φ1$ MC ELS INlRANSniVE P. VAN ACKER •Η ••^•^••^•^•1 MODELS FOR INTRANSITIVE CHOICE "OMNIS ELECTIO EST EX NECESSITATE" IThomas Aquinas, Summa Тксо^од^саа , 1а2ае.13,6| PROMOTOR: PROF. DR. TH.G.G. BEZEMBINDER. MODELS FOR INTRANSITIVE CHOICE PROEFSCHRIFT TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE SOCIALE WETENSCHAPPEN AAN DE KATHOLIEKE UNIVERSITEIT TE NIJMEGEN, OP GEZAG VAN DE RFCTOR MAGNIFICUS PROF. DR. A.J.H. VENDRIK, VOLGENS BESLUIT VAN HET COLLEGE VAN DECANEN IN HET OPENBAAR TE VERDEDIGEN OP VRIJDAG 25 MAART 1977, DES NAMIDDAGS TE 4.00 UUR DOOR PETER OTTO FRANS CLEMENS VAN ACKER GEBOREN TE UKKEL (BELGIO JITVOERING FN DRUK' DYNAPRINT, BRUSSFL. Θ COPYRIGHT ESCHER-STI CHT Ι MG. ^І.С. ESCHER, "WATERFALL". (GEMEENTEMUSEUM, THE HAGUE, HOLLAND) T/u' rívbatc abuüt the anumption o¿ Oumi¿tivÍty tuim upon thv intzApneXation o-í cMtcUn leal mild expa'u't'iicei. [TULLOCK (1964, p. 401)1 5 CONTENTS CHAPTER I. ON THE NOTIONS OF PREFERENCE AND CHOICE 1.1. Introduction 7 1.2. Mathematical preliminaries 10 1.3. Axioms for preference relations 16 1.4. Binary choice sbructures 19 1.5. On the relationship between preference and choice 26 1.6. A survey of models for intransitive choice 37 1.7. The rationality issue 46 CHAPTER IT. ON VARIOUS MODIFICATIONS OF THE ADDITIVE DIFFERENCE MODEL 11.1. Introduction 55 11.2. General concepts 58 11.3. Conditions on unidimensional oreference relations 64 11.4. Preference aggregating functions 71 11.5. Conditions on preference aggregating functions 79 11.6. General options underlying the models 89 11.7. The models 91 CHAPTER Ш. TRANSITIVITY REVISITL'D 111.1. Introduction 107 111.2. The transitivity issue in the osychological littérature 110 111.3. More about some formal aspects of the theory 115 111.4. On simulation based upon analogies to the construction of sensory magnitude scales 121 111.5. On simulation based upon analogies to models of similarity 128 111.6. On the general impossibility of simulation 132 EPILOGUE 139 REFERENCES 141 SUMMARY 149 SAMENVATTING 155 CURRICULUM VITAC 160 7 CHAPTER I ON THE NOTIONS OF PREFERENCE AND CHOICE 1.1. INTRODUCTION. In the literature the terms "preference" and "choice" have often been employed informally, with the result that either their connection has been left largely undefined or that these terms have been proclaimed synonymous, preference then being identified with choice. This study is precisely concerned with the possible relationships between the notions of preference and choice. Although these notions require a rigorous and formal treatment, an attempt is made in the present Section to give an intuitive idea of the mam problems. Formal statements are presented in the subsequent sections of this Chapter. In the traditional approach to problems of choice it is quite customary to rely on a complete ordering axiom. The formal properties of ordering relations will be discussed in Section 1.2. Suffice it to say here that the complete ordering of a non-empty set of objects involves a ranking relation R with two specific properties, сompletenei i and tKano-t-t-LV-Ltij, to be presented formally in Section 1.3 together with other properties commonly used as axioms for preference relations. Completeness requires that an agent must be able to tell, for any two objects χ and у, that he prefers χ to и (symbolically, xRy) or that he prefers у to χ (yRx) , or possibly both. If both xRt/ and i/Rx hold, we can declare χ and tj as "indifferent" and refer to this as xli/. Hence, formally, completeness of the ranking relation is written as follows: XRÍ/, i/Rx or xly. If an agent chooses χ rather than t/ it is presumably because he prefers it. And an agent with a preference relation (i.e., a ranking relation R) that is complete knows his mind over every pair. So far so good, but if preference is operationally üefmed as choice, 8 then it seems unthinkable that the requirement of completeness can ever be empirically violated. Such a definition makes indeed for a simple but rather trivial theory of preference. The transitivity assumption goes as follows: for any three objects x,w,z such that an agent prefers χ to y and y to z, this agent is supposed to prefer χ to ζ. Formally, the transitivity condition states that кЯу and yRz together imply xRz. If an agent has a preference ranking over a set of objects and if choices are made in accord with these preferences, then the agent will generate transitive choices. Thus, when preference is identified with choice, the transitivity axiom guarantees an internally coherent choice pattern: if an agent chooses χ in preference to ij, and ij in preference to z, then he will choose χ in preference to z. In some straightforward sense this sounds "rational" and, indeed, transitivity has often been regarded as particularly indicative of rational behaviour, as we shall see in a subsequent section. Ihere is considerable evidence in the literature that indicates that agents, when faced with repeated choices between two given alternatives χ and y, are not perfectly consistent with their choices in that they choose χ in some instances and y in others. The findings of psychologists show that there is much of this "inconsistency" in human and animal behaviour. Hence, if an agent has chosen χ over y and ι/ over ζ in two separate trials, it may happen, for all we know, that, when presented with the pair χ and z, he chooses ζ to χ, violating the transitivity assumption. A triple к, y, ζ such that xRi/ and ij4z, yet zRx, is said to be intransitive and intransitive triples are indeed unavoidable when the pairwise choices are inconsistent over trials. These inconsistencies have led many authors to characteri7e choice in a probabilistic fashion and to de^ tic preference in terms of pairwise choice probabilities or, in practice, in terms of their estimates. In the most popular definition, preference is simply identified with a greater frequency of choice so that the inconsistency of the choices is incorporated into the preference relation and, indeed, transitivity of the preference relation is saved as long as the pairwise choice probabilities satisfy certain probabilistic consistency conditions, to be indicated in Section 1.4. As Section 1.5 tries to show, this is not the end of the story. The 9 whole issue of transitivity cannot be settled by imputing every observed violation of transitivity to the inconsistency of the pairwise choices. This would not solve the problem of the existence of "true" or genoíi/p-tc intransitivity of choices, that is, of the existence of choices that are really intransitive and not an epiphenomenon of inconsistency. We shall argue that inconsistency should be distinguished from intransitivity with which it has so often been confounded in the literature. This is the condition ¿-(.we qua иои for true intransitivity to enter theories of human or animal behaviour. By the same token we show how, quite naturally, truly intransitive choices may be related to intransitive preferences. Section 1.6 is concerned with models for intransitive choices. Our conception is that most of what appears to be observed intransitivity is but appâtent intransitivity, namely, inconsistent choice behaviour. In the absence of a model that guides the prediction of genotypic intransitivities, it may indeed be very difficult to detect consistent violations of transitivity. This is a call for models that provide genotypic intransitivities and that makes those violations plausible m terms of an analysis of the choice process. Although there has been numerous theoretical studies of choice behaviour, there are but a few exceptional models of choice that purport true (genotypic) intransitivity. Yet, despite the almost universal acceptance of the transitivity assumption, there exists experimental evidence that shows that, under appropriate experimental conditions, some behaviour may be genotypically intransitive. It is beyond doubt that we need proper choice models capable of predicting true intransitivity, the psychological phenomenon we are searching for. The transitivity assumption is not to be dismissed lightly though. It has ruled supreme in both the psychological and economic literature and transitivity is still a necessity for almost every psychological or economic theory. Its overwhelming prestige is probably due to the alleged relation between transitivity and rationality. In many instances the word "rationality" was even used as a synonym for "transitivity". Quite recently, there has been increasing recognition that the hypothesis of so-called rational behaviour should not be simply confounded with the hypothesis of transitivity. In Section 1.7 support is lent to this evolution, as we set forth our discomfort with the prevailing notion of rationality. As the modern approach to problems of preference and choice mdkes good use of the algebra of binary relations, we have to edge our way through a few mathematical preliminaries before proceeding to the formal arguments. 1.2. MATHEMATICAL PRELIMINARIES. In this Section we present, with an eye to subsequent use, some logical properties of general and specific binary relations. Many of these are well known, though several may have been obscured by the confusion that prevails in the scientific vocabulary. In order to treat the subject properly, we present a synoptic table with the terminology we have decided to adopt, along with a few alternative names used m the literature. Let A be a non-empty, not necessarily finite, set. A binary relation S over A is nothing but a specific subset of the Cartesian product ΛχΑ. If a particular ordered pair (*,«) is in the subset we write xSij or, alternatively, (x,i/)<-S; if the ordered pair (x.y) does not belong to the subset we write xStj or {K,IJ)</S.