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Article Excerpt
The authors propose two variants of lexicographic preference rules. They obtain the necessary and sufficient conditions under which a linear utility function represents a standard lexicographic rule, and each of the proposed variants, over a set of discrete attributes. They then: (i) characterize the measurement properties of the parameters in the representations; (ii) propose a nonmetric procedure for inferring each lexicographic rule from pairwise comparisons of multiattribute alternatives; (iii) describe a method for distinguishing among different lexicographic rules, and between lexicographic and linear preference models; and (iv) suggest how individual lexicographic rules can be combined to describe hierarchical market structures. The authors illustrate each of these aspects using data on personal-computer preferences. They find that two-thirds of the subjects in the sample use some kind of lexicographic rule. In contrast, only one in five subjects use a standard lexicographic rule. This suggests that lexicographic rules are more widely used by consumers than one might have thought in the absence of the lexicographic variants described in the paper. The authors report a simulation assessing the ability of the proposed inference procedure to distinguish among alternative lexicographic models, and between linear-compensatory and lexicographic models.
Key words: lexicographic preferences; noncompensatory preference models; linear models; optimization techniques; greedy algorithm; approximation algorithms; utility theory; conjoint analysis; hierarchical clustering; market segmentation; hierarchical market structures
1. Introduction
A lexicographic rule orders alternatives over attributes in the same way that a dictionary orders words over letters. A consumer using the rule evaluates alternatives first on the most important attribute, and if there are ties, on the secondmost important attribute, and so forth. For example, a person buying a personal computer (PC) displays lexicographic preferences if he or she strictly prefers Microsoft Windows to any other operating system; among Windows-based systems, he or she always prefers PCs equipped with the Intel's latest microprocessor; and then uses, in sequence, brand name, memory, and hard-disk space to break ties among still-tied alternatives.
A lexicographic rule only orders alternatives. It therefore does not require that consumers make, or be able to make, cardinal (interval-scaled) judgments. However, it demands stronger judgments than are necessary for binary classifications, such as those obtained using conjunctive, disjunctive, or subset-conjunctive rules (Gilbride and Allenby 2004, Jedidi and Kohli 2005).
There is good evidence that people use lexicographic rules. Drolet and Luce (2004) note that consumers use them when they have emotional reasons to avoid trade-offs. Slovic (1975) reports the use of lexicographic rules for breaking ties among equally valued alternatives. Tversky et al. (1988) examine the rules that consumers use for choice and matching tasks involving public policies, job applicants, and benefit plans. Although each alternative in their study is described using only two attributes, they find that choice is more lexicographic than matching. Yee et al. (2007) report that approximately two-thirds of their subjects use lexicographic rules for evaluating Smart-Phones. A growing literature in psychology also documents the use of these rules in the formation of judgments over perceptual cues (e.g., Gigerenzer et al. 1991; Martignon and Hoffrage 1999, 2002; Broder 2000). For more evidence of, and details about, the use of lexicographic rules in consumer research, we refer the reader to Colman and Stirk (1999), Dhar and Nowlis (1999), Roedder-John (1999), Gonzalez-Vallejo et al. (1996), Kahn and Baron (1995), Westenberg and Koele (1994), Ford et al. (1989), and Payne et al. (1988).
In contrast with the above literature characterizing the use of lexicographic rules, there is only limited work in marketing and consumer research on the mathematical representation, inference, and testing of lexicographic preference structures. A result known at least since Debreu (1954) says that it is impossible to construct a utility function representing lexicographic preferences over two or more real-valued attributes. (1) However, lexicographic utility functions can exist over discrete attributes. Martignon and Schmitt (1999) give a numerical sequence for representing lexicographic preferences over binary attributes. Martignon and Hoffrage (2002) prove that a sequence of weights [w.sub.1], ..., [w.sub.n] represents a lexicographic order over binary cues [x.sub.1], ..., [x.sub.n], if [w.sub.i] > [w.sub.i-1] + ... [w.sub.n], for all i = 1, ..., n - 1. In particular, they prove that if [w.sub.i] = [1/2.sup.i], then a lexicographic ordering of cues is identical to the ordering of the scores produced by the function [w.sub.1][x.sub.1] + ... + [w.sub.n][x.sub.n]. Kohli (1999) observes that all number systems are lexicographic, and can be used to represent, among other structures, lexicographic preferences over discrete attributes. He shows that number systems in which the base (radix) changes from one digit to the next can represent alternatives defined over attributes with varying numbers of levels. Substantial theoretical research has also examined conditions under which continuous lexicographic utility functions can exist (e.g., Fishburn 1974, 1975), the possibility of representing such preferences by multiple functions (e.g., Bridges 1983, Chateauneuf 1987, Wakker 1988, Knoblauch 2000), and the formulation of models for probabilistic lexicographic preferences (e.g., Tversky 1972, Manrai and Sinha 1989).
The present paper obtains the general conditions under which a linear model is necessary and sufficient for representing lexicographic preferences over discrete and/or finitely divisible attributes. The representations described by Martignon and Schmitt (1999) and Kohli (1999) are obtained as special cases. We show that these representations--indeed, all representations of lexicographic preferences by a linear model over a finite number of discrete attributes--are not unique, in the sense that they allow a larger set of transformations for the parameter values than are permitted in a linear model representing interval-scaled preferences. We obtain the associated invariance conditions for the parameters. We then introduce two variants of lexicographic preference models. We call these satisficing and binary lexicographic models. (2) Consumers can use the two variants separately or together. A linear utility function continues to be sufficient for representing each of these variants. We describe how each of the lexicographic models are related to each other, and to a linear compensatory model, in a partially nested structure. We propose a method for inferring each of the four lexicographic preference models (standard, satisficing, binary, and binary-satisficing) from ranking or paired-comparisons data. Computationally, the proposed procedure is substantially less demanding than an enumeration of all possible lexicographic orderings of the attributes and levels. We then propose a method for assigning a linear model, or one of the lexicographic models, to a consumer based on a sequence of tests comparing nested models. Finally, we examine how individual-level lexicographic rules can be combined to construct hierarchical clusters (segments) and aggregate market structures. There is a substantial marketing literature on market structure analysis using brand-switching data (e.g., Grover and Srinivasan 1987, Kannan and Wright 1991, Russell and Kamakura 1994, Urban et al. 1984). To our knowledge, the present approach is the first to use conjoint data for constructing aggregate, hierarchical market structures.
Organization of the Paper. Section 2 discusses the representation of lexicographic preferences over discrete attributes by a linear utility function, and discusses the invariance properties of the parameters. Section 3 describes the two variants of lexicographic preferences and their representation by utility functions. Section 4 examines the inference of lexicographic preference structures from paired-comparisons data, which can be directly elicited from consumers or inferred from rankings or ratings of multiattribute alternatives. Section 5 reports empirical tests of the alternative lexicographic models; it also describes the proposed approach for constructing hierarchical clusters (segments) and aggregate market structures using conjoint data. Section 6 presents the results of a simulation designed to assess the accuracy of the proposed algorithm for identifying the lexicographic rules. It examines the accuracy of a procedure for distinguishing among alternative lexicographic rules, and between lexicographic and linear models. It also reports the computational efficiency of the algorithm used for inferring the lexicographic rules.
2. Representation Over Finite Attributes
Let m [greater than or equal to] 2 denote the number of attributes. We assign the integers 1, ..., m, to the attributes in decreasing order of their importance to a given consumer. Let attribute k have [n.sub.k] [greater than or equal to] 2 levels, for all k = 1, ..., m. We arrange the levels of attribute k in increasing preference order, sequentially assigning to them the values
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Let
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
denote the (ordered) set of values assigned to the levels of attribute k.
Let M denote a set of alternatives. Let x and x' denote two alternatives in M. Let [x.sub.k] and [x'.sub.k] denote the kth elements of x and x', respectively. We say that x is lexicographically preferred to x' if there exists some k, 1 [less than or equal to] k [less than or equal to] m, such that
[x.sub.k] > [x'.sub.k] and [x.sub.l] = [x'.sub.l], for all l = 1, ..., k - 1. (1)
That is, x is preferred to x' on the most important attribute for which the two alternatives are not equally preferred.
Let
min [DELTA][x.sub.k] = min{[a.sup.k.sub.j] - [a.sup.k.sub.j-1] | j = 2, ..., [n.sub.k]}
denote the smallest difference in the successive values of [x.sub.k], and let
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
denote the difference between the largest and smallest values of [x.sub.k], for all k = 1, ..., m. The following theorem characterizes the necessary and sufficient conditions under which a linear model represents lexicographic preferences. A proof for the theorem appears in the appendix.
THEOREM 1. Let
u(x) = [[beta].sub.1][x.sub.1] + ... + [[beta.sub.m][x.sub.m], (2)
where [x.sub.k] is the kth-most important attribute, k = 1, ..., m. Then u(x) represents lexicographic preferences over the m attributes if, and only if,
[[beta].sub.k]min[DELTA][x.sub.k] > [m.summation over (j=k+1])][[beta].sub.j]max[DELTA][x.sub.j], for all k = 1, ..., m. (3)
We illustrate the main point of Theorem 1 with an example using m = 3 attributes, each with [n.sub.k] = 3 levels. For each attribute k, we can select any increasing sequence of nonnegative values [a.sup.k.sub.1], [a.sup.k.sub.2], and [a.sup.k.sub.3]. Suppose we set [a.sup.k.sub.1] = [a.sub.1], [a.sup.k.sub.2] = [a.sub.2], and [a.sup.k.sub.3] = [a.sub.3]. Then
min[DELTA][x.sub.k] = min{[a.sub.2] - [a.sub.1], [a.sub.3] - [a.sub.2]} and max[DELTA][x.sub.k] = [a.sub.3] - [a.sub.1].
Let [x.sub.1] denote the most important attribute, [x.sub.2] the secondmost important attribute, and [x.sub.3] the least important attribute. Consider the utility function
u([x.sub.1], [x.sub.2], [x.sub.3]) = [[beta].sub.1][x.sub.1] + [[beta].sub.2][x.sub.2] + [[beta].sub.3][x.sub.3], (4)
where [[beta].sub.k] > and [x.sub.k] [member of] {[a.sub.1], [a.sub.2], [a.sub.3]}, for k = 1,2,3. Theorem 1 says that (4) represents lexicographic preferences if, and only if, the following conditions are satisfied:
[[beta].sub.2]min[DELTA][x.sub.2] > [[beta].sub.3]max[DELTA][x.sub.3]; and (5)
[[beta].sub.1] min [DELTA][x.sub.1] > [[beta].sub.2] max [DELTA][x.sub.2] + [[beta].sub.3] max [[DELTA][x.sub.3]. (6)
Condition (5) requires that the smallest change in utility for attribute 2 be no smaller than the maximum possible change in utility obtained by changing [x.sub.3]. Condition (6) requires that the smallest change in utility for attribute 1 be no smaller than the maximum possible change in utility obtained when both [x.sub.2] and [x.sub.3] are changed at the same time. If satisfied, these conditions ensure that an alternative with a preferred level of attribute 1 has a higher utility value, regardless of the levels of the less-preferred attribute.
Note that the specific values of [a.sub.1], [a.sub.2], [a.sub.3] play no role in the above example. We can, if we wish, choose these values from any increasing sequence of nonnegative, real numbers, and then constrain the [[beta].sub.k] values so that the conditions in (5) and (6)--more generally, the conditions in (3)--are satisfied. For example, suppose we choose [a.sub.1] = 0, [a.sub.2] = 10, and [a.sub.3] = 99 in the above example, where k = [n.sub.k] = 3. Then min [DELTA][x.sub.1] = min [DELTA][x.sub.2] = [a.sub.2] - [a.sub.1] = 10, max [DELTA][x.sub.2] = max [DELTA][x.sub.3] = [a.sub.3] - [a.sub.1] = 99, and so the utility function (4) represents lexicographic preferences if 10[[beta].sub.2] > 99[[beta].sub.3] (i.e., [[beta].sub.2] > 9.9[[beta].sub.3]) and if 10[[beta].sub.1] > 99[[beta].sub.2] + 99[[beta].sub.3] (i.e., [[beta].sub.1] > 9.9([[beta].sub.2] + [[beta].sub.3])). The important thing to note is that this is a weaker relationship between the parameters than is obtained in a linear model representing interval-scaled preferences, where for any fixed measurement scales for [x.sub.k] and [x.sub.1], the ratio [[beta].sub.k]/[[beta].sub.1] must be a constant, for all k, 1 = 1, 2, 3. The reason for this difference, which we discuss in more detail below, is that (2) and (3) specify an ordinal utility function. As a result, the [beta] parameters permit not only multiplicative transformations, which alone are allowed for cardinal (interval-scaled) utility functions, but the larger class of transformations described by (3). We therefore caution against interpreting the values of [[beta].sub.k][x.sub.k] as the part-worths in a conjoint model. Loosely put, one cannot extract the same information from the parameters of an ordinal, lexicographic utility function as one can from the parameters of a cardinal utility function. To further illustrate this point, consider the following two utility functions, each of which represents the same (lexicographic) ordering of alternatives:
u = [x.sub.1]/[n.sub.1] + ... + [x.sub.m]/[n.sub.1] ... [n.sub.m],
where [x.sub.k] [member] {0, ..., [n.sub.k-1]}, for all k = 1, ..., m; (7)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)
The successive, within-attribute values of [[beta].sub.k][x.sub.k] differ in (7) by the same amount, 1/([n.sub.1] ... [n.sub.k]); but they differ in (8) by the unequal amounts, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all j = 1, ..., [n.sub.k]. Thus, a part-worths interpretation for the [[beta].sub.k][x.sub.k] is not appropriate for a lexicographic utility function.
Kohli (1999) notes that (7) represents a number system in which the base (radix) changes from one digit to the next, and gives the example of time measurements (hours, minutes, seconds) to illustrate its use in an actual measurement system. He also shows how lexicographic structures can be represented by wave functions. Martignon and Hoffrage (2002) examine the special case of (7) in which [n.sub.i] = 2, for all i = 1, ..., m. They prove that (7) assigns...
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