Imperceptible Difference and Categorization:
A Rational Problem Simplification Strategy∗
André d’Almeida Monteiro†
Gávea Investmentos
First version: September, 2003 This version: February, 2006
Abstract
This paper proposes a multidimensional rational preference that incorporates imperceptible difference on a countable alternative set whose elements are described by n
attributes based on the notions of similarity and categorization. Preference invariants
are investigated as well as conditions that assure consistency among multidimensional
similarity and n similarities over the attributes. If the conditions for the existence of the
proposed preference are provided, imperceptible difference induces a rational strategy
to simplify a problem choice: reduction of the alternative set by categorization.
∗
I have benefited from discussions with Dionísio Dias Carneiro, Vinicius Carrasco, Carlos Kubrusly, José
Alexandre Scheinkman and Wei Xiong. The first version of this paper was written when I was visiting the
Bendheim Center for Finance and The Economics Department of Princeton University in 2003.
†
e-mail: [email protected]. Tel: + 55 21 3206-9018
1
1
Introduction
When faced with certain types of decision tasks, the decision maker may be indifferent
among some elements just because they are “very close” to each other. It is not a question
of incompleteness of the preference or lack of decision. He cannot express his strict preference
among them because their differences are not enough to be perceived by him. This is called
imperceptible or just noticeable difference.
The difference between two elements may be imperceptible for the decision maker for
some reasons. One reason is his incapacity to recognize that two objects are different. Some
people, for instance, do not perceive the difference between blue and red. Luce (1956)
provides a classical example: a person is not able to detect the difference caused by one
grain of sugar in a cup of coffee.
Perception accuracy may also vary among people because of expertise (training and
inherit ability). There is no judgement but lack of discrimination capacity or accuracy. MasColell, Whinston and Green (1995) give an example: a person is deciding on the color of his
apartment walls. He may be indifferent between two shades of gray because their difference
is imperceptible for him. However, the choice of these shades makes the difference for an
interior designer. A sommelier is able to recognize the difference between two years of the
same wine, whereas a person who is not used to drink wine probably is not able to distinguish
between them.
Another reason comes from a judgment of merit about the difference motivated by some
source of cost in evaluating a large-number alternative set. In this case, the decision maker
recognizes that two objects are different, but judges their difference as not noteworthy or
negligible. Bernstein (1995) and Swensen (2000) report that some investors allocate funds
among groups of similar assets, instead of among individual assets. These groups are called
styles and this investment process, style investing. These investors’ motivation is their
incapacity to evaluate all the available assets (only in NYSE there are approximately 3,600
2
listed stocks). Proceeding like that, they perceive the difference among assets within the
same style as noteworthy compared with the cost of evaluating all the assets.
Although it seems to be reasonable that decision makers take into consideration the notion
of “very close” in their choices, imperceptible difference brings two problems for a classical
rational preference: the failure of transitiveness and the perception discontinuity at the mark
of “very close”.1 Both are related to the agglomerative nature of imperceptible difference: its
capacity to create chains of elements in which differences between two immediate neighbors
are imperceptible. If the decision maker is not indifferent among all elements in the same
chain, the symmetric part of the preference is not transitive. The mark of “very close”
generates a discontinuity point in the judgment: all of a sudden, after two alternatives having
achieved a certain level of difference, the decision maker is able to distinguish between them.
In addition, this perception discontinuity mark seems to be meaningless in terms of decision
theory.
Luce (1956) was the first to axiomatize imperceptible difference. He introduces the
concept of semiorder. The main idea is that the decision maker judges two elements as
indifferent if their associated utility values are close enough to each other. In other words,
indifference is not transitive.
The main goal of the present paper is to reconcile imperceptible difference and rational
preference by using the notions of similarity and categorization. Similarity here has the
meaning of proximity measured on the alternative attributes. Categorization is the grouping
of objects into non-empty and mutually exclusive categories based on similarity. More precisely, a category is a subset of elements that are more similar to each other than to elements
of other subsets. This definition is crucial for the whole paper. Categorization is a choice
behavior deeply documented and investigated by the psychological literature: the decision
makers exhibit the tendency to simplify a problem through the reduction of the number of
1
A binary relation is said to be a rational preference, or a complete pre-order, if and only if it is complete
and transitive.
3
alternatives by categorizing them.2 It is also in the core of pattern recognition literature.3
The major contribution of the present paper is to show that imperceptible difference can
be incorporate into a rational multidimensional preference on a countable alternative set if
and only if the decision maker perceives his alternatives through categories generated by
similarity. If the alternative set can be partitioned into categories, imperceptible difference
is a transitive relation, and the perception discontinuity mark just captures the disconnectedness existing in the alternative set, that is, the differences among the categories. Let the
proposed preference be called Similarity Based Preference (SBP).
The present paper also contributes in giving choice rationality for certain kinds of categorization. Usually categorization arises to deal with a source of cost that leads the decision
maker to reduce his alternative set by grouping it such as monitoring, research, store, time
and subjective cost. However, the present paper proves that, even in the absence of any
source of cost, the decision maker is allowed to categorize the alternative set and to be indifferent among the elements within the same category. In the sequence, he can take just one
element per category to the following steps of the decision making process. Therefore, it is
shown that imperceptible difference may be able to induce a rational strategy to simplify a
problem choice by reducing the alternative set.
The preference is modeled as follows. There are no probabilities involved in the decision
and the modeling is descriptive. Every element of the alternative set is described by n
attributes and all of them are relevant for the choice and the choice is exhausted by them.
The notion of imperceptible difference is formalized by a binary relation, called similarity.
Similarity is a function of two decision maker’s parameters: the way he measures difference
between elements based on their n attributes (a metric function, d), and his perception
accuracy for differences (a threshold, ε). The decision maker evaluates two elements as
similar if and only if their difference is imperceptible — below his perception accuracy.
2
3
For example, see Medin and Aguilar (1999).
See, for example, Duda and Hart (2002) and Theodoridis and Koutroumbas (1999).
4
According to Armstrong (1950, page 122), “the imperfect powers of discrimination of the
human mind whereby inequalities become recognizable only when of sufficient magnitude.”
Gilboa and Lapson (1995) advocate the existence of such thresholds for physical perception
problems. Rubinstein (1988) applies a threshold to model decision maker perception on
prices and probabilities of a lottery. Here, the threshold will be used in a general decision
problem.
As the modeling is descriptive, the reasons that lead the decision maker to not distinguish
between two elements do not matter and the decision maker’s perception parameters (d and
ε) are taken as given.
SBP puts the question of “very close” in a different place than a semiorder does. In
a semiorder relation, two elements are not distinguishable if their utility values are close
enough. Then, the decision maker evaluates the very-close question in the real line. On the
other hand, under SBP approach, two elements are imperceptible different if they are close
enough based on their attributes. The very-close question is, then, evaluated in the space of
the alternative attributes. There is also a point of primitiveness here: if the decision maker
judges the difference between two elements as imperceptible, he should not be able to attach
different utility values for them.
Fixed the asymmetric part of the preference, the analysis of invariances of SBP for the
decision maker’s perception parameters brings two interesting conclusions. First, two decision makers with different perception of the world (different d and ε) may end up perceiving
exactly the same categories. Second, if the decision maker varies his perception accuracy
by enough, the resulting SBPs form an unique hierarchical structure. That is, in order to
reduce the number of indifference classes, the decision maker can only merge the already
existing indifference classes.
The decision maker deals with the notion of similarity simultaneously judging the differences of the n attributes. Since it is reasonable that he also evaluates similarity on each
attribute, n similarity relations are defined, one on each attribute independently. Assuming
5
that there is no association among similarity relations on the attributes, the decision maker’s
multidimensional judgment and his n judgments over each attribute should be consistent.
Thus, two conditions are required for the system of n + 1 similarity relations to be said
consistent: similarity in all the n attributes should imply multidimensional similarity and
vice-versa. The first condition is due to Gilboa and Lapson (1995). If the system is consistent, the n + 1 perception accuracy thresholds are mutual-dependent and more structure is
required from d.
Nevertheless, if there are channels of association among similarity relations on the attributes (as the correlated similarity proposed by Aizpurua et al., 1991, for example), the
second condition becomes meaningless and there is room only for the first condition. In this
case, SBP is flexible enough to accommodate interdependence among similarity relations on
attributes.
The papers that most relate to the present one, regarding the idea of similarity, are
Rubinstein (1988) and Vila (1997). Rubinstein (1988) defines similarity relations on the
two dimensions of lotteries space independently by using semiorders. Vila (1997) extends
Rubinstein results to a general three-dimensional choice problem.4
The imperceptible-difference-driven categorization can be viewed as a selection process
of the alternative set: only a subset of the alternative set is considered in the next steps of
the decision process. In this sense, this paper relates to Kreps (1979) and Ergin (2003). The
idea of categorization has been explored in the economics literature by some authors such
as Esteban and Ray (1994), Barberis and Shleifer (2003), Keely (2003), Fryer and Jackson
(2004) and Peng and Xiong (2005).
The rest of this paper is divided into five sections. The second section introduces the
basic concepts and builds the necessary and sufficient conditions for the existence of SBP.
The third one explores the connections between imperceptible difference and categorization.
4
Recently, some authors have used the concept of similarity in game theory, such as Camerer, Hisa and
Ho (2002), Chen and Khoroshilov (2003), and Sarin and Vahid (2004).
6
The next section focuses on the invariances of SBP for the parameters of the decision maker’s
perception. The fifth section analyzes the connections among the similarity relations and
preferences in the multidimensional space and on each attribute dimension. The sixth section
concludes the paper. All the proofs are provided in the Appendix.
2
Similarity Based Preference
This section establishes the decision making setup and introduces the basic definitions: difference between elements and sets, perception accuracy, similarity, chains, sub-chains and
group of chains. It also presents the necessary and sufficient conditions for the existence of
a SBP.
The decision maker faces a countable alternative set X. Each x ∈ X is a n-tuple:
x = (x1 , ..., xn ), n ∈ N. The decision maker perceives the difference between a pair of
elements of X through a metric function: d : X → R+ . That is, (X, d) is metric space. It is
a constraint on the way he evaluates differences.
The diameter of a set is defined as the maximum distance between its own elements.
The distance between two sets is the minimum distance between elements belonging to the
different sets. Let X1 and X2 be subsets of a set X.
diam (X1 ) = max d (x, y)
x,y∈X1
d (X1 , X2 ) =
min
x∈X1 ,y∈X2
d (x, y)
The difference between two elements in X is imperceptible for the decision maker if
and only if it is lower than his perception accuracy ε. This concept is formalized by a
binary relation called similarity. Assume that the decision maker’s parameters, d and ε, are
exogenous.
7
Definition 1 Two elements x, y ∈ X are said to be similar, xSd,ε y, if and only if their
difference measured by d is lower or equal to ε.
xSd,ε y ⇐⇒ d (x, y) ≤ ε
(1)
where ε ∈ ]0, diam (X)[.
Two alternatives are similar, if their difference is imperceptible. The higher the perception
accuracy, the smaller the threshold ε is. The definition of a metric function requires that
ε ≥ 0. Since the case of ε = 0 and ε = diam (X) are not interesting, ε lies on the open
interval ]0, diam (X)[. Naturally, the meaning of ε depends on the metric d. The following
proposition characterizes similarity relation.
Proposition 1 Similarity, Sd,ε , is a binary relation on X, Sd,ε ⊂ X × X, which is
complete, reflexive and symmetric regardless d or ε. Additionally, it is not transitive in
general, and this property depends on (X, d) and on ε.
An important remark is that Sd,ε fails to be an equivalence relation only because it is not
transitive in general. The core idea of this section is to find the cases where this relation is
transitive and to explore the preferences in which imperceptible difference means indifference
in terms of choice theory. The similarity relation has an agglomerative nature: starting from
a pair of similar elements, it can create chains of similarity.
Definition 2 A chain generated by the similarity relation Sd,ε , called Sd,ε -chain, is a
non-empty subset of X composed of a sequence of pairs of similar elements:
Sd,ε -chain = {x1 Sd,ε x2 , x2 Sd,ε xj−k , ..., xj−1 Sd,ε xj }.
The transitivity problem is clearly seen through a chain: the decision maker may judge
x1 and x2 as similar, x2 and xj−k as well, and may not judge x1 and xj−l in this way. Then,
imperceptible difference is not transitive, and it cannot mean indifference.
8
A Sd,ε -chain is cycle if x1 = xj . A sub-Sd,ε -chain is a Sd,ε -chain contained by another one.
Two chains are connected if they share at least one element. A Sd,ε -group is a collection
of connected Sd,ε -chains. Two Sd,ε -groups are said disconnected if they do not share any
Sd,ε -chain.
The next theorem establishes the necessary and sufficient conditions for the existence of
a SBP. Under these conditions, similarity relation is transitive and, consequently, imperceptible difference is compatible with rational preferences. Let be a weak order on X, an
asymmetric and negatively transitive binary relation. Let R be a equivalence relation on X.
Let Ω (R) be the collection of weak orders on X such that xRy ⇐⇒ (not x y, not y x)
for all x, y ∈ X.
Theorem 1 The union between the similarity relation Sd,ε and a weak order is a
rational preference d,ε on X, represented by an ordering-preserving real-valued function
ud,ε , if and only if the decision maker perceives X through disconnected Sd,ε -groups; these
groups are composed only by cycle sub- Sd,ε -chains; and Ω (Sd,ε ). Additionally, fixed a
weak order belonging to Ω (Sd,ε ), d,ε is unique.
The intuition behind the above conditions is that the decision maker has to perceive
the set X through categories and the asymmetric and symmetric parts of the SBP are
compatible. Theorem 1 controls the similarity chains such that they become indifference
classes of a preference by requiring spatial characteristics from them on the attribute space.
This spatial requirement makes an indifference class a category. This point will be explored
in the next section.
A SBP is properly specified by d,ε . Its primitive is its symmetric part, which is determined by d and ε. Thus, two decision makers may have the same perception parameters
(the same indifference classes) and end up with different rankings because of the different
weak orders belonging to Ω (Sd,ε ). The fulfillment of the conditions depends not only on X,
but also on d and ε. SBP is not locally insatiable exactly because of perception inaccuracy:
9
given x ε X, there is no y ε X such that d (x, y) ≤ ε and x y or y x.
SBP is a complete preorder, not a semiorder. SBP puts the question of “very close”
in a different place than a semiorder does. In a semiorder relation, two elements are not
distinguishable if their utility values are close enough. Then, the decision maker evaluates the
very-close question in the real line. On the other hand, under SBP approach, two elements
are imperceptible different if they are close enough based on their attributes. The very-close
question is, then, evaluated in the space of the alternative attributes. There is also a point
of primitiveness here: if the decision maker judges the difference between two elements as
imperceptible, he should not be able to attach different utility values for them.
Theorem 1 deals only with the case in which the indifference classes of a preference
are solely generated by similarity. This assumption can be relaxed and the Theorem 1
restated. Another equivalence relation R may be defined on X that contains the similarity
relation, xRy =⇒ xSd,ε y for all x, y ∈ X. In other words, the evaluation of similarity
should be the first step in the decision process. If the weak order is redefined based on
R, xRy ⇐⇒ (not x y, not y x) for all x, y ∈ X, the conditions of Theorem 1 are still
satisfied. Thus, d,ε is the union of Sd,ε , R and .
Vila (1997) extends Rubinstein results to a general three-dimensional choice problem
and concludes (page 287) that “transitive preferences can not be consistent with similarity
relations”. The fact that this conclusion disagrees with Theorem 1 is not surprising, since
the decision making setups are not comparable. Rubinstein’s similarity is defined on each
dimension independently as semiorders. Here, similarity is originated on a multidimensional
space and the present paper models the case in which similarity is an equivalence relation.
The expression chose by Rubinstein to represent similarity is different from the definition of
similarity in (1). Notice that Rubinstein’s similarity can reduce the number of dimensions
that really matter for the choice, whereas SBP allows for the reduction of the number of
alternatives.
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ε = max diam(C i )
i
C2
α = min d (C i , C j )
C1
i≠ j
C3
α>ε
Figure 1: The indifference classes (C1 , C2 and C3 ) of a SBP
3
SBP and Categorization
This section explores how the notion of category can shed light to the transitivity and
perception discontinuity problems presented in imperceptible difference. It also investigates
the cases where categorization can be rationally induced by imperceptible difference.
The first two conditions of Theorem 1 can be restated if similarity is seen as the distance
between elements: X has to be partitioned in such way that the maximum diameter among its
categories, ε, is strictly inferior than the minimum distance between two different categories,
α. Figure 1 illustrates this condition on an alternative set whose attributes are in the R2
and d is the Euclidean metric. Let Ci be the i-th category of X.
The condition above implies that every element belonging to the same category is only
similar to all others in the same category. The equivalence in (2) formalizes this idea.
α > ε ⇐⇒ x, y ∈ Ci and z ∈ Cj , i = j ⇒ d (x, y) < d (x, z) and d (x, y) < d (y, z)
11
(2)
The existence of categories makes similarity and imperceptible difference transitive. The
disconnectedness inherent to the presence of categories makes ε meaningful in terms of
decision making regardless the merit of perception accuracy.
Returning to the Mas-Colell, Whinston and Green (1995) example, if the person perceives
the difference among shades of the same color as smaller than the difference among shades
of different colors, then each color is an indifference class and imperceptible difference can
be modeled by a rational preference. Each color may be viewed as a category and ε reflects
the discontinuity inherent to different colors.
Imperceptible difference, in its turn, gives rationality for special cases of categorization. In
costly choices, categorization arises as a tool to control cost by reducing the opportunity set
that will really be manipulated by the decision maker. Theorem 1 can also be understood as
a representation theorem for categorization in costless choices. Because the elements inside a
same category are so alike in comparison with other elements, the decision maker is rationally
allowed to be absolutely indifferent among alternatives inside a category. His opportunity set
is no longer the entire X, but it is composed by one alternative per category. In this sense,
imperceptible indifference, under Theorem 1 conditions, leads to an opportunity-set-selection
process. Therefore, imperceptible difference is able to rationally simplify the choice.
Consider an investor in the US stock market, who applies the mean-variance investment
criterion, building his efficient portfolio with no cost. His forecasts for mean, variance and
covariances of next period return of a stock are based on the company fundamentals. He
evaluates similarity between two companies based on their fundamentals: he measures differences between the fundamentals through a metric d an has a perception accuracy of ε.
Although he knows that the companies are different, if they are similar, he produces the
same forecasts for the moments of their expected returns. Because the companies are similar, they contribute exactly in the same manner to the investor’s portfolio. This analysis is
extended for the whole set of stocks. If he can partition the companies into categories, he
is indifferent among the stocks in the same category. Therefore, he is allowed to build his
12
efficient portfolio considering just one stock per category, simplifying his job.
4
Decision Maker’s Parameters
This section works on the influence of the decision maker’s perception parameters on the
resulting SBP. Proposition 2 tells that more than one value of perception accuracy thresholds
can produce the same SBP. Although the perception accuracy changes, similarity relation
may be unchanged and, consequently, the indifference classes are the same.
Proposition 2 The preference d,ε is invariant for any decision maker’s perception
accuracy threshold belonging to the interval:
Td,ε = [diammax , dmin [
(3)
where diammax is the maximum diameter among the disconnected Sd,ε -groups and dmin is
the minimum distance among them.
At first sight, the previous proposition makes suspicious the uniqueness of a SBP claimed
in Theorem 1. It is not the case, since a decision maker, at least at same time, can have only
one difference measure and only one perception accuracy threshold. Notice that the interval
is defined by four elements of X.
If the decision maker varies his perception accuracy beyond the limits of (3), he may
have preferences with different number of indifferent classes. Because the similarity relation
changes, the previous weak order is no longer valid. From a theoretical point of view, the
asymmetric part of the preference has to be consistent regardless the number of indifference
classes. Therefore, the preference structure has to count with an underlying strict order, o .
Theorem 2 claims that as long as the decision maker perception accuracy shrinks (the
threshold enlarges), if a different SBP exists, its indifference classes are the results of merges
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C 5ε
ε
C 4ε
C 3ε
ε
C1
γ
C 2γ
C1γ
C 2ε
Figure 2: Indifference classes of two SBPs with the same d and thresholds ε and γ.
of previous SBPs indifference classes. This structure represents an unique hierarchy over the
possible sets of indifference classes.
Theorem 2 A decision maker who has his perception accuracy strictly decreased, keeping
the same underlying strict order o and difference measure d, has different SBPs only if they
form a strictly decreasing hierarchical structure.
The above hierarchical structure should not be taken for granted: it strongly depends
on the definition of both distance between two subsets and similarity relation. Figure 2
shows the indifference classes of two different SBPs with the same d (the Euclidean metric)
on alternative set whose attributes are in the R2 . The SBPs differ only on the perception
accuracy thresholds, ε < γ. Notice the hierarchical structure involving the indifference
classes: C1γ = C1ε ∪ C2ε ∪ C3ε , C2γ = C4ε ∪ C5ε . On the other hand, nothing interesting happens
in the case of strictly increasing perception accuracy.
14
The next proposition states that SBP is invariant for same transformations over the
metric d if the threshold is properly adjusted. The key point is the notion of equivalence
between metrics. Two metrics, d1 and d2 , on the same set X are equivalent if a subset is
open in (X, d1 ) if and only if it is open in (X, d2 ). Let I : (X, d1 ) → (X, d2 ) be the identity
mapping of (X, d1 ) onto (X, d2 ).
Proposition 3 Consider a SBP d1 ,ε and a metric d2 on X. There exist a SBP d2 ,γ
that is equal to d1 ,ε if d2 is equivalent to d1 , Td2 ,γ = I (Td1 ,ε ) and the weak order is the
same.
Proposition 3 tells that, once the topological neighborhoods are the same, the four elements of X which determine Td1 ,ε and Td2 ,γ are the same. The next result states that decision
makers with different perception of the world (different d and ε) may end up perceiving the
same categories. And, if both share the same weak order on X, the resulting SBP are the
same. It is a corollary of Proposition 3.
Corollary 1 Consider a decision maker A who has a preference d1 ,ε on X. Consider
another decision maker B who has a preference d2 ,γ on X .
These two preferences are
equal if d2 is equivalent to d1 , Td2 ,γ = I (Td1 ,ε ) and the weak order is the same.
5
Similarities and Preferences on Attributes
The SBP on X is originated taking into consideration all the n alternative attributes simultaneously. The multidimensionality of the problem brings a question of decision theory
consistency between the overall judgment and the judgments on each attribute. This section
investigates the linkages between the (multidimensional) similarity relation and the n similarity relations on the attributes based on the hypothesis that there is no association among
the latter n similarity relations. The final question to be addressed is the connections among
SBPs on X and on each attribute.
15
The bridges between the multidimensional space and the n attribute spaces are orthogonal
projections and norms. Therefore, it is necessary to equip the metric space with linear and
geometric structures. The price paid is a stronger requirement over d.
Let χ be a real linear space over R and ; : χ × χ → R an inner product on χ defined in
n
the usual way. Then, (χ, ; ) is a real-valued inner-product space. Let B = {B i }i=1 be its
orthonormal base and X be a countable subset of vectors belonging to it. The norm and
the metric d on X are induced by the inner product: d (x, y) ≡ x − y ≡ x − y; x − y2 .
Then, each x ∈ X is redefined as a n-dimension vector: x = [x1 , ..., xn ]. Assume that χ = Rn
and each attribute is represented by one dimension.
The definition of similarity relation on the attributes follows Definition 1. Because χ =
Rn , the decision about the difference measure on the attribute dimension is natural: the
usual metric on the real line. Then, the difference measure is not a parameter of similarity
on the attributes. Define X i ⊂ R as the set composed by the i-th attribute of the elements
in X, that is X i = (xi , y i , z i , ...) .
Definition 3 Let x, y ∈ X. They are said similar regarding to attribute i, xi Sρi i y i , if and
only if the absolute value of their difference is lower or equal to ρi .
where ρi ∈ ]0, diam (X i )[.
xi Sρi i y i ⇐⇒ xi − y i ≤ ρi
(4)
This definition is extended to all attributes. As these n relations are special cases of Sd,ε ,
Proposition 1 applies to them. Notice that the thresholds ρi ’s might not be equal and each
Sρi i would result in the ε-difference similarity defined by Rubinstein (1988), on page 148,
if X was the unitary hypercube in Rn . The conditions for the existence of a SBP on each
attribute in isolation are stated by Theorem1 in its one-dimensional-similarity version.
Assume that there is no association among similarity relations on the attributes. One
example of association among similarity is the correlated similarity introduced by Aizpurua
16
et al. (1991). The authors model the lottery choice with similarity. However, they do not
define similarity relations on prize and probability dimensions independently. The idea is
that similarity relation on the probability space depends on the prize: the decision maker’s
discrimination capacity on probabilities increases as the prize increases.
Once the hypothesis above has been made, it makes sense to require that the judgment
about similarity on the multidimensional space and on the n attributes be consistent. In order
to formalize this idea, two conditions will be imposed on the n + 1 similarity relations. The
first one is due to Gilboa and Lapson (1995). They argue that it seems to be unreasonable
that the decision maker who does not separately perceive difference among the n attributes
of x and y judges them as perceptible different. The second condition is the converse of the
first one: if the decision maker does not perceive the difference between x and y, he does not
perceive the differences across their n attributes separately.
These two consistence conditions make the n + 1 perception-accuracy thresholds mutualdependent. Proposition 4 analyzes the first condition, where the n similarity relations are the
primitives. That is, the values of ρi s are given and the proposition imposes a restriction over
ε. Proposition 5 deals with the converse case: the threshold ε is given and the proposition
restricts the n thresholds ρi s. Both propositions require more structure from d.
Proposition 4 Let x and y be two vectors in X which are similar regarding all their
n attributes, xi Sρi i y i for i = 1, . . . , n, and d is a metric function induced by a norm on X.
n
Then, x and y are similar, xSd,ε y, only if ε ≥ ρi .
i=1
Proposition 5 Let x and y be two similar vectors in X, xSd,ε y, and d a metric function
induced by an real-valued inner product on X. Then, x and y are similar regarding all their
n attributes, xi Sρi i y i , for i = 1, . . . , n only if ρi ≤ ε, for i = 1, . . . , n.
Proposition 4 demands a stricter condition regarding the perception accuracy thresholds
and less structure from the difference measure than Proposition 5. Definition 4 selects the
tougher part of the last two propositions and establishes the conditions on both the n + 1
17
thresholds and d in order for the system to be consistent. Equipped with a consistent system
of similarity relations, similarity can consistently transit from the multidimensional space to
each attribute and vice-versa.
Definition 4 A system of n + 1 similarity relations, composed by Sd,ε and n Sρi i , is said
to be consistent if and only if the decision maker’s difference measure d is a metric function
n
induced by an real-valued inner product on X and ε ≥ ρi .
i=1
Relaxing the assumption that the judgments about similarity on the attributes are independent, the second condition becomes meaningless. Requiring only the Gilboa and Lapson’s
condition, SBP is flexible enough to accommodate interdependence among similarity relations on attributes.
Now, it is possible to explore the connections among preferences. Since the primitive
of SBP is its symmetric part, the first step is to check the links among the indifference
classes of d,ε and of the n preferences iρi . Notice that the disconnectedness of two
Sd,ε -groups does not guarantee the disconnectedness of the associated Sρi i -chains (generated
by orthogonal projections) on each attribute dimensions. Therefore, even if the system
of similarity relation is consistent, the existence of d,ε does not necessarily provide the
conditions for the existence of the n iρi preferences. Consider the Figure 2 for a simple
example. If the indifference classes of d,ε were only C1ε , C2ε and C3ε , the projections of their
points on the horizontal axis do not fulfill the conditions of Theorem 1. Conversely, the
existence of the n iρi preferences doe not guarantee the existence of d,ε .
6
Conclusion
The present paper reconciled imperceptible difference and rational preference by using the
notions of similarity and categorization. It was shown that imperceptible difference can
be incorporate into a rational multidimensional preference on a countable alternative set if
18
and only if the decision maker perceives his alternatives through categories generated by
similarity.
Similarity is a binary relation that models imperceptible difference by controlling the
decision maker’s capacity for perceiving difference between alternatives based on their n attributes. It is function of the way the decision maker measures difference (a metric function)
and his perception accuracy for differences (a threshold). Two alternatives are similar if and
only if their difference is imperceptible — lower than the threshold. A category is defined as
a subset of elements that are more similar to each other than to elements of other subsets.
This definition was crucial for the whole paper.
If the alternative set can be partitioned into categories, imperceptible difference is a
transitive relation. And the perception discontinuity mark just captures the disconnectedness
existing in the alternative set, that is, the differences among categories. Let the proposed
preference be called Similarity Based Preference (SBP). Its primitive is similarity relation.
SBP is not a semiorder. SBP puts the very-close question to the decision maker in the
n-dimensional space of the attributes.
Imperceptible difference, in its turn, gives rationality for special cases of categorization.
Theorem 1 can also be understood as a representation theorem for categorization in costless
choices. Imperceptible difference is able to rationally simplify the choice under Theorem 1
conditions leading to an opportunity-set-selection process: the decision maker’s opportunity
set is no longer the entire X, but it is composed by one alternative per category.
Fixed the asymmetric part of the preference, the analysis of invariances of SBP for
the decision maker’s perception parameters brings two interesting conclusions. First, two
decision makers with different perception for differences of the alternatives (different d and
ε) may end up having the same SBP. Second, if the decision maker varies his perception
accuracy by enough, the resulting SBPs form a hierarchical structure. That is, in order to
reduce the number of indifference classes, the decision maker can only merge the already
existing indifference classes. In addition, this hierarchical structure is unique.
19
The linkages between the (multidimensional) similarity relation and the similarity relations on each n attributes, based on the hypothesis that there is no association among the
latter n similarity relations, were explored. In order to assure the consistent transit of similarity from the multidimensional space to each attribute and vice-versa a consistent system
was defined. A system composed by n + 1 similarity relations is consistent if and only if
the decision maker’s difference measure d is a metric function induced by a real-valued inner
product on X, and the threshold ε is greater than the sum of the n similarity-thresholds on
the attributes. Finally, even if the system of similarity relation is consistent, the existence
of d,ε does not assure the existence of the n iρi preferences and vice-versa.
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22
Appendix
Proposition 1 Similarity, Sd,ε , is a binary relation on X, Sd,ε ⊂ X × X, which is
complete, reflexive and symmetric regardless d or ε. Additionally, it is not transitive in
general, and this property depends on (X, d) and on ε.
Proof: Because d is a metric on X, the relation applies for all x, y ∈ X: d (x, y) ≤ ε
or d (x, y) > ε. Thus, it is a complete relation. It is reflexive: xSd,ε x ⇐⇒ d (x, x) = 0 ≤ ε.
It is symmetric, since d is a symmetric function by definition. These three properties are
independent of d and ε. It is not generally true that d (x, y) ≤ ε and d (y, z) ≤ ε =⇒
d (x, z) ≤ ε. Thus, it is not transitive in general and its property depends on (X, d) and ε.
Theorem 1 The union between the similarity relation Sd,ε and a weak order is a
rational preference d,ε on X, represented by an ordering-preserving real-valued function
ud,ε , if and only if the decision maker perceives X through disconnected Sd,ε -groups; these
groups are composed only by cycle sub- Sd,ε -chains; and Ω (Sd,ε ). Additionally, fixed a
weak order belonging to Ω (Sd,ε ), d,ε is unique.
Proof: To perceive the set X through subsets means to partition X into these subsets.
Necessary condition. Suppose that there exist m disconnected Sd,ε -groups. The distance
between two disconnected Sd,ε - groups is strictly greater than ε. Take one of the Sd,ε -groups.
If all of its sub-Sd,ε -chains are cycle, then all elements belonging to this Sd,ε -group are
similar to each other. This argument is applied to every m disconnected Sd,ε -groups. In a
set partitioned in this way, similarity is transitive and Proposition 1 can be used to show that
it is an equivalence relation. Thus, since X is countable , Sd,ε induces a countable quotient
space, X/Sd,ε . The indifference classes of X/Sd,ε are the disconnected Sd,ε -groups. Let be
a weak order (an asymmetric and negatively transitive binary relation) on X. Let Ω (Sd,ε ) be
the collection of weak orders on X such that xSd,ε y ⇐⇒ (not x y, not y x) for all x, y
∈ X. Let d,ε be the union of Sd,ε and a ∈ Ω (Sd,ε ). This union is complete and transitive.
Therefore, d,ε is a rational preference relation on X. Fishburn (1970), Theorem 2.2, shows
23
that there is an ordering-preserving real-valued function that represents the similarity-based
preference such that: x d,ε y ⇐⇒ ud,ε (x) ≥ ud,ε (y) .
Sufficient condition. Suppose that d,ε is rational. Since the weak order is asymmetric,
the similarity relation Sd,ε must quotient X out. Then, it has necessarily to be an equivalence
relation. Proposition 1 states that Sd,ε is reflexive, symmetric, and, in general, intransitive.
According to Definitions 1 and 2, X has to be composed of disconnected Sd,ε -groups and
these ones have to be composed only by cycle sub- Sd,ε -chains in order to be transitive. Fixed
, d and ε, the uniqueness of d,ε results from the fact that the collection of all equivalence
relations on a set is in a one-to-one correspondence with the collection of all partitions of
X.
Proposition 2 The preference d,ε is invariant for any decision maker’s perception
accuracy threshold belonging to the interval:
Td,ε = [diammax , dmin [
where diammax is the maximum diameter among the disconnected Sd,ε -groups and dmin is
the minimum distance among them.
Proof: Suppose the number of disconnected Sd,ε -groups is m. By Theorem 1, the diameter of a Sd,ε - group may be strict less than ε. Let diammax be the maximum diameter
j
of the disconnected Sd,ε -groups: diammax = max diam Sd,ε
− group ≤ ε. On the other
1≤j≤m
hand, the same theorem implies that the difference between two disconnected Sd,ε -groups is
strictly greater than ε. Define dmin as the minimum difference among the disconnected Sd,ε i
j
groups: dmin = mind Sd,ε
− group, Sd,ε
− group > ε. As long as the similarity relation is
i=j
redefined by any real value belonging to the interval Td,ε , defined by (2), the m disconnected
Sd,ε -groups are the same and they observe the conditions of Theorem 1. Thus, the following
two rational preferences, d,ε = Sd,ε ∪ and d,γ = Sd,γ ∪ , are equal: if γ ∈ Td,ε .
24
Theorem 2 A decision maker who has his perception accuracy strictly decreased, keeping
the same underlying strict order o and difference measure d, has different SBPs if they form
an unique strictly decreasing hierarchical structure.
Proof: As the decision maker’s perception accuracy decreases, the threshold ε increases.
Let ξ = {εi }ni=1 be a strictly increasing sequence of positive real numbers belonging to
the allowed interval for ε: ]0, diam (X)[. A strictly decreasing hierarchical structure means
that x d,εi y ⇒ x d,εj y and d,εi = d,εj for i < j and all x, y ∈ X. Suppose that
Theorem 1 applies for each element of ξ in order to generate a similarity-based preference.
Let P = {d,εi }ni=1 be the collection of these n preferences. Assume that they are different.
The underlying strict order o is the same across the preferences, that is, o is a strict order
on the n quotient spaces X/Sd,εi , i=1, . . ., n. Then, the source of the difference has to be
their symmetric parts. More precisely, the n quotient spaces must differ among them. Let
M = {mi }ni=1 be the sequence composed of the number of indifference classes associated with
each d,εi . Since d is kept unchanged, the perception accuracy thresholds are the responsible
for the difference. Proposition 2 requires that εi ∈
/ Td,εj for i = j. For any x, y ∈ X and
εi , i=1,. . . , n-1, xSd,ε y ⇒ xSd,εi+1 y, because d (x, y) ≤ εi ⇒ d (x, y) ≤ εi+1 . It means
that elements belonging to a same disconnected Sd,ε -group are in the same disconnected
Sd,εi+1 -group, for i=1,. . . , n-1. Because the uniqueness of the m-partition that satisfies the
conditions of Theorem 1, mi > mi+1 . Therefore, the increasing of the perception accuracy
threshold implies only in merger of the existing indifference classes. The uniqueness of the
m-partition can be used again to show that P = {d,εi }ni=1 is unique.
Proposition 3 Consider a SBP d1 ,ε and a metric d2 on X. There exist a SBP d2 ,γ
that is equal to d1 ,ε if d2 is equivalent to d1 , Td2 ,γ = I (Td1 ,ε ) and the weak order is the
same.
Proof: Let m be the number of indifference classes of d1 ,ε . By definition, if d1 ˜ d2 ,
the topologies of (X, d1 ) and (X, d2 ) are equal. Then, d1 (x, y) < d1 (x, z) and d1 (x, y) <
25
d1 (y, z) ⇔ d2 (x, y) < d2 (x, z) and d2 (x, y) < d2 (y, z) for all x, y, z ∈ X. By (2), there exist
a γ ∈ ]0, diam (X)[ such that the similarity relation Sd2 ,γ on X is an equivalence relation and
generates the same indifference classes of d1 ,ε . By (2) yet, γ = max diam Sdj2 ,γ − group .
1≤j≤m
Since Sd2 ,γ = Sd1 ,ε , Sd2 ,γ ⇐⇒ (not x y, not y x) for all x, y ∈ X.
Thus, d2 ,γ =
Sd2 ,γ ∪ =d1 ,ε . Proposition 2 tells that d2 ,γ is invariant for any perception accuracy
threshold belonging to Td2 ,γ . Let C i = Sdi 1 ,ε − group = Sdi 2 ,γ − group for i = 1, ..., m.
Td2 ,γ = max maxi d2 (x, y) , min min
d2 (x, y) and
1≤j≤m x,y∈C
i=j x∈C i ,y∈C j
Td1 ,ε = max maxi d1 (x, y) , min min
d1 (x, y) . Let I : (X, d1 ) → (X, d2 ) be the
i
j
1≤j≤m x,y∈C
i=j x∈C ,y∈C
identity mapping of (X, d1 ) onto (X, d2 ). Kubrusly (2000), in Corollary 3.19 (page 108),
shows that d1 ˜ d2 ⇔ I is continuous. Therefore,
I [Td1 ,ε ] = max maxi I [d1 (x, y)] , min min
Id1 (x, y) =
1≤j≤m x,y∈C
i=j x∈C i ,y∈C j
max maxi d2 (x, y) , min min
d2 (x, y) = Td2 ,γ .
i
j
1≤j≤m x,y∈C
i=j x∈C ,y∈C
Corollary 1 Consider a decision maker A who has a preference d1 ,ε on X. Consider
another decision maker B who has a preference d2 ,γ on X . These two preferences are equal
if d2 is equivalent to d1 , Td2 ,γ = I (Td1 ,ε ) and the weak order is the same.
Proof: Immediate application of Proposition 3.
Proposition 4 Let x and y be two vectors in X which are similar regarding all their n
attributes, xi Sρi i y i for i = 1, . . . , n, and d a metric function induced by a norm on X. Then,
n
x and y are similar, xSd,ε y, only if ε ≥ ρi .
i=1
n
Proof: Let x, y ∈ X. Since X is a linear space and B = {B i}i=1 its orthogonal
n
n
i
i
i
i
i
i
basis, (x − y) =
(x − y ) B . Since X is a normed space, x − y = (x − y ) B ≤
n
i=1
i
i
i=1
i
(x − y ) B . As x and y are similar in terms of all their n attributes, B is orthonormal
i=1
and the metric d is induced by the norm .in X, d (x, y) ≤
n
ρi . Therefore, in order to x
i=1
26
and y be similar in the multidimensional space, according to Definition 1, ε ≥
n
ρi .
i=1
Proposition 5 Let x and y be two vectors in X which are similar, xSd,ε y, and d a
metric function induced by a real-valued inner product on X. Then, x and y are similar
regarding all their n attributes, xi Sρi i y i for i = 1, . . . , n, only if ρi ≤ ε, for i = 1, . . . , n.
Proof: Let x; y be an inner product on X ⊂ Rn and d (x, y) ≡ x − y ≡ x − y; x − y2 .
By definition, xSd,ε y ⇐⇒ d (x, y) ≤ ε. As the distance d is induced by the norm .,
d (x, y) ≡ x − y ≤ ε. Define the absolute value of the difference between x and y in
(x−y),Bi each dimension as the projection of x − yon it: projBi x − y = x−yBi x − y =
|cos ((x − y) , B i )| x − y, for all i = 1, . . . , n. Then, xi Sρi i y i ⇐⇒
|xi − y i | = |cos ((x − y) , B i )| x − y ≤ ρi . Since x − y ≤ ε and |cos ((x − y) , B i )| ≤
1, if ρi ≤ ε, then xi Sρi i y i for all i = 1, . . . , n.
27