DOCSLIB.ORG

Rationality and the Speed of Decision Making

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rationality and the Speed of Decision Making Rationality and the speed of decision-making Michael Mandler Department of Economics Royal Holloway College, University of London Egham, Surrey TW20 0EX United Kingdom Abstract turns out to be unambiguous: the rational agents can on some domain always use a strictly shorter check- We consider agents who choose by proceeding list of criteria than the irrational agents can and on through an ordered list of criteria and give no domain do they have to use a longer list. Ratio- the lower bound on the number of criteria nal agents can in fact make preference discriminations that are needed for an agent to make deci- with great e¢ ciency; the number of discriminations sions that obey a given set of preference rank- they can make is an exponential function of the num- ings. Agents with rational preferences can ber of criteria at their disposal. And the speed advan- always use lists with the lower-bound num- tage of rationality holds even when compared to agents ber of criteria while any agent with nonra- whose sole source of irrationality is incompleteness. tional preferences must on some domains use Our conclusions depart from Herbert Simon’s view strictly more criteria. We preview some of that the assumption of utility maximization places ‘a the results in Mandler (2009) and explain in heavy (often unbearable) computational burden on the more detail the order-theoretic link between decision maker’(Simon (1990)). For checklist users, in rationality and rapid decision-making. contrast, it is the rational agents who have the easier computational task and therefore make decisions more quickly. The di¢ culty with Simon’sand kindred views 1 INTRODUCTION is that they take utility to be a set of external facts that an agent has to uncover. If instead we take an One way to choose between alternatives is to divide agent’s decision-making procedure as primitive, then the domain of possible alternatives into various cate- the computational burden of decision-making becomes gories and then decide on an ordering of those cate- a function of the procedure alone. We may then ask gories. If this step does not discriminate su¢ ciently if the procedures that lead to rational decisions carry one can then move on to a second categorization, and a heavier or lighter burden than the irrational proce- so on. For any pair of alternatives, an agent can pro- dures. ceed in this fashion through a sequence of orderings, choosing the alternative that is recommended by the The greater decision-making e¢ ciency of rational pref- …rst ordering that actually ranks the pair. While this erences derives from a fundamental fact about linear choice procedure, which we call a checklist of criteria, orders. On domains that consist of at least four ele- does not look like classical economic behavior it does ments, linear orders are the only binary relations such describe one of the rough-and-ready methods that peo- that any two subsets with the same number of elements ple use to make decisions. are order-isomorphic. In our setting, a rational pref- erence relation induces a linear ordering over its indif- A checklist user’s decisions implicitly de…ne a prefer- ference classes and the order isomorphism fact implies ence relation: for any pair of alternatives that are fed that no matter how we constrain the criteria that an into the choice procedure, we can label the chosen al- agent can use, the agent will be able to divide up his ternative to be strictly preferred, and if no criterion in indi¤erence classes into order-isomorphic subsets that a checklist discriminates between the alternatives we a single criterion can then order simultaneously. As can label the pair unranked. Which types of checklist we will see, this is the source of the speed advantage users can make decisions the most rapidly, the rational of rational agents. agents whose preferences are complete and transitive or the irrational agents? The answer to this question Checklists of criteria bear a close resemblance to Chip- man’s (1960) lexicographic theory of utility, which section 3. was designed to provide representations for preferences For example, X might be a large set of cars, crite- that do not have classical utility functions (such as lex- rion 1 could divide cars into cheap, mid-priced, and icographic preferences). In Chipman, the preference expensive categories, criterion 2 could divide cars into between two items is determined by the …rst utility in sedans, convertibles, SUV’s,and all others, and …nally a sequence of real-valued utility functions that ranks criterion 3 might divide cars by their country of ori- the items. In our model a sequence of criteria replaces gin. After the categorization step, the agent must the sequence of utilities and, to gauge decision-making …gure out how to order each set of categories. Then, e¢ ciency, we will use an accounting system that mea- to decide between a pair of cars, the agent proceeds in sures the discriminatory capacity of criteria. It turns sequence through the criteria: if one car has the bet- out that even the least discriminating criteria, when ter price ranking, then the agent buys that car; if not, strung together in a sequence, can generate a vast set the agent goes on to the next categorization, and so of preference discriminations. The absence of a mea- on until the agent comes to a categorization that does sure of discriminatory capacity in Chipman has hid rank the pair. If the pair is ranked by none of the from view the power of his representation concept. criteria, the agent declares either car to be acceptable. More concretely, the present work follows in the foot- We can de…ne a preference relation from a checklist steps of the checklists studied in Mandler, Manzini, by seeing for any pair in X which element of the pair and Mariotti (2008). The MMM checklists are con- is chosen. Formally, a preference is another asym- strained to be rational, however, which precludes the metric binary on X. Given a preference , we say rational versus irrational horse race that we pur- that C = (C ; :::; C ) is a checklist for if and only sue here. Kalai, Rubinstein, and Spiegler (2002), 1 L if, for all x; y X, Apesteguia and Ballester (2005, 2008), and Manzini 2 and Mariotti (2007) also consider choice behavior x y i with 1 i L such that x Ci y and that can be decomposed into ‘criteria.’ KRS and , 9 Apesteguia and Ballester (2005) have an agenda sim- not y Cj x for all j < i: ilar to the present paper but they seek out the most Notice that it is only the …rst criterion that ranks a concise set of rationales for decisions, not the most e¢ - pair that determines preference; if the …rst criterion cient use of discriminatory capacity. The introduction Ci that ranks and x and y has x Ci y then some later of an accounting system for discriminatory capacity of criterion Ck can have y Ck x and still x y. We will criteria marks out a distinct agenda. also say that a preference has the checklist C if C is In addition to previewing some of the results in Man- a checklist for . A checklist for a preference amounts dler (2009), we will use this forum to explain in more to a mild extension of Chipman’s (1960) de…nition of detail the order-theoretic connection between rational- a lexicographic utility. ity and the speed of decision-making. Checklists of properties. Some seemingly crude se- quential decision procedures may not at …rst appear 2 CHECKLISTS OF CRITERIA to qualify as checklists, for example, the model in Mandler, Manzini, and Mariotti (2008). Suppose an Fix some domain of alternatives X. We consider agent chooses between two items x and y by proceed- agents who make decisions between pairs of alterna- ing through a list of ‘properties,’P1;P2;P3; ::: , where tives x; y X by proceeding through a checklist of each Pi is a subset of X. The agent …rst checks if x 2 criteria C1; :::; CL, where each Ci is an asymmetric bi- and y have property 1 (i.e., if x P1 and y P1). If nary relation on X, and choosing the alternative that either or x or y has property 1 but2 the other2 does not is ranked superior by the …rst Ci that actually orders then the agent chooses the item that does. If both x x and y. One should think of each criterion as di- and y have property 1 or neither does then the agent viding X into a relatively small number of categories proceeds to property 2, and so on. A list of properties or equivalence classes. Outside of asymmetry, crite- quali…es as a checklist of criteria since the agent could ria are unrestricted; they could be rational orderings, make the same decisions by using a C1 that ranks any they could cycle, or they could leave some alternatives item in P1 as strictly superior to any item in X P1, n unranked. a C2 that ranks item in P2 as strictly superior to any item in X P2, and so on. An agent might construct a criterion Ci in two steps, n …rst a division of X into categories that partition X Checklists can be seen as a decision procedure for two- and then a ranking (though not necessarily a complete element choice sets, but they can also be used to decide ranking) of these categories. Formally, the categories from a larger choice set A by letting the Ci sequentially are the equivalence classes of Ci, which we de…ne in eliminate Ci-dominated items from A.
Load more

Recommended publications
  • Simple Laws About Nonprominent Properties of Binary Relations
    Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt jochen.burghardt alumni.tu-berlin.de Nov 2018 Abstract We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like \not irreflexive, but asymmetric". We considered the laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties. Keywords: Binary relation; Quine-McCluskey algorithm; Hypotheses generation arXiv:1806.05036v2 [math.LO] 20 Nov 2018 Contents 1 Introduction 4 2 Definitions 8 3 Reported law suggestions 10 4 Formal proofs of property laws 21 4.1 Co-reflexivity . 21 4.2 Reflexivity . 23 4.3 Irreflexivity . 24 4.4 Asymmetry . 24 4.5 Symmetry . 25 4.6 Quasi-transitivity . 26 4.7 Anti-transitivity . 28 4.8 Incomparability-transitivity . 28 4.9 Euclideanness . 33 4.10 Density . 38 4.11 Connex and semi-connex relations . 39 4.12 Seriality . 40 4.13 Uniqueness . 42 4.14 Semi-order property 1 . 43 4.15 Semi-order property 2 . 45 5 Examples 48 6 Implementation issues 62 6.1 Improved relation enumeration . 62 6.2 Quine-McCluskey implementation . 64 6.3 On finding \nice" laws . 66 7 References 69 List of Figures 1 Source code for transitivity check . .5 2 Source code to search for right Euclidean non-transitive relations . .5 3 Timing vs. universe cardinality .
    [Show full text]
  • PROBLEM SET THREE: RELATIONS and FUNCTIONS Problem 1
    PROBLEM SET THREE: RELATIONS AND FUNCTIONS Problem 1 a. Prove that the composition of two bijections is a bijection. b. Prove that the inverse of a bijection is a bijection. c. Let U be a set and R the binary relation on ℘(U) such that, for any two subsets of U, A and B, ARB iff there is a bijection from A to B. Prove that R is an equivalence relation. Problem 2 Let A be a fixed set. In this question “relation” means “binary relation on A.” Prove that: a. The intersection of two transitive relations is a transitive relation. b. The intersection of two symmetric relations is a symmetric relation, c. The intersection of two reflexive relations is a reflexive relation. d. The intersection of two equivalence relations is an equivalence relation. Problem 3 Background. For any binary relation R on a set A, the symmetric interior of R, written Sym(R), is defined to be the relation R ∩ R−1. For example, if R is the relation that holds between a pair of people when the first respects the other, then Sym(R) is the relation of mutual respect. Another example: if R is the entailment relation on propositions (the meanings expressed by utterances of declarative sentences), then the symmetric interior is truth- conditional equivalence. Prove that the symmetric interior of a preorder is an equivalence relation. Problem 4 Background. If v is a preorder, then Sym(v) is called the equivalence rela- tion induced by v and written ≡v, or just ≡ if it’s clear from the context which preorder is under discussion.
    [Show full text]
  • LECTURE 1 Relations on a Set 1.1. Cartesian Product and Relations. A
    LECTURE 1 Relations on a set PAVEL RU˚ZIˇ CKAˇ Abstract. We define the Cartesian products and the nth Cartesian powers of sets. An n-ary relation on a set is a subset of its nth Carte- sian power. We study the most common properties of binary relations as reflexivity, transitivity and various kinds of symmetries and anti- symmetries. Via these properties we define equivalences, partial orders and pre-orders. Finally we describe the connection between equivalences and partitions of a given set. 1.1. Cartesian product and relations. A Cartesian product M1 ×···× Mn of sets M1,...,Mn is the set of all n-tuples hm1,...,mni satisfying mi ∈ Mi, for all i = {1, 2,...,n}. The Cartesian product of n-copies of a single set M is called an nth-Cartesian power. We denote the nth-Cartesian power of M by M n. In particular, M 1 = M and M 0 is the one-element set {∅}. An n-ary relation on a set M is a subset of M n. Thus unary relations correspond to subsets of M, binary relations to subsets of M 2 = M × M, etc. 1.2. Binary relations. As defined above, a binary relation on a set M is a subset of the Cartesian power M 2 = M × M. Given such a relation, say R ⊂ M × M, we will usually use the notation a R b for ha, bi∈ R, a, b ∈ M. Let us list the some important properties of binary relations. By means of them we define the most common classes of binary relations, namely equivalences, partial orders and quasi-orders.
    [Show full text]
  • Problem Set Three: Relations
    Problem Set Three: Relations Carl Pollard The Ohio State University October 12, 2011 Problem 1 Let A be any set. In this problem \relation" means \binary relation on A." Prove that: a. The intersection of two transitive relations is a transitive relation. b. The intersection of two symmetric relations is a symmetric relation, c. The intersection of two reflexive relations is a reflexive relation. d. The intersection of two equivalence relations is an equivalence relation. Problem 2 Background. For any binary relation R on a set A, the symmetric interior of R, written Sym(R), is defined to be the relation R \ R−1 on A. For example, if R is the relation that holds between a pair of people when the first respects the other, then Sym(R) is the relation of mutual respect. Another example: if R is the entailment relation on propositions, then the symmetric interior is truth-conditional equivalence. Prove that the symmetric interior of a preorder is an equivalence relation. Problem 3 Background. If v is a preorder, then Sym(v) is called the equivalence relation induced by v and written ≡v, or just ≡ if it's clear from the context which preorder is under discussion. If a ≡ b, then we say a and b are tied with respect to the preorder v. Also, for any relation R, there is a corresponding asymmetric relation called the asymmetric interior of R, written Asym(R) and defined to be 1 RnR−1. For example, the asymmetric interior of the love relation on people is the unrequited love relation.
    [Show full text]
  • Relations-Complete Rev: C8c9782 (2021-09-28) by OLP/ CC–BY Rel.2 Philosophical Reflections Sfr:Rel:Ref: in Section Rel.1, We Defined Relations As Certain Sets
    Chapter udf Relations rel.1 Relations as Sets sfr:rel:set: In ??, we mentioned some important sets: N, Z, Q, R. You will no doubt explanation sec remember some interesting relations between the elements of some of these sets. For instance, each of these sets has a completely standard order relation on it. There is also the relation is identical with that every object bears to itself and to no other thing. There are many more interesting relations that we'll encounter, and even more possible relations. Before we review them, though, we will start by pointing out that we can look at relations as a special sort of set. For this, recall two things from ??. First, recall the notion of a ordered pair: given a and b, we can form ha; bi. Importantly, the order of elements does matter here. So if a 6= b then ha; bi 6= hb; ai. (Contrast this with unordered pairs, i.e., 2-element sets, where fa; bg = fb; ag.) Second, recall the notion of a Cartesian product: if A and B are sets, then we can form A × B, the set of all pairs hx; yi with x 2 A and y 2 B. In particular, A2 = A × A is the set of all ordered pairs from A. Now we will consider a particular relation on a set: the <-relation on the set N of natural numbers. Consider the set of all pairs of numbers hn; mi where n < m, i.e., R = fhn; mi : n; m 2 N and n < mg: There is a close connection between n being less than m, and the pair hn; mi being a member of R, namely: n < m iff hn; mi 2 R: Indeed, without any loss of information, we can consider the set R to be the <-relation on N.
    [Show full text]
  • Simple Laws About Nonprominent Properties of Binary Relations Jochen Burghardt
    Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt To cite this version: Jochen Burghardt. Simple Laws about Nonprominent Properties of Binary Relations. 2018. hal- 02948115 HAL Id: hal-02948115 https://hal.archives-ouvertes.fr/hal-02948115 Preprint submitted on 24 Sep 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Simple Laws about Nonprominent Properties of Binary Relations Jochen Burghardt jochen.burghardt alumni.tu-berlin.de Nov 2018 Abstract We checked each binary relation on a 5-element set for a given set of properties, including usual ones like asymmetry and less known ones like Euclideanness. Using a poor man's Quine-McCluskey algorithm, we computed prime implicants of non-occurring property combinations, like \not irreflexive, but asymmetric". We considered the laws obtained this way, and manually proved them true for binary relations on arbitrary sets, thus contributing to the encyclopedic knowledge about less known properties. Keywords: Binary relation; Quine-McCluskey algorithm; Hypotheses generation Contents 1 Introduction 4 2 Definitions 8 3 Reported law suggestions 10 4 Formal proofs of property laws 21 4.1 Co-reflexivity . 21 4.2 Reflexivity . 23 4.3 Irreflexivity .
    [Show full text]
  • Ontological Foundations for Conceptual Part-Whole Relations: the Case of Collectives and Their Parts
    Ontological Foundations for Conceptual Part-Whole Relations: The Case of Collectives and Their Parts Giancarlo Guizzardi Ontology and Conceptual Modeling Research Group (NEMO), Federal University of Espírito Santo (UFES), Vitória (ES), Brazil [email protected] Abstract. In a series of publications, we have employed ontological theories and principles to evaluate and improve the quality of conceptual modeling grammars and models. In this article, we advance this research program by conducting an ontological analysis to investigate the proper representation of types whose instances are collectives, as well as the representation of part-whole relations involving them. As a result, we provide an ontological in- terpretation for these notions, as well as modeling guidelines for their sound representation in conceptual modeling. Moreover, we present a precise qualifi- cation for the parthood relations of member-collective and subcollective- collective in terms of formal mereological theories of parthood, as well as in terms of the modal meta-properties of essential and inseparable parts. Keywords: ontological foundations for conceptual modeling, part-whole rela- tions, representation of collectives. 1 Introduction In recent years, there has been a growing interest in the application of Foundational Ontologies, i.e., formal ontological theories in the philosophical sense, for providing real-world semantics for conceptual modeling languages, and theoretically sound foundations and methodological guidelines for evaluating and improving the individ- ual models produced using these languages. In a series of publications, we have successfully employed ontological theories and principles to analyze a number of fundamental conceptual modeling constructs such as Types, Roles and Taxonomic Structures, Relations, Attributes, among others (e.g., [1,2]).
    [Show full text]
  • Variable Intervals Model
    Mathematical Social Sciences 38 (1999) 21±33 Variable intervals model Marc-Arthur Diaye* Universite de Paris 1, LAMIA, Paris, France Received 1 June 1997; received in revised form 1 July 1998; accepted 1 August 1998 Abstract The variable intervals model is a generalization of Fishburn's Intervals model. It fully characterizes the complete acyclic relation when the alternatives set is countable. In the uncountable case, a perfect separability condition has to be added. 1999 Elsevier Science B.V. All rights reserved. Keywords: Acyclicity; Representation; Underlying preference JEL classi®cation: C00; D00 1. Introduction A number of models are now in use for the modeling of preferences. The most famous are the semiorder model and the interval order model (a review of the variants of the semiorder model can be found in Fishburn, 1997). However, a common point in all of these models is that their asymmetric parts are transitive. The purpose of this paper is to study the relationship between the variable intervals model (®rst formulated by Abbas and Vincke, 1993), which is a generalization of the intervals model (Fishburn, 1970), and the complete acyclic binary relations. The paper is organized as follows. Section 2 reviews some basic de®nitions in ordered sets theory. Section 3 is devoted to the variable intervals model. This model characterizes the complete acyclic relations when the set of alternatives is countable. In the uncountable case, a particular separability condition has to be added. According to the variable intervals model, an agent having a complete acyclic preference has an underlying true preorder (complete and transitive) preference.
    [Show full text]
  • Parthood and Naturalness 1. Introduction There Is a Familiar Distinction That One May Draw Among Features of the World. Some P
    Parthood and Naturalness Abstract: Is part of a perfectly natural, or fundamental, relation? Philosophers have been hesitant to take a stand on this issue. One of reason for this hesitancy is the worry that, if parthood is perfectly natural, then the perfectly natural properties and relations are not suitably “independent” of one another. (Roughly, the perfectly natural properties are not suitably independent if there are necessary connections among them.) In this paper, I argue that parthood is a perfectly natural relation. In so doing, I argue that this “independence” worry is unfounded. I conclude by noting some consequences of the naturalness of parthood. 1. Introduction There is a familiar distinction that one may draw among features of the world. Some properties and relations characterize the fundamental nature of the world, what the world is like “at bottom” – these are the ones that ground (in some intuitive sense) everything else. Call these properties and relations perfectly natural.1 Canonical examples of perfectly natural properties are those posited by an ideal physics (such as two grams mass), and canonical examples of perfectly natural relations are the spatiotemporal distance relations (such as three feet from). Other properties and relations characterize the superficial and gerrymandered features of world – features that obtain because of how things are at a more fundamental level. Canonical examples include properties such as grue and bleen, and relations such as richer than and happier than. Now consider the relation part of. Is part of perfectly natural? Prima facie, parthood seems to be among the properties and relations that comprise the fundamental structure of the world.2 But philosophers have been surprisingly hesitant to take part of as perfectly natural.
    [Show full text]
  • Universal Semiorders
    1 Universal Semiorders ∗ 2 Alfio Giarlotta Department of Economics and Business, University of Catania, Catania 95129, Italy Email: [email protected] Stephen Watson Department of Mathematics and Statistics, York University, Toronto M3J 1P3, Canada Email: [email protected] May 5, 2016 3 Abstract 4 A Z-product is a modified lexicographic product of three total preorders such that the middle 5 factor is the chain of integers equipped with a shift operator. A Z-line is a Z-product having 6 two linear orders as its extreme factors. We show that an arbitrary semiorder embeds into 7 a Z-product having the transitive closure as its first factor, and a sliced trace as its last 8 factor. Sliced traces are modified forms of traces induced by suitable integer-valued maps, 9 and their definition is reminiscent of constructions related to the Scott-Suppes representation 10 of a semiorder. Further, we show that Z-lines are universal semiorders, in the sense that they 11 are semiorders, and each semiorder embeds into a Z-line. As a corollary of this description, 12 we derive the well known fact that the dimension of a strict semiorder is at most three. 13 Key words: Semiorder; interval order; trace; sliced trace; Z-product; Z-line; Scott-Suppes 14 representation; order-dimension. 15 1 Introduction 16 Semiorders are among the most studied categories of binary relations in preference modeling. 17 This is due to the vast range of scenarios which require the modelization of a preference structure 18 to be more flexible and realistic than what a total preorder can provide.
    [Show full text]
  • Partially Ordered Sets
    Dharmendra Mishra Partially Ordered Sets A relation R on a set A is called a partial order if R is reflexive, antisymmetric, and transitive. The set A together with the partial order R is called a partially ordered set , or simply a poset , and we will denote this poset by (A, R). If there is no possibility of confusion about the partial order, we may refer to the poset simply as A, rather than (A, R). EXAMPLE 1 Let A be a collection of subsets of a set S. The relation ⊆ of set inclusion is a partial order on A, so (A, ⊆) is a poset. EXAMPLE 2 Let Z + be the set of positive integers. The usual relation ≤ less than or equal to is a partial order on Z +, as is ≥ (grater than or equal to). EXAMPLE 3 The relation of divisibility (a R b is and only if a b )is a partial order on Z +. EXAMPLE 4 Let R be the set of all equivalence relations on a set A. Since R consists of subsets of A x A, R is a partially ordered set under the partial order of set containment. If R and S are equivalence relations on A, the same property may be expressed in relational notation as follows. R ⊆ S if and only if x R y implies x S y for all X, y in A. Than (R, ⊆) is poset. EXAMPLE 5 The relation < on Z + is not a partial order, since it is not reflexive. EXAMPLE 6 Let R be partial order on a set A, and let R -1 be the inverse relation of R.
    [Show full text]
  • Preferences on Intervals: a General Framework
    Preferences On Intervals: a general framework Meltem Ozt¨ urk¨ Alexis Tsoukias` LAMSADE-CNRS, LAMSADE-CNRS, Universite´ Paris Dauphine Universite´ Paris Dauphine 75775 Paris Cedex 16, France 75775 Paris Cedex 16, France [email protected] [email protected] Abstract a number of properties the decision maker should be aware of. The paper presents a general framework for inter- In this paper we focus our attention on both cases, but with val comparison for preference modelling purposes. particular attention to the situations where the elements of Two dimensions are considered in order to establish the set A can or are actually represented by intervals (of the such a framework: the type of preference structure reals). In other terms we are interested on the one hand to the to be considered and the number of values associ- necessary and sufficient conditions for which the preference ated to each interval. It turns out that is possible statements of a decision maker can be represented through to characterise well known preference structures as the comparison of intervals and on the other hand on general special cases of this general framework. models through which the comparison of intervals can lead to the establishment of preference relations. Comparing intervals is a problem relevant to several differ- 1 Introduction ent disciplines. We need intervals in order to take into account Preferences are usually considered as binary relations applied intransitivity of indifference due to the presence of one or on a set of objects, let’s say A. Preference modelling is con- more discrimination thresholds (an object 10cm long is “re- cerned by two basic problems (see [Vincke, 2001]).
    [Show full text]