that c(s) = z, x i z or x = z, UCS(x, i, S) UCS(x, i, T ) and LCS(z, i, S) LCS(z, i, T ). Suppose the hypothesis holds. 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 ranks better than z and worse than x in attribute i. Observe that as we go from set S to set F we are just removing alternatives that rank better than z in attribute i, by (B) that means that z is not eliminated in set F, and by (A) x is not eliminated in set F. But then UCS(x, i, F ) UCS(x, i, T ) and LCS(x, i, F ) LCS(x, i, T ) which implies that by condition (B), x is not eliminated in set T 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. And that means if c(t ) y then c(t ) is revealed to be preferred to y. Lets define the relation P with this observation. Definition 5. xp y iff there exists a set T containing y such that c(t ) = x, x y and y is revealed to be non-eliminated by attribute i in set T for all i K. Since the underlying preference is transitive we can conclude that if xp y and yp z then x is revealed to be preferred to z. Which means that if we denote by P R the transitive closure of P, then x is revealed to be preferred to y if xp R y. It turns out that P R is actually all the revealed preference we can get. Theorem 1. Suppose c is CAF then x is revealed to be preferred to y iff xp R y. x is revealed to be considered in set S iff x is revealed to be non-eliminated by attribute i in set S for all i 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. 9

10 S {x, y} {x, z} {y,z} {x, y, z} Revealed Preference x z Revealed N 1 (S) {x, y} {x, z} {z} {z} Revealed N 2 (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 8 Example 8. 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. The table above lists the revealed preference, revealed non-elimination from Definition 4, revealed consideration and the actual consideration sets. Note that N i (S) corresponds to the revealed nonelimination by attribute i. Lets look at the set {x, z}. As the choice is x, we know that x is revealed to be non-eliminated by attribute 1 and attribute 2. As z ranks higher than x in attribute 1 by Definition 4 this trivially implies that z is also revealed to be non-eliminated by attribute 1. Finally, as z is chosen in {x, y, z} it is revealed to be non-eliminated by attribute 2 in set {x, y, z}, but as we go from {x, y, z} to {x, z} the upper contour set of z in attribute 2 shrinks, by Definition 4 this implies that z is revealed to be non-eliminated by attribute 2 in set {x, z}. This is all the revealed non-elimination we can get in this set. Revealed consideration follows by Theorem 1. Furthermore as x is chosen and z is revealed to be non-eliminated by every attribute, by Definition 5 we have xp z and by Theorem 1, x is revealed to be preferred to z. It turns out that this is the only revealed preference we can get from 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, E 2 (S) = for all S, E 1 (xz) = E 1 (xy) =, E 1 (xyz) = {x, y} and E 1 (yz) = {y}, leading to Γ(xyz) = Γ(yz) = {z}, Γ(xy) = {x, y} and Γ(xz) = {x, z}. An example that is not consistent with CAF will follow shortly at the end of this section. We have seen what we can confidently conclude about the preference and the consideration set if the choice is consistent with CAF. Now the question is when can we conclude that the choice is consistent with CAF. The following theorem provides the answer. Theorem 2. c is CAF iff P is acyclic. 10

11 Attribute 2 t z y x Attribute 1 Figure 2: Attribute rankings for Example 9 The proof is in Appendix B, here I will provide a sketch of the proof. We have already seen that the condition is necessary. To show sufficiency, observe that since P is acyclic it can be extended to a complete, transitive and strict preference relation. Then for each attribute i and set S, we can include every alternative that can be eliminated (that is not revealed to be non-eliminated by attribute i in set S) in E i (S). c(s) maximizes in the resulting Γ(S), since for any set S if an alternative x is revealed to be non-eliminated by every attribute we have c(s)p x. To finish the proof all we need to do is to show that the resulting Γ is an attribute filter, for details of this step I refer the reader to the proof. Comparison to Other Models As discussed in the Introduction, Masatlioglu et al. (2012) and Lleras et al. (2010) consider similar models in which the DM chooses the most preferred alternative in her consideration set. The difference is that the consideration set is assumed to be an attention filter in Masatlioglu et al. (2012) and a consideration filter in Lleras et al. (2010) (see Section 2 for the definitions). The corresponding choice procedure is called choice with limited attention (CLA) and choice with limited consideration (CLC), respectively. In Section 2, 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. Example 9 shows that observed choice might be CAF, but neither CLA nor CLC and Example 10 shows that the observed choice might be CLA and CLC while it might fail to be consistent with CAF. The examples also show independence of CAF from Rationalization (Cherepanov et al. (2013)) and Categorize then Choose (Manzini and Mariotti (2012)) 7 Example 9. Suppose that Z = {x, y, z, t}, there are two attributes and the attribute rankings are given in Figure 2. 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. 7 Note that the data I consider is richer than the choice data, which all the other models mentioned here consider. 11

12 Attribute 2 x y z Attribute 1 c(xy) = c(yz) = y and c(xyz) = c(xz) = x Figure 3: Data corresponding to Example 10 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 then if the choice is a CLC then y is considered in {x, y, z, t} implies that y is 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 6. Weak WARP c satisfies weak WARP if x y, {x, y} S T, C(xy) = C(T ) = x, then C(S) 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, Example 10 below shows that most of these models are not a special case of CAF. Example 10. Suppose that Z = {x, y, z}, there are two attributes and the attribute rankings are given in Figure 3. Suppose the DM has the following preference, y x z (unobservable) and uses the following choice procedure. She only considers the top alternative on each attribute ranking and maximizes her preference on the resulting consideration set. The corresponding data is in Figure 3. This choice is not consistent with CAF. To see this, observe that y is revealed to be considered in the sets {x, y} and {y, z}. But as we go from the set {x, y} to the set {x, y, z} the lower contour set of y in attribute 2 expands, which means that y is revealed to be non-eliminated by attribute 2 in set {x, y, z}. Similarly, as we go from the set {y, z} to the set {x, y, z} the lower contour set of y in attribute 1 expands, which means that y is revealed to be non-eliminated by attribute 1 in the set {x, y, z}. Then y is revealed to be considered in {x, y, z}, which implies xp y. 12

13 x is revealed to be non-eliminated by attribute 1 in {x, y, z} and as we go from {x, y, z} to {x, y} the upper contour set of x in attribute 1 shrinks which implies that x is revealed to be non-eliminated by attribute 1 in {x, y}. As c(xy) = y and x is better than y in attribute 2, x is also revealed to be non-eliminated by attribute 2 in {x, y}. Then x is revealed to be considered in {x, y}, which implies yp x. But then xp yp x, hence the choice is not consistent with CAF. Rationalization and Categorize then Choose are completely characterized by Weak WARP, which has no bite when Z = 3, so they are consistent with this example. Similarly, CLA and CLC do not put any restrictions on c when Z = 3, so they are also consistent with this example. Hence with Example 9, this example proves that CAF is independent from CLA,CLC, Rationalization and Categorize then Choose. 4 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. A typical data that corresponds to the attraction effect can be found in Figure 4. 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 the compromise effect can be found in Figure 4. Attribute 2 Attribute 2 x x y y z z c(xy) = c(xz) = x, c(yz) = y, c(xyz) = y Attribute 1 c(xy) = c(xz) = x, c(yz) = z, c(xyz) = y Attribute 1 Figure 4: Data corresponding to the attraction effect (on the left) and the compromise effect (on the right) 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 rationalized with the following preference, x y z and the following elimination sets, E i (S) = for i = 1, 2 if S < 3, E 1 (xyz) = x and E 2 (xyz) = z. The elimination sets satisfy the restrictions and c maximizes the preference in the corresponding consideration set. The idea is simple, in the binary choice between x and y nothing is eliminated, so the DM maximizes her preference in this set. But when we include z, it makes x look relatively poor on attribute 1, which leads x to be eliminated and y is chosen. Now lets look at the compromise effect. This data can be rationalized with the following elimination sets, E i (S) = for i = 1, 2 if S < 3, E 1 (xyz) = x and E 2 (xyz) = z 13

14 and the following preference, x z y. Here the DM chooses the preferred alternative in binary sets, but when faced with all of the alternatives, she eliminates the extreme alternatives, leading to the choice of the intermediate one. It is no use for a model to explain such anomalies if it doesn t put reasonable restrictions on what is consistent with the model. The data that corresponds to the opposites of attraction and compromise effects can be found in Figure 5. We will now see that neither the opposite of attraction nor the opposite of compromise can be explained with CAF. Note that the data does not include the choices from every set, this is because no matter what the choice is in these sets it will be inconsistent with CAF. Attribute 2 Attribute 2 x x y y z z Attribute 1 Attribute 1 c(xy) = y and c(xyz) = x c(xy) = y and c(xyz) = x Figure 5: Data corresponding to the opposite of attraction effect (on the left) and the opposite of compromise effect (on the right) The following argument demonstrates that both the opposite of the attraction effect and the opposite of compromise is not compatible with CAF. First note that y is revealed to be non-eliminated in the set {x, y} and 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 is revealed to be non-eliminated by attribute 2 in {x, y, z}. As x is chosen in {x, y, z} and as x ranks lower than y in attribute 1, y is also revealed to be non-eliminated by attribute 1 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 is revealed to be non-eliminated by attribute 1 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 is revealed to be non-eliminated by attribute 1 in {x, y}. As x ranks higher than y in attribute 2 and y is revealed to be non-eliminated by attribute 2 in {x, y}, x is also revealed to be non-eliminated by attribute 2 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. 5 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. 14

15 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. Definition 7. A choice function c is a choice through attribute filters with unobservable attributes if there exists a complete, transitive and strict preference relation over Z, a set of attributes K = {1,..., k}, a linear order i for each i = 1,...k and an attribute filter Γ such that c(s) is the -best element in Γ(S). Proposition 1. Any c is a choice through attribute filters with unobservable attributes. Proof of Proposition 1. 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. 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 S. We can do this for every alternative x S \ c(s) and for every set S Z. The described procedure leads to the consideration set Γ(S) = c(s) for all S. Take any linear order. Now I will show that (Γ, ) rationalizes c as CAF with unobservable attributes. As Γ(S) = c(s) for all S, c(s) trivially maximizes Γ(S). We just need to show that Γ is indeed an attribute filter. As at each attribute ranking either nothing is eliminated or only the lowest ranking alternative in set S is eliminated it satisfies (A). By construction, if x is eliminated in set F by attribute i, then x is the worst alternative in set F according to i and every alternative that is not in F ranks lower than x according to i. Now suppose that x is not eliminated by attribute i in set S and set T is obtained from set S by adding alternatives that rank worse than x in attribute i and/or removing alternatives that rank better than x in attribute i. x is only eliminated at the attribute ranking where it is the alternative that ranks the worst in the set, this means that x is not eliminated by attribute i in set T if any alternative that ranks worst than x is added to the set. Furthermore x is only eliminated at the attribute ranking and in sets, where every alternative not in the set ranks below x, this means that x is not eliminated by attribute i in set T if any alternative that ranks better than x is removed from the set. So, x is not eliminated by attribute i in set T. This shows (B), therefore Γ is an attribute filter and (, Γ) rationalizes c as CAF with unobservable attributes. Pre-Attribute Filters As mentioned in the Introduction a model with Condition (A) alone has no empirical content as any 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 8. Pre-Attribute Filter Γ is a pre-attribute filter if there exists a mapping E i : Z Z for each i K, where E i (S) S, Γ(S) = S \ ( i K E i (S)), Γ(S) and each E i satisfies the following for any set S 15

16 (A) If x E i (S) and y S is such that x i y then y E i (S) If the choice is rationalizable with a pre-attribute filter I will call it choice through pre-attribute filters (CPF). Definition 9. Choice Through Pre-Attribute Filters (CPF) A choice function c is a choice through pre-attribute filters (CPF) if there exists a complete, transitive and strict preference relation over Z and a pre-attribute filter Γ such that c(s) is the -best element in Γ(S). Proposition 2. Any c is a CPF. Proof of Proposition 2. Take any c. We will start by constructing the preference. Let be any complete, strict and transitive preference relation such that whenever x dominates y (i.e. x i y for all i K) we have y x. For every set S, let E i (S) include everything that ranks below c(s) in attribute i, i.e. E i (S) = {z S c(s) i z}. The resulting consideration set is a pre-attribute filter, furthermore everything in S that does not dominate c(s) is eliminated. Hence, c(s) maximizes in Γ(S). Remark 2. The preference used in the proof has the peculiar property that if x dominates y then y x. In many settings it might be reasonable to assume that the preference of the DM is monotonic, i.e. x y if x i y for all i K. This raises the question of whether the proposition would still be valid if we assume that the preference is monotonic. Lets say that a choice function satisfies monotonicity if whenever x, y S and x dominates y then c(s) y. Then c would be a CPF with monotonic preferences iff c is monotonic. To see this first suppose c is a CPF with monotonic preferences. If x dominates y then x cannot be eliminated without eliminating y and since x is preferred to y, y can never be chosen if x is available. Hence, c satisfies monotonicity. For the other way, suppose c is monotonic. Then we can take any that respects monotonicity and use it in conjunction with the elimination sets provided in the above proof to rationalize the choice as CPF with monotonic preferences. Remark 3. This also raises the question of whether the main results of the paper are robust to the assumption of monotonicity. Indeed they are. If we assume that the DM has monotonic preferences then we have to expand the relation P to include the preferences implied by monotonicity, i.e. xp y if x i y for all i K. With this expanded P both of the main theorems would still be valid. 8 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 is a finite set of alternatives, Z denotes the set of all nonempty subsets of Z. Let K = {1, 2,..., k} denote the set of attributes. For each i K, there exists a complete preorder i on Z that denotes the ranking of Z on attribute i K. 8 I am not providing the proof for this claim as the same proofs in Appendix B would work with the expanded P. 16

17 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, c, K and i for i = 1,.., k is observable. UCS(x, i, S) (LCS(x, i, S)) denotes the upper contour set (lower contour set) of x in set S according to i, i.e. UCS(x, i, S) = {z S z i x}. The definition of an attribute filter below reduces to Definition 1 when there is no indifference in attribute rankings, note that if x i y 9 and x E i (S) then the definition requires y to be in E i (S). 10 Definition 10. Γ is an attribute filter if there exists a mapping E i : Z Z for each i K, where E i (S) S, Γ(S) = S \ ( i K E i (S)), Γ(S) and each E i satisfies the following for any set S (A) If x E i (S) and y S is such that x i y then y E i (S) (B) If x / E i (S) and set T containing x is such that UCS(x, i, S) UCS(x, i, T ) and LCS(x, i, S) LCS(x, i, T ) then x / E i (T ) The DM has a complete, transitive and reflexive preference on Z. Finally, here is the definition of CAF. Definition 11. Choice Through Attribute Filters (CAF) A choice correspondence c is a choice through attribute filters (CAF) if there exists a complete, transitive 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)} I will start with the identification of the consideration set. Definition 12. x T is revealed to be non-eliminated by attribute i in set T if there exists z T and a set S including x such that z c(s), x i z, UCS(x, i, S) UCS(x, i, T ) and LCS(z, i, S) LCS(z, i, T ). Suppose the hypothesis holds. 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 ranks strictly better than z and strictly worse than x in attribute i. Observe that as we go from set S to set F we are just removing alternatives that rank better than z in attribute i, by (B) that means that z is not eliminated in set F, and by (A) x is not eliminated in set F. But then UCS(x, i, F ) UCS(x, i, T ) and LCS(x, i, F ) LCS(x, i, T ) which implies that by condition (B), x is not eliminated in set T 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. Definition 13. xsy iff there exists a set T such that y / c(t ), x c(t ) and y is revealed to be non-eliminated by attribute i in set T for all i K. xry iff there exists a set T, x c(t ) and y is revealed to be non-eliminated by attribute i in set T for all i K. 9 x i y iff x i y and y i x. 10 There are other definitions that would reduce to the original definitions when there is no indifference, which differ in how we treat indifference in attribute rankings. 17