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by: Ms. Juvenal Kertzmann


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Psychological Bulletin 1986 Vol 99 No 2 166180 Measurement The Theory of Numerical Assignments University of California lrvine Copyright 1986 by the American Psychological Association I Louis Narens R Duncan Luce Harvard University In this article we review some generalizations of classical theories of measurement for concatenation eg mass or length and conjoint structures eg momentum of mass velocity pairs or loudness of intensity frequency pairs The earlier results on additive representations are brie y surveyed Gen eralizations to nonadditive structures are outlined and their more complex uniqueness results are described The latter leads to a de nition of scale type in terms of the symmetries automorphisms of the underlying qualitative structure The major result is that for any measurement onto the real numbers only three possible scale types exist that are both rich in symmetries but not too redundant ratio interval and another lying between them The possible numerical representations for conca tenation structures corresponding to these scale types are completely described The interval scale case leads to a generalization of subjective expectedutility theory that copes with some empirical violations of the classical theory Partial attempts to axiomatize concatenation structures of these three scale types are described Such structures are of interest because they make clear that there is a rich class of nonadditive concatenation and conjoint structures with representations of the same 0033290986500 nc 75 scale types as those used in physics Many scientists and philosophers are well aware of what the physicist E P Wigner in 1960 called the unreasonable e ec tiveness of mathematics in the natural sciences Some like Wigx ner have remarked on it a few like the ancient philosopher Pythagoras c 582500 BC have tried to explain it Today as throughout much of history it is still considered a mystery There is however a part of applied mathematical science that is slowly chipping away at a portion of the mystery This sub eld usually called measurement theory focuses on how numbers enter into science Part of the eld searches for rules axioms that allow one to assign numbers to entities in such a way as to capture their empirical relations numerically Another part attempts to use such qualitative axioms to understand to some degree the nature and form of a variety of empirical relations among various dimensions Such relations when stated numerically are com monly called laws In recent times a few leading mathema ticians philosophers physicists statisticians economists and psychologists have developed new processes for measurement This work has resulted in the detailed mathematical development of new structures has provided scientists with a greater under standing of the range of mathematical structures they are likely to encounter and use in their science and has generated some longlasting controversies that are only now beginning to be re solved For surveys that go into far more technical detail than this article see Krantz Luce Suppes and Tversky 1971 in press Narens 1985 Pfanzagl 1968 1971 and Roberts1979 This work was supported in part by National Science Foundation Grant IST8305819 to Harvard University Correspondence concerning this article should be addressed to R Duncan Luce Department of Psychology and Social Relations William James Hall Harvard University 33 Kirkland Street Cambridge Mas sachusetts 02138 166 Origins of Measurement Theory Empirical Structures for Concatenations The origin ofmodern measurement theory can be traced back at least to the investigations in the late 19th century by H v Helmholtz the eclectic physicianphysicist into the formal nature of certain basic physical attributes such as mass and length which he recognized as having the same intrinsic mathematical structure as the positive real numbers together with addition and their natural order 2 We denote tliis system by Re 2 In such cases one can observe a natural empirical ordering re lation over a set of objects where the order re ects qualita tively the degree or amount of the tobemeasured attribute that is exhibited by the objects One can also nd a natural empirical operation a that combines any two objects exhibiting the at tribute into a composite object that also exhibits the attribute For example for mass one can use an equalarm pan balance in a vacuum to establish the order To be sure it is rare now to order masses in this way but conceptually such a procedure un derlies mass measurement en two objects x and y are placed in separate pans and the balance fails to tilt they are said to exhibit mass to the same degree that is to be equivalent in mass which is written x y Otherwise the object in the pan that drops say x is said to have the greater mass which is written x gt y Placing two objects x and y in the same pan constitutes the operation of combining and the result is denoted x u y In the abstract model the combining operation goes under the ge neric name of concatenation If we let X denote the set of all objects under consideration including all the combinations that can be formed using a then the potential observations from the pan balance yield the mathematical structure 96 X z We call this a qualitative structure whereas a possible representing structure such as Ref 2 gt is called a numerical representing structure MEASUREMENT THE THEORY OF NUMERICAL ASSIGNMENTS 167 One reason for studying the abstract nature of such measure ment is that the same mathematical system can apply to a wide variety of attributes We have already mentioned that the struc tures under consideration serve as a basis for measuring a number of the basic physical quantities mass length duration and charge Less obvious see below is that much the same structure underlies the measurement of probability Additive Representations Helmholtz 1887 stated physically plausible assumptions about the structure assumptions about 2 about o and about their interplay and showed that when the assumptions are true measurement can be carried out in the following sense There exists a mathematical mapping p called a homomorphism from X into positive real numbers such that for each x and y in X a x z y if and only if 06 2 ltpy and b ltpx u y 006 02 We usually say that under the mapping go the ordering relation maps 2 into 2 and the qualitative operation a into Such homomorphisms of 6 into ltRe 2 are called additive rep resentations For a formal statement of the concept of homo morphism see Appendix 1 Equally important he showed that such structurepreserving measures are relatively unique Any two differ only by a numerical multiplicative factor and multi plying any one measure by a positive numerical factor yields another measure These facts are often summarized by saying the measurement is unique once a unit has been selected The more contemporary summary statement is that the set of all such homomorphisms forms a ratio scale1 Such a complete de scription of the uniqueness of the representation is called a uniqueness theorem Axioms for Extensive Quantities In 1901 Holder a mathematician published an improved ver sion of the theory in which among other things he introduced the highly important concept of an Archimedean ordered group In this work he made signi cant use of an axiom dating back to the Greek mathematician Archimedes d 212 BC which captures the idea of commensurability within a physical attribute by asserting that no object is in nitely larger than another for any physical attribute Archimedes had introduced it in part to provide a more rigorous basis for the notion of a continuum and in part to avoid some of the paradoxes described by the philosopher Zeno In our notation this property may be for mulated as follows For the sake of concreteness consider the measurement of length for a set X of rods for which the ordering z is determined by placing two rods side by side and observing which spans the other and concatenation is determined by plac The term scale is used loosely in the literature and with much am biguity and imprecision Many authors for example refer to the usual set of representations for length as a ratio scale for length and speak of the scale type of length measurement as being ratio while simulta neously referring to individual representations as scales as in the meter scale for length We have chosen to disambiguate by calling the entire set of representations a scale and by using the term representation for the other use of scale Within this usage concepts like a ratio Scale for quot and the scale type of quot are sensible and retain their usual meanings ing two rods end to end to form another rod For each rod 6 nd another rod say x1 equivalent in length to x Then nd a rod x1 equivalent to xl u x and another 6 equivalent to x a x and so on The sequence x1 x2 xl is called a standard sequence based on x The Archimedean axiom asserts that for any two rods x and y there is some member xquot of the standard sequence based on x that is larger than y Or put another way every bounded subsequence of a standard sequence is nite In addition to the Archimedean axiom Holder assumed 96 X z satis es ve other properties closely resembling the following 1 Weak ordering The relation is transitive x z y and y z 2 imply x z z for all x y z in X and connected either xz yory xholds forallxyinX 2 Monotonicity The ordering and operation interlock in such a way that the concatenation of objects preserves the or dering that is for all x y w z in X if x z y and z z w then x o z z y a w 3 Restricted solvability For each x y in X if x gt y there exists some 2 in X such that x gt y z This together with the other axioms implies the existence of arbitrarily small objects 4 Positivity All objects combine to form something larger than either of them alone that is for all x y in X both xoygtx and xeygty 5 Associativity If one is combining three or more objects it matters not at all how the grouping by pairs occurs so long as their order is maintained that is for all x y z in X Xyozxuy02 An Archimedean structure satisfying Properties 1 5 is referred to as extensive and using Holder s method each such structure can be shown to have a representation exactly like Helmholtz s into the ordered set of positive real numbers with addition It should be noted that the axioms are of two quite distinct types Axioms 1 2 4 5 and the Archimedean property must hold if an additive representation exists that is they are necessary conditions given the representation Axiom 3 is said to be struc tural because it limits our attention to a subset of structures possessing additive representations Re nements Diference Sequences and Partial Operations Throughout this century Holder s axiomatization has been re ned and generalized For example by recasting the Archi medean axiom in terms of difference sequences satisfying the recursive relation x 1 a it x o v for some 0 gt M Roberts and Luce 1968 formulated necessary and su icient conditions for an additive representation and Narens 1974 showed that the Archimedean axiom can be dropped if one is willing to permit additive representations into a generalization of the real number system called the nonstandard real numbers A particularly im portant modi cation for measurement was the generalization to concatenations that are not necessarily de ned for every pair of objects Luce amp Marley 1969 There are at least two good rea sons for modifying the theory to deal with such partially de ned operations One is that it is usually impractical to concatenate arbitrarily large objects pan balances collapse and rods con 168 LOUIS NARENS AND R DUNCAN LUCE catenate properly only on at platforms which necessarily are bounded Another is that some important systems are inherently bounded from above and do not even in theory permit un limited concatenation Qualitative probability is one example Here uncertain events are ordered by a relation of more likely than and the union of disjoint events is taken to be concaten ation that is if A and B are events with A 0 B D then a is de ned to be A a B A U B Measurement in this situation consists in nding a function P from uncertain events into the closed unit interval such that P preserves the more likely than relation and for disjoint A and B PA B PA u B PA PB In the literature the term extensive is often applied to the gen eralization where not all concatenations are de ned as well as to closed operations as in Holder s original system Nature of Fundamental Measurement 1940 Commission Report Only Extensive Measures With the successful axiomatization of extensive structures and the recognition of their importance for the foundations of physics a curious debate ensued during the 19205 and 30s about what else is measurable Some philosophers of physics especially Campbell 1920 1928 but also Bridgman 1922 1931 and later Ellis l966 expounded the position that measurement from rst principles is necessarily extensive in character Campbell referred to scales resulting from such measurements as fun damental all else being derived Thus momentum density and all other physical measures whose units can be expressed as products of powers of the fundamental units of mass length time temperature and charge were treated as derived Although these derived measures were clearly a crucial part of the total measurement structure of physics especially as formulated in dimensional analysis no very careful analysis was provided of them They together with a basis of extensive measures form the nite dimensional vector space of physical measures that is rou tiner invoked in dimensional analysis However this vector space was not developed from entirely qualitative observations rather it was postulated as descriptive of the way numerical physical measures interlock Ellis in particular clearly understood that something more was needed and although he hinted at the so lution he failed to work it out At the same time psychologists and economists were pursuing other approaches to measurement that more or less explicitly ran afoul of the dictum that fundamental measurement rests an associative monotonic operations of combination The debate reached its intellectual nadir with the 1940 Final Report of a Commission of the British Association for Advancement of Sci ence Ferguson et al 1940 in which a majority declared fun damental measurement in psychology to be impossible because no such empirical operations could be found Campbell a mem ber of the Commission and probably a major force in its creation 8 years earlier wrote Why do not psychologists accept the nat ural and obvious conclusion that subjective measurements of loudness in numerical terms are mutually inconsistent and cannot be the basis of measurementquot Stevens 39s Reply Scale Type Not Addition Stevens whose work on loudness measurement with Davis in 1938 was in part at issue was independently considering the same question in a series of discussions in the late 19305 with a distinguished group of scientists and philosophers G D Birkho R Camap H Feigl C G Hempel and G Bergmann Out of this arose his now widely accepted position that a key feature of measurement is not only the empirical structure and its repre sentation but the degree of uniqueness of the representation as is re ected in the group of transformations that take one rep resentation into another In contrast to Campbell Stevens claimed that the nature of the transformations taking one rep resentation into another was the important feature of the rep resentation not the particular details of any axiomatization of it In his 1946 and 1951 publications Stevens singled out four groups of transformations on the real or positive real numbers as relevant to measurement onetoone strictly monotonic in creasing af ne and similarity see Table 1 And he introduced the corresponding terms of nominal ordinal interval and ratio to refer to the families of homomorphisms or scales related by these groups Later he added a fth group the power group Table 1 Measurement Scales Transformations of R Scale R x M x Absolute Re or Re x 4 kquot Discrete Re or Re k xed and positive it ratio ranges over integers x r rx Ratio Re or Re r r ranges over positive reals x v kquotx 5 Discrete Re k xed and positive n interval ranges over integers s ranges over reals x sx quot Log discrete Re k fixed and positive n interval ranges over integers s ranges over reals x rx s Interval Re r ranges over positive reals s ranges over reals x a sx Log interval Re r and s range over positive reals x fx Ordinal Re or Re f ranges over strictly increasing functions from R onto R x x Nominal Re or Re f ranges over one to one functions from R onto R Note Suppose 36 X 3 S1 I S and 7 R 2 R1 Rquot are relational structures R Re or R Re and a is the set of representations of 96 into 71 of is called the scale from 96 into it MEASUREMENT THE THEORY OF NUMERICAL ASSIGNMENTS 169 x H sx s gt O r gt 0 applicable only to measurement in the positive reals and he referred to the corresponding scale as log interval As late as 1959 he remarked about this latter scale that apparently it has never been put to usequot which as we shall see re ects a common misunderstanding of classical physics which in fact is full of loginterval scales that are conventionally treated as ratio scales by making speci c choices for the exponents Although these groups of transformations played an important role in geometry and physics and seemed to encompass much of what was then known about measurement structures Stevens offered no argument as to why these and not others should arise and thus his analysis was more descriptive than analytical By the 1950s it began to be clear that there are measurement struc tures that do not t the scheme As we shall see below consid erable progress toward understanding this question has been made in the past 4 years Having characterized scales by the type of transformation in volved Stevens went on to emphasize that scienti c propositions he was especially concerned about statistical ones formulated in terms of measured values must exhibit invariance of meaning under the admissible transformations characterizing the scale type As Luce 1978 showed this concept of a meaningful prop osition was a generalization of the familiar assumption in di mensional analysis that physical laws must be dimensionally in variant under changes of units A full understanding of the con cept of meaningful scienti c proposition still remains a challenge It is by no means clear what the circumstances are for which invariance under admissible transformations is an adequate cri terion for meaningfulness nor is it known what other criteria should be used when it is not However these involved issues are a matter for another article Narens amp Luce in press Stevens s second thrust was to devise an empirical procedure for the measurement of subjective scales in psychophysics that does not presuppose an associative operation The method which he dubbed magnitude estimation has been moderately widely used because it produces quite systematic results Nevertheless it has proved extremely difficult to defend his assumption that the method of magnitude estimation actually results in ratio scales Although he recognized more than anyone else at the time the signi cance of scale type in contrast to the particular structures exhibiting it he seemed not to appreciate that in fact the concept of scale type is a theoretical one that can only be formulated precisely in terms of an explicit axiomatic model of an empirical process He failed to acknowledge that it takes more than one s intuitions to establish that a measurement process such as magnitude estimation leads to a ratio scale Early Alternatives to Extensive Measurement At about the same time and continuing through the next two decades others were working on alternative measurement ax iomatizations that accorded better with Stevens s view of the scope of measurement than with those of the British philosophers and physicists Four of these developments are worth mentioning Beginning as early as Holder 1901 difference measurement has been axiomatized Here one has an ordering of pairs of ele ments and the representation is as numerical differences or ab solute values of differences eg see Krantz et al 1971 Chap 4 Because these structures are typi ed by line intervals identi ed by their end points it is clear that they can readily be reduced to extensive measurement and so they were not really considered an important departure from the dictum that fundamental mea surement is equivalent to extensive measurement The second was the investigation into structures having an operation that is monotonic with respect to the ordering but that is neither positive nor associative In particular Pfanzagl 1959 axiomatized structures that satis ed the condition of bisymmetry x o y a u a 1 x n u a y a 0 which is a generalization of associativity He showed that such structures have a linear rep resentation of the form qpx u y apx bgo y c where a and b are positive When a is also idempotent x o x x for all x then c 0 and b l a and the model is one for any process of forming weighted means An important physical example is the temperature that results when two gases of different tem peratures are mixed in xed proportions In addition of course averaging is important throughout the social sciences The third development which was conceptually closely related to this although technically quite different in detail was the earlier axiomatization by Von Neumann and Morgenstem 1947 of expected utility Here the operation was in essence a weightng with respect to probabilities of a chance event and its comple ment Strictly speaking this is a form of derived measurement because numbers probabilities are involved in the underlying structure however by the mid1950s purely qualitative theories were developed of which the most famous is that of Savage 1954 The resulting large literature on this topic has almost without exception led to interval scale representations of some form of averaged utilities Conjoint Measurement Perhaps the clearest demonstration of nonextensive structures for which intervalscalable fundamental measurement is possible was the creation in the 19605 of the theory of additive conjoint measurement Although the earlier examples had convinced many specialists that the scope of fundamental measurement is broader than Campbell had alleged it was only with the intro duction of conjoint measurement with its simple techniques and its possible applicability throughout the social sciences as well as physics that this view became widely accepted A con joint structure simply consists of an ordered structure that can be factored in a natural way into two or more ordered sub structures Typical examples of such structures are the ordering by mass of objects characterized by their volume and density the loudness ordering provided by a person for pairs of sounds one to each ear and the preference ordering provided by an animal for amounts of food at certain delays Observe two things about the above examples First the fac torizable orderings are very closely related to the concepts of tradeoffs and indifference curves that are widely used throughout science in each case the equivalence part of the ordering de scribes the tradeoff between the factors that maintains at a con stant value the amount of the attribute in question be it mass loudness or preference Second no empirical concatenation op eration is involved in a conjoint structure Yet as Debreu 1960 showed by using a mix of algebraic and topological assumptions and as Luce and Tukey 1964 showed using weaker and entirely algebraic assumptions such structures can sometimes be rep resented multiplicatively on the positive real numbers 170 LOUIS NARENS AND R DUNCAN LUCE More formally assume that there are two factors and let A denote the set of elements forming the rst one and P those forming the second one Thus the set A X P which is composed of all ordered pairs a p with a any element in A and p any element in P and is called the Cartesian product of A and P is the set of objects under consideration The set A X P is assumed to be ordered by the attribute in question Let t denote this ordering For example if A consists of various possible amounts of a food and P consists of the various possible delays in receiving the food then an attribute of interest is the preference of some animal or breed for various amount duration pairs Thus a p gt b q in this case means that the amount a at duration 1 is preferred or indifferent to the amount b at duration q The interesting scienti c questions are What properties do we nd or expect to satisfy and are these such that they lead to a nice numerical representation of the data The two most basic assumptions often made about 2 are that it is a weak order see Axioms jbr Extensive Quantities and Ap pendix 2 and that it exhibits a form of monotonicity that in this context is called independence One important consequence of independence is that the order induces a unique order on each of the factors What independence says is that if the value of one factor is held xed then the ordering induced by 2 on the other factor does not depend on the value selected for the xed one or put more formally for all a and b in A and p and q in P a p 2 1717 if and only if a q z b q and a p e a q if and only if b p z b 4 Note that in the rst statement the value from the second factor P is the same on both sides of an inequality whereas in the second statement the xed value is from the rst factor A The orderings induced in this fashion on A and P are denoted re spectively A and z and are de ned by a A b if and only if for some and so for all p a p z b p and p 1 q if and only if for some and so for all a 6111 2 a q It is easy to verify that they are weak orders if z is On the assumption of weak ordering and independence the next question is under what additional conditions do there exist real valued mappings 11 on A and tip on P and a function F of two real variables that is strictly monotonic in each such that for all a and b in A andp and q in P 217 b q Fill114 11041 2 F II41 Ilpq The two it functions are in some sense measures of the two components of the attribute and F is the rule that describes how these measures trade otf in measuring the attribute The rst case to be studied in detail was the one of interest in classical physics namely the one for which the ps map onto the if and only if positive real numbers and F displays a multiplicative trade oh so that a I e b q if and only if lIAGWAIJ Z town1W1 Moreover the representations form a loginterval scale see Table l which means that for each positive a and 1WAlpp is an equally good representation and any two multiplicative repre sentations are so related In psychology and economics a different but equivalent rep resentation is more usual it is additive rather than multiplicative and is de ned on all of the real numbers This representation is obtained simply by taking the logarithm of mi in the above multiplicative representation Because of this in the social sci ences the qualitative theory is usually referred to as additive conjoint measurement even when a multiplicative represen tation is being used and we follow this practice in the remainder of this article Recoding Conjoint Structures as Concatenation Ones The proofs of the original conjoint measurement theorems although correct were not especially informative and in partic ular failed to make clear that the problem could be reduced mathematically although not empirically to extensive mea surement This was established rst by Krantz 1964 who de ned an operation on A X P and later by Holman 1971 who de ned an operation on just one component The latter con struction has the advantage of generalizing to nonadditive struc tures Suppose for the moment that the structure is suf ciently regular eg continuous so that any equation of the form a p b q can be solved for the fourth element when the other three are speci ed This condition is called unrestricted solvability Turning to Figure 1 x no in A and 170 in P and consider any a and b in A such that in terms of ordering 5 induced by t on the A component a gtA a0 and b gtA do The goal is to nd a way to add together the intervals from a0 to a and from no to b The strategy is to map the an to b interval onto a comparable interval on the second component that begins at 10 and then to map the latter interval back onto the rst factor but this time with a as its starting point The map to the second factor is achieved by solving for the element called 1rb in the equation an 1rb b pa And the return mapping is achieved by solving for the element called a o b in the equation a a b p0 a 1rb What Holman discovered was that a necessary condition for the conjoint structure to be additive is for this induced concatenation operation a to be associative This in turn is equivalent to the following property called the Thom sen condition holding throughout the conjoint structure when ever both a r 5 q and c 1 b r hold then so does a 1 I 4 In essence this says that the common terms c and r cancel out as is true of the corresponding simple additive equations involving real numbers A further condition an Archimedean one is also needed in order to prove the existence of an additive representation Ba sically that axiom simply says that the induced operation meets the usual Archimedean property of extensive measurement al though it can be stated directly in terms of without reference to the operation So in sum conditions that are suf cient to MEASUREMENT THE THEORY OF NUMERICAL ASSIGNMENTS 171 construct an additive representation of a conjoint structure am weak ordering independence the Thomsen condition unre stricted solvability and the Archimedean property What Krantz 1964 and Holman 1971 did was to show that despite the fact that there is no empirical operation visible in an additive conjoint structure the tradeoff formulated in that structure can be recast as an equi ent associative mathematical operation This allowed the earlier representation theorem for extensive structures to be used to prove the existence of an additive conjoint representation This construction is such that it can actually be mimicked in practice by constructing standard sequences and using these to approximate within a speci ed error the desired measure In the early 19705 such constructions were carried out for loudness by Levelt Riemersma and Bunt 1972 and by Falmagne 1976 Generalizations Restricted Solvabilit y and Nonassociativity Since the early 19605 many variants of extensive and additive conjoint measurement have been used by scientists in a number of elds We are not able to go into the details here but the contributions of J C Falmagne P C Fishburn D H Krantz R D Luce F S Roberts 1 Suppes and A Tversky deserve special note because they repeatedly emphasized the need to understand explicitly how measurement arises in science and clearly demonstrated its potency in a number of theoretical and experimental domains The original theory of additive conjoint measurement and its reduction to extensive measurement was quickly seen to be too restrictive in two senses First in many social science situations involving tradeoffs even ones with continuous factors mn restricted solvability fails to hold For example the loudness of a pure tone depends both on signal intensity and frequency which is the reason for loudness as well as gain controls on an ampli er but the limits on human hearing are such that it is not always possible to match in loudness a given tone by adjusting the frequency of another tone of prescribed intensity The reasons for this have to do with the processing limits of the human car What does hold however is a form of restricted solvability which says for example that with b in A and p and q in P given then there is an element a in A that solves the equivalent a p b q provided that we know there exist elements a and a in A such that a 39 p gt b q gt a 1 So for example letting the first component be the intensity of a tone and the second its frequency if b q is a given tone and p is a given frequency then the question is whether there is an intensity a so a 7rb bp aobp Factor P aom b rib Po a0 aOb Factor A Figure 1 A graphic depiction of the solutions 1rb and a bin a conjoint structure whose components are mapped on a continuum The solid curves are indifference curves Various values on factor A are denoted do a b and a b and those on factor P by P0 and 1rb 172 LOUIS NARENS AND R DUNCAN LUCE that the tone a p is equal in loudness to b q Although it is not always possible to nd such an intensity it is certainly plau sible that it exists whenever there are intensities a and a so that tone a39 p is louder and the tone aquot p is less loud than b q It turns out although the argument is more complex that one can still prove the existence of an additive representation with restricted solvability substituted for unrestricted solvability see Krantz et a1 1971 Chap 6 In terms of Holman s induced operation mentioned above this change of assumption renders the operation a partial one that is one that is de ned only for some pairs of elements Because it is possible to work out a version of extensive measurement for partial operations see Rc nements Di rence Sequences and Partial Operations indeed such is necessary to understand probability as fundamental measure ment it is still possible to carry out the construction for the conjoint structure Second the property of additivity captured in the Thomsen condition does not always hold Fortunately Holman s de nition of an operation or a partial operation in the case of restricted solvability does not in any way depend on the Thomsen con dition Thus in general any conjoint structure gives rise to a concatenation structure in which the induced operation satis es monotonicity Such induced operations are associative in exactly those cases in which the Thomsen condition holds Moreover one can show that a very great Variety of nonassociative operations arise as induced operations of conjoint structures This in itself was adequate reason to study nonassociativc concatenation structures which began in the mid1970s In trying to understand the uniqueness of nonassociative representations a more com plete theory of scale types described in Scale Type General De nition had to be developed Narens and Luce 1976 showed that any concatenation struc ture meeting all of the conditions for an extensive structure except for associativity has a numerical representation in terms of some nonassociative numerical operation Their proof was not con structive Rather it rested on the classic result of the mathema tician Cantor 1895 to the effect that a totally ordered set X a weakly ordered set in which equivalence is equality is isomorphic to a subset of real numbers under 2 if and only if it includes a subset Y comparable to the rational numbers in the sense that it is countable Ycan be put in onetoone correspondence with the integers and order dense in X which means that for any two distinct elements of X it is possible to nd at least one element from Y that is between them The key to the proof was to use the axioms of the nonassociative structure to show the existence of such a countable dense subset Since then Krantz has de veloped a constructive proof Krantz et al in press At the time Narens and Luce 1976 were much concerned by their failure to characterize fully the family of representa tions the scale They were able to show that when two homo morphisms into the same numerical system agree at a point then under weak conditions they are identical This result does not however establish how two different homomorphisms relate That question was resolved by Cohen and Narens 1979 who showed that the group of automorphisms of this kind of con catenation structure and so the group of transformations that describe its scale type see Table 1 can be ordered in such a way that the Archimedean axiom holds Thus by what Holder had established the transformation group is isomorphic to a subgroup of the multiplicative group of the positive real numbers When the subgroup is actually the entire group we have What Stevens called a ratio scale The other subgroups had not been previously encountered as measurement Scales but Cohen and Narens were able to give numerical examples of each type We return to questions of scale type later see Scale Type General De nition Distribution of Concatenation Operations in Conjoint Structures Once it is realized that conjoint measurement which treats those structures Campbell spoke of as derived is just as free from prior measurement as is extensive measurement a problem arises that understandably went unrecognized by the earlier in vestigators An attribute such as mass can be fundamentally measured in more than one way For example the mass ordering of substances S and volumes V yields a conjoint representation ILsIlv which is a measure of mass Figure 2 At the same time the usual extensive structure of concatenation of masses leads to the standard additive measure pm Obviously 475 must be an increasing function of pm because both measures preserve the mass ordering Furthermore because volumes can also be concatenated an extensive measure of volume rpv also exists and the conjoint measure of volume uv is a monotonic in creasing function of it From what is known about physical mea surement a particular its call it p5 can be chosen so that to rpsgw This is the representation that is customarily used for this conjoint structure and the particular substance measure 995 is called the density of the substance Note however that from the point of view of conjoint measurement for each positive real 1 and arpsquotltpvquot is an equally valid representation and so any is an equally valid measure of density Thus by selecting the exponent B to be 1 or equivalently by identifying WV With W we have by at altered what is really a loginterval repre sentation density into one that appears to be a ratio scale This means that in order to force density actually to be a ratio scale more physical structure than the ordering of the density volume pairs is needed As we noted earlier Stevens 1959 failed to recognize the use of such conventions when he remarked that loginterval scales were scarce Quite the contrary they are ex ceedingly common but are often lost sight of by the practice of making certain arbitrary choices of exponents The reason why the extensive and conioint measures of the same attribute are often powers of each other is that the two structures are interlocked qualitatively by what are called laws of distribution In the example above two such laws hold one between mass and the conjoint structure and the other between volume and the conjoint structure Such laws take the following form for an operation on a component Let k be the conjoint ordering of A X P 2 the order induced on the rst component and a a concatenation operation on A such that A 5 ad is an extensive structure Following Narens and Luce 1976 and Ramsay 1976 we say that a is distributive in the conjoint structure provided that the following condition holds for all a b 6 din A and p q in P whenever a 1 0 a and b p d 4 a v4 1712 6 A d a then MEASUREMENT THE THEORY When the operation is on A X P a somewhat dilTerent but equiv alent formation is needed it is not di tcult to show that if the extensive and conjoint measures on the A component are related b n o 39 39 39 39 quot39 39 must hold It derives from tlte usual numerical distribution x yz tz yz A major observation of Narens and Luw me was that the 39 distributive interlock is a qualitative with the properties of extensive and additive contoint structures underlies the entire structure of physical dimensions In fact Narcns l98la mtablished a far more general result than the one mentioned ab e 0n need tiv 39s su icient thatthe structure converse is also true The condition which together involving the concatenation memtion has a ratio stale rep sentation exactly what that entails is described in Scale Type General De nilian and Possible Itopresentment afCtmmlenalian Simlures Moreover one need not assume that the conjoint I m it n 39 39 39 39 mm the other assumptions General Representation Theory Raprcrt39ematians and Scales As the various examples of measurement discussed above npr peared alter World War 11 it began w 39 that CONTAINERS Volumes M V was lt v 32 mm 7 A Ol NUMERICAL ASSIGNMENTS 173 they are all special cases of a general method of measurement that has come to he referred o as re resentational theoryquot This View whose earliest explicit Formulations were probably holds that measurement is possible whenever the following ob tains First the underlying empirical situation is characterized a th These primitivts are erupt nations on X that characterize the empirical situation under consideration Second there are rcslrictions axioran the structure that re ect truths about the empirical situation These are to be considered as putative empirical laws Third there is speci ed a numerically based relational structure 7 2 R where 39 ubsct o be timbers and the R are relations and operations of comparable types to the cor responding cmpirical ones Finnlly the fourth feature which accomplishes mt asuremcnl is the proof or the existence of a 39 39 to R We refer to quotC as the empirical or qualitative ttrurmre 72 as the representing slrut mre and the stru preserving map ing as a homo morphism ora representation The collection olallhcrmomorph isms into the same representing structure is referred to as a scale see Footnote l The b 39e aim of representational theory is rst to use the l 9quot II 4 kt S E 3 LIQUIDS Substances v 3 vzs t 0 174 LOUIS NARENS AND R DUNCAN LUCE Yuz A Figure 3 Observer B lives on Object l and perceives Object 2 as having velocity y whereas observer A perceives Object 1 and its resident observer B as having velocity x and Object 2 as having velocity 2 The concate nation operation is de ned by x y z the existence or representation theorem and second to char acterize how these mappings homomorphisms that constitute the scale relate to one another this is called the uniqueness theorem In the classical case of extensive measurement it is shown that a nonempty scale exists and is characterized as a ratio scale in the sense that p and p are both in the scale if and only if there is a positive real constant r such that ltp M It should be realized that the representing structure is not itself unique there always are a variety of alternative ones and different ones are used for different purposes Velocity provides an example of this Suppose X is a set of constant velocities in a given direction that are ordered by the distances traveled in a xed time interval Concatenation of velocities x and y is the velocity that is obtained by superimposing x on y That is x o y is the velocity of a body that an observer on another body moving at velocity x would judge to have the velocity y Figure 3 In classical physics X is taken to be all possible velocities whereas in relativistic physics it is convenient to restrict X to velocities less than that of light Except for that difference the two structures are assumed to be extensive however in their measurement very di erent representations are used In the classical case the usual additive representation is used but in the relativistic case one selects c gt O to represent the velocity of light and maps X k into 0 0 2 63 where 63 is de ned as follows for all u and v in 0 c l uvcz It can be shown that these two numerical representing structures are in fact isomorphic the isomorphism being u tanh u c u in 0 c If in the relativistic case n were represented additively the velocity of light would be assigned the value 00 The real reason for changing the representation from an additive one is not to avoid co but rather to maintain the usual relation among velocity distance and duration namely that the former is pro portional to distance traversed divided by the duration uEBCv Homogeneity and Uniqueness With the results about nonassociative structures as a stimulus and working within the general representational framework Na rens 198 1b proposed a method for classifying scale types which has proved useful in describing the possible representations that can arise Although the two concepts needed homogeneity and uniqueness are formulated in a rather abstract way only the former seems illusive So we focus On it both here and in the next section Many of the most familiar mathematical structures used in science such as Euclidean space exhibit the property of being homogeneous Like homogenized milk each part of the space looks like each other part This is the general intuitive concept Every element in the domain of the structure is from the point of view of the properties de ning the structure its primitives just like every other element There is no way of singling out an individual element as different from the others To formulate this precisely and generally two things are needed a a very general concept of what we mean by a structure and b the concept of an automorphism of a structure The latter permits us to say when the structure looks the same from two points of view To describe the situation a very general model of measurement is used First 96 X z 5 Squot is a relational structure that characterizes the empirical situation in the sense that k is a total ordering of X ie a weak ordering for which indifference is actually equality and SI Sl are other empirical re lations Second it R 2 R1 Rquot is the representing numerical structure And third 6 is a scale for 96 In many important scienti c applications R is either the real numbers or the positive real numbers and the elements of e are isomorphisms of 96 onto 71 We assume this situation throughout the rest of this article unless stated otherwise An automorphism is simply an isomorphism of a structure with itself that is a onetonne map of the structure with itself that preserves all of the primitives lntuitively an automorphism corresponds to what we usually refer to as a symmetry of the structure namely a mapping of the structure so that things look the same before and after the mapping is completed So for example if the structure is a sphere we know that it is sym metrical in the sense that it looks exactly the same before and after any rotation about its center Thus for the sphere rotations are automorphisms The general concept applies of course to any relational structure It is easy to verify that for each auto morphism a of 96 and for each P in 5 the mapping p a a where at denotes function composition ie for x in X go a ctx ltpax is also in A and if d is also in of then vi ltp a go39 is an automorphism of 96 Thus there is a oneto one correspon dence between the scale of and the automorphism group and so a classi cation of the one is equivalent to a classi cation of the other The following classi cation of the automorphism group in terms of its richness called homogeneity and of its redun dancy called uniquenessquot has proven to be very useful The structure is said to satisfy I point homogeneity if and only if for each x y in X there exists an automorphism a of the structure such that not y This means that the structure exhibits a good deal of symmetry because the automorphisms of a structure capture its symmetries In geometry this concept is equivalent to the concept of l transitivity which has been extended there to apply to any M distinct points mapped by a continuous trans formation to any other M distinct points of 9G in which case it is called M transitivity For measurement the generalization that is relevant is that each set of M ordered elements can be mapped by some automorphism into any other set of M com parably ordered elements This latter condition is called Mpoint MEASUREMENT THE THEORY OF NUMERICAL ASSIGNMENTS 175 homogeneity ForM gt i it is different from the geometric concept of Mtransitivity When a structure is M point homogeneous for every positive M it is said to be 00point homogeneous It is to quot l point U quot quot just to homogeneity but we are careful to distinguish clearly other values of Mpoint homogeneity To capture the idea of redundancy in the automorphism group we say that the structure satis es Npoint uniqueness if and only if whenever two automorphisms agree at N distinct points then they agree everywhere If the structure is not Npoint unique for any nite N it is said to be 00point unique Several simple observatiOns Suppose a structure is in nite Mpoint homogeneous and Npoint unique Then M s N if M 39 s M then the structure is M point homogeneous and if N z N then it is N point unique Thus in particular all M point homogeneous structures M 2 1 are lpoint homogeneous that is homogeneous Testing for Homogeneity Although homogeneity is a concept about the structure it is in fact usually not obvious how to recast it in terms of qualitative properties that can be readily studied empirically In some cases particularly when there is a primitive binary operation such logical equivalences are known see Possible Representations of Concentration Structures It should be mentioned that the proofs of such equivalences are usually not easy and generally require much mathematical machinery or the use of a nontrivial rep resentation uniqueness result Quite often homogeneity need not be explicitly stated because it follows as a consequence of a representation theorem For example in the extensive case for which there is a representation onto the positive real numbers lpoint homogeneity easily follows from the existence and uniqueness results for additive representations In such cases or in ones in which an empirical equivalent is known homogeneity does not pose a serious empirical problem Yet in many impor tant scienti c applications no such structural equivalences are known in such cases homogeneity is simply postulated directly as a theoretical concept Nonetheless because of its power it is often easy to devise simple tests to show that it does not hold even though we may not know how to test a irmatively for when it does hold The following is one of the more useful such tests Suppose X X S S is a relational structure and that P is a Property one place relation about X that is de nable from the primitives S 8 using ordinary rst order predicate logic It can be shown that if 96 is lpoint homogeneous then either Px is false for every x in X or Px is true for every x in X The following examples illustrate its use Suppose X is a qual itative structure for probability see Re nements Diszzrence Se quences and Partial Operations and A z B stands for A is at least as likely as B Consider the predicate PA B A z B Observe that PA is true for A the sure event and false for A the null event Thus We know that qualitative probability is not homogeneous This contrasts with the usual extensive models for length and mass which are homogeneous As a second example consider a structure 0 1 R in which 0 1 is the for all open interval of real numbers between 0 and 1 and R is the ternary relation on 0 1 de ned by Rx y 2 if and only if x y z are in 0 1 and x y 2 Consider the predicate Px Because P13 is true and P23 is false the structure 0 1 R is not homogeneous As we shall see in the next section the only other important case of nite point homogeneity is 2point Unlike lpoint ho mogeneity it has proved very dif cult to nd qualitative equiv alences to 2point homogeneity that are empirically realizable and hold across a wide range of interesting structures So in practice one either simply postulates it as a theoretical assump tion or derives it usually through a complicated mathematical argument from the particular primitive relations under consid eration As with lpoint homogeneity there are ways to show that it fails a Because structures that are 2point homogeneous are also lpoint homogeneous see the end of Homogeneity and Uniqueness the de nitional test for lpoint homogeneity can be invoked b Because a lpoint unique structure cannot be 2point homogeneous see the end of Homogeneity and Unique ness it suf ces to show the structure is lpoint unique and sometimes that is easy to do c As we describe in Possible Rep resentations of Concatenation Structures the special case of a 2point homogeneous structure with a primitive monotonic op eration necessarily has a very restrictive form of numerical rep resentation and it may be possible to show by empirical tests that such a representation is simply too restrictive to model the empirical situation there exists a 2 such that Rx x 2 Scale Type General De nition Recall that in in nite structures there is a largest value K of homogeneity and a smallest value L of uniqueness These are referred to respectively as the degree of homogeneity and uniqueness This pair of numbers K L is useful for classifying the type of scale exhibited by a structure it is called the scale type It is easy to verify that if 6 is a ratio scale then cf is of type 1 1 if a is an interval scale then S is of type 2 2 and if A is an ordinal scale then A is of type 00 co Narens 19813 1981b established the following converse of these observations Suppose a structure has a representation onto the real numbers If its scale is of type 1 1 then a representing structure can be found such that its representations form a ratio scale and if the scale is of type 2 2 then it has a representing structure such that its representations form an interval scale In addition he showed that it is impossible for the scale to be of type M M for 2 lt M lt 00 There are 00 00 cases that do not have ordinal scale representations however this does not much matter because the 00 point homogeneous cases including the ordinal scalable onessimply do not arise in empirical situations for which there is a reasonable amount of structure Alper 1984 1985 has shown that the only cases of structures with representations onto the positive real numbers and of scale type K L with 0 lt K lt 176 DOUIS NARENS AND R DUNCAN LUCE Land 1 SL lt ooaretheonesinwhichK1andL 2 and in that case a discrete interval scale Table 1exists These results give considerable insights into why so few scale types have arisen in the development of the sciences The whole issue of how intelligently to classify structures with eitherK 0 orL no is wide open Possible Representations ofConcatenation Structures For the important and widely applicable case of concatenation structures of the form 96 X z a where z is a weak ordering on X and a is a binary operation on X comparable results to those given for general structures hold without the assumption that 26 can be mapped onto the real or positive real numbers Luce and Narens 1983 1985 have shown that if such a con catenation structure is of scale type K L with K gt 0 and L lt 00 then only types 1 l 1 2 and 2 2 can occur The latter two necessarily are idempotent and the 1 1 type is either idempotent weakly positive x a x gt x for all x or weakly negative x a x lt x for all x An important sulficient condition for L to be nite is that the structure have a representation onto the positive real numbers for which the numerical operation is continuous Continuity of an operation is usually judged to be an acceptable scienti c idealization For these three scale types it is desirable to describe all possible candidate numerical rep resenting structures So using Narens s 1981a 198 lb results it suf ces to consider concatenation structures on the positive reals with ratio loginterval or log discrete interval scales Table 1 Suppose that 63 denotes the representing operation Luce and Narens l 985 extending the results of Cohen and Narens 1979 have shown that in all these cases there exists a function f from the positive real numbers into itself such that f is strictly increas ing xx is strictly decreasing and the operation is given by x 99 y y xy It is worth noting that the only cases in which the above mentjoned homogeneous structure can be positive x a y gt x xoygt yornegativexaylt xxnyltyarethel lones with f1 1 All the remaining structures are intensive in the sensethatxux xandifx gt ythenx gtxygt y Formal properties of concatenation structures are summarized in Ap pendix 3 Clearly the above operation is invariant under ratio scale transformations The 1 2 and 2 2 cases simply impose additional restrictions on f For example consider the equation such that for all x gt 0 X fix The l 1 case is characterized by the equation holding if and only if p l the 1 2 case by its holding if and only if for some xed k gt O and variable integer n p k and the 2 2 case by its holding for all p gt 0 In this situation the 2 2 case is equiv alent to the existence of constants c d 0 lt c d lt 1 such that if x gt y x if xcylc x GB y X N y xquoty d if x lt y The last representation called the dual bilinear representation shows that the 2 2 case is highly restrictive and that all 2 2 operations are nothing more than two pieces of two bisymmetric operations So far as we know the dual bilinear representation except for c d has not arisen in physics but recently Luce and Narens 1985 have used it to formulate a generalized theory of expected utility which appears to overcome a number of the empirical discon rmations of the classical theories of the subject This is described in the next section Before turning to that we consider two further questions ax iomatization of general concatenation structures and conditions equivalent to homogeneity Narens and Luce 1976 showed that concatenation structures satisfying all of the axioms of extensive structures except possibly associativity had a numerical repre sentation Such structures called PCSs play an important role in measurement theory Luce and Narens 1985 have provided a comparable axiomatization for general intensive structures Much is known about axiomatizing homogeneity for concaten ation structures First on the assumption of a representation onto the real numbers certain basic algebraic properties such as associativity bisymmetry and right autodistributity x n y a 2 x z a y a 2 all force homogeneity to hold2 Second for 1 adds variety of PCSs homogeneity is equivalent to the following structural condition For all elements x and y and all positive integers n xquot yn xn yna where x denotes the nth element of a standard sequence based on x see Axioms for Extensive Quantities Luce 1986 has shown that a closely related although perhaps less useful cri terion exists for homogeneity in intensive structures The third method for establishing homogeneity is to axiomatize directly all concatenation structures of39 a given type For the 2 2 case this is equivalent to axiomatizing the dual bilinear representa tions which was done in Luce 1986 These techniques for characterizing concatenation structures by scale type can of course be extended to nonadditive conjoint structures as Luce and Cohen 1983 showed however matters are a bit more complex than one might first anticipate In par ticular automorphisms of the conjoint structure need not always factor into automorphisms of the orderings induced on the com ponents and even when they do the scale types are not usually the same We do not go into these complex details here Dual Bilinear Utility A theory of preferences among gambles can be based on the idea that gambles can be concatenated in a special way to form other gambles and that rationality considerations need be applied only to the simplest concatenations of gambles with gambles Luce amp Narens 1985 When rationality considerations are more broadly invoked even marginally this theory reduces to the usual subjective expected utility model used throughout the social sciences 2 It is worth noting that many times when an axiom is added to a general concatenation structure that has a numerical representation it can be formulated numerically as a functional equation In some cases solutions are available in the literature a good starting point for nding such solutions is Acz l 1966 MEASUREMENT THE THEORY OF NUMERICAL ASSIGNMENTS 177 More speci cally suppose x and y are gambles and A is an event Then x A y denotes the gamble in which x is the outcome when A occurs and y when A fails to occur It is assumed that there is a preference ordering over gambles The Luce and Narens model ends up with a utility function U over the gambles ie a realvalued function such that g g g if and only if Ug 2 Ug2 and two weighting functions S and 539 de ned over events such that Ux eA y UXSA Uy1 SA if UX gt Uy U06 if Ux Uy UxS A Uy1 S A if Ux lt U y The weighting functions S and 5 need not be probability functions This model is called the dual bilinear utility model The standard subjective expected utility model SEU arises when S S a nitely additive probability measure over the set of events see Fishburn 1981 for a detailed summary of SEU The dual bilinear model may seem a little arti cial at rst How ever it follows from an admost universally used assumption about utility functions namely that the representations of an individ uals utility function over gambles form an interval scale together with some very natural and weak assumptions about A and 2 An additional reason for considering the dual bilinear model is that it is weaker than the SEU model and there is an abundance of empirical data showing that SEU fails to describe behavior A summary of many of the problems was given by Kahneman and Tversky 1979 Basically the failures are concerned with three types of rationality The rst is transitivity of preference which has been shown to fail under some circumstances by Lich tenstein and Slovic 1971 1973 and Grether and Plott 1979 No model such as the present one which associates utility with gambles can account for this The second type of failure has to do with what Luce and N arens call accounting equations and Kahneman and Tversky refer to as the framing of gambles An example of an accounting equation that is implied by the dual bilinear model is x AY Byx By Aya where A and B are independent events such as A is an even number coming up on a roll of a die and B is a red number coming up on a turn of a roulette wheel Observe that x is the outcome on both sides if in independent realizations of the events both A and B occur in that order on the left and in the opposite order on the right An example of another accounting equation IS xu y m 2 XVI 2 u y u 2 where successive A mean independent realizations of the event A eg the rst A refers to an even number coming up on a roll of a die the second A as an even number coming up on a di brent roll of the same die etc This holds in the bilinear model if and only if S S which is true for SEU but not in general for the dual bilinear model The earliest discussion of failures of accounting equations was by Allais 1953 see also Allais amp Hagen 1979 A third type of failure is also one of the accounting type but it is more subtle because it involves a kind of monotonicity of events Suppose C is an event that is disjoint from events A and B then the assertion is onyzxwy In essence then the pair of gambles on the right is obtainable from the pair on the left by shifting the assignment of outcomes over C and y to x Ellsberg 1961 pointed out that this often fails for people s preferences and this has been repeatedly con rmed In the dual bilinear model this equivalence holds if and only if the two weights exhibit the property that for C disjoint from A and B S A a S IB if and only if STA U C 2 S39fB U C This is true of the SEU model because of the Ss are probabilities and so S A U C S IA S iC The basic distinction between the two types of accounting equations has to do with forcing the two weights to be identical in one case and to be probability functions in the other It should be noted that this model is in many ways similar to and more completely speci ed than the prospect theory of Kahneman and Tversky 1979 as was discussed in some detail in Luce and Narens 1985 As yet no empirical studies have been reported that are targeted directly at the dual bilinear model if and only if x oAUcy z x BUC y ior Conclusions In summary a great deal is now known about the scales for inherently symmetrical onedimensional attributes and about how they interlock to form the systematic structure of multidi mensional physical quantities Perhaps the major milestones of the past 25 years are these First the development of conjoint structures which not only provided a deep measurement analysis of the numerous nonextensive derived structures of physics but also provided a measurement approach that appears to have applications in the nonphysical sciences and has laid to rest the claim that the only possible basis for measurement is extensive structures Second the development of the distributive interlock between ratio scale concatenation structures and conjoint struc tures which serves to explain why physical measures are all in terlocked as products of powers of a few ratio scales Third the growing recognition of the importance of automorphism groups in classifying measurement structures and the explicit de nition of scale type in terms of degree of homogeneity and degree of uniqueness Fourth the application of that classi cation to or dered structures with a concatenation operation and to conjoint structures thereby providing a complete catalogue for these sit uations of the possible representing structures for the homoge neous cases A number of important problems remain unresolved For one we do not have an adequate axiomatization of the general class of homogeneous intensive structures except for the interval scal able ones Second we do not have comparable results for non homogeneous structures even ones with concatenation opera tions This is not an esoteric question because any totally ordered structure with a partial operation such as probability when looked at the right way has only one automorphism the identity map Thus in such cases the automorphism group fails to em 178 body any structural information Nevertheless despite their lack of global symmetry such structures often appear to be quite regular in other aspects and this needs to be captured in some fashion and studied Third there are some important cases of interlocking concatenation and conjoint structures that are not covered by the distribution results mentioned perhaps the most striking example being relativistic velocity as a component of the distance conjoint structure with time as the other component Because many psychological attributes appear to be bounded understanding this physical case may be more pertinent than it rst might seem References Acz l J 1966 Lectures onfunctional equations and their applications New York Academic Press Allais M 1953 Le comportement de 1 homrne rationnel devant 1e risque critique des postulats et axiomes dc l cole Americaine The behavior of the rational man in the face of risk A critique of the postulates and axioms of the American school Econometrica 21 503546 Allais M amp Hagen O Eds 1979 Expected utility hypothesis and the Allais paradox Dordrecht The Netherlands Reidel Alper T M 1984 Groups of homeomorphisms of the real line Un published Bachelor of Science thesis Harvard University Cambridge Massachusetts Alper T M 1985 A note on real measurement structures of scale type in m 1 Journal of Mathematical Psychology 29 73 81 Bridgman P 1922 Dimensional Analysis New Haven Yale University Press 2nd ed published 1931 Campbell N R 1920 Physics the elements Cambridge England Cambridge University Press Reprinted 1957 as Foundations of Sci ence New York Dover Campbell N R 1928 An account of the principles of measurement and calculation London Longsman Green Cantor G 1895 Beitrage zur Begriindung dcr transliniten Mengenlehre Contributions to the founding of the theory of trans nite numbers Math Ann 46 481 512 Cohen M amp Narens L 1979 Fundamental unit structures A theory of ratio scalability Journal of Mathematical Psychology 20 193 232 Debrcu G 1960 Topological methods in cardinal utility theory In K J Arrow S Karlin amp P Suppes Eda Mathematical Methods in the Social Sciences 1 95 9 pp 16 26 Stanford CA Stanford University Press Ellis B 1966 Basic concepts of measurement London Cambridge University Press Ellsberg D 1961 Risk ambiguity and the Savage axioms Quarterly Journal of Economics 75 643 669 Falmagne JC 1976 Random conjoint measurement and loudness summation Psychological Review 83 65 79 Ferguson A Chairman Meyers C S ViceChairman Bartlett R J Secretary Banister H Bartlett F C Brown W Campbell N R Craik K J W Drever 1 Guild J Houstoun R A Irwin J 0 Kaye G W C Philpott S J F Richardson L F Shabe J H Smith T Thouless R II amp Tucker W S 1940 Quantitative estimates of sensory events The Advancement of Science The Report of the British Association for the Advancement of Science 2 33 1 349 Fishburn P C 1981 Subjective expected utility A review of normative theories Theory and Decision 13 139199 Grether D amp Plott C 1979 Economic theory of choice and the pref erence reversal phenomenon American Economic Review 69 623 638 Helmholtz H von 1887 Zihlen und Messen erkenntnistheoretisch LOUIS NARENS AND R DUNCAN LUCE betrachet Phllosophische Au sdtz Eduard Zeller gewidmet Leipzig Reprinted in MssenschaftlicheAbhandlungen 1895 Vol 3 pp 356 391 C L Bryan trans Counting and Measuring Princeton NJ Van Nostrand 1930 Holder 0 1901 Die Axiome der Quantitat und die Lehre vom Mass The axioms of quantity and the theory of mass Sitchsische Alcademie Wissenschuften zu Leipzig Mathematisch Physische Klosse 53 1 64 Holman E W 1971 A note on conjoint measurement with restricted solvability Journal of Mathematical Psychology 8 489 494 Kahneman D amp Tversky A 1979 Prospect theory an analysis of decision under risk Econometrica 47 263491 Krantz D H 1964 Conjoint measurement The Luce Tukey axiom atization and some extensions Journal of Mathematical Psychology 1 248 277 Krantz D H Luce R D Suppes P amp Tversky A 1971 Foundations of Measurement Vol 1 New York Academic Press Krantz D II Luce R D Suppes P amp Tversky A in press Foun dations of Measurement Vol 2 New York Academic Press Levelt W J M Riemersma J B amp Bum A A 1972 Binaural additivity of loudness British Journal ofMathematical and Statistical Psychology 25 5168 Lichtenstein S amp Slovic P 1971 Reversals of preferences between bids and choices in gambling decisions Journal of Experimental Psy chology 89 46 55 Lichtenstein S amp Slovic P 1973 Responseinduced reversals of pref erences in gambling An extended replication in as Vegas Journal of Experimental Psychology 101 16 20 Luce R D 1978 Dimensionally invariant laws correspond to mean ingful qualitative relations Philosophy of Science 45 1 16 Luce R D 1986 Uniqueness and homogeneity of real relational struc tures Manuscript submitted for publication Luce R D amp Cohen M 1983 Factorizable automorphisms in solvable conjoint structures 1 Journal of Pure and Applied Algebra 27 225 261 Luce R D amp Marley A A J 1969 Extensive measurement when concatenation is restricted and maximal elements may exist In S Morgenbesser P Suppes amp M G White Eds Philosophy science and methods Essays in honor of Ernest Nagel pp 235449 New York St Martin s Press Luce R D amp Narens L 1983 Symmetry scale types and general izations of classical physical measurement Journal of Mathematical Psychology 27 44 85 Luce R D amp Narens L 1985 Classi cation of concatenation mea surement structures according to scale type Journal of Mathematical Psychology 29 172 Luce R D amp Tukey J W 1964 Simultaneous conjoint measurement A new type of fundamental measurement Journal of Mathematical Psychology 1 l27 Narens L 1974 Measurement without Archimedean axioms Philos ophy of Science 41 374393 Narens L 1976 Utility uncertainty tradeoi structures Journal of Mathematical Psychology 13 296 322 Narens L 1981a A general theory of ratio scalability with remarks about the measurementtheoretic concept of meaningfulness Theory and Decision 13 1 70 Narcns L 1981b On the scales of measurement Journal of Mothe matical Psychology 24 249 275 Narens L 1985 Abstract measurement theory Cambridge MA MIT Press Narens L amp Luce R D 1976 The algebra of measurement Journal of Pure and Applied Algebra 8 197 233 Narens L amp Luce R D in press Meaningfulness and invariance concepts In J Eatwell M Milgate amp P Newman Eds The new Palgrave A dictionary of economic theory and doctrine New York Macmillan MEASUREMENT THE THEORY OF NUMERICAL ASSIGNMENTS 179 Pfanzagl J 1959 A general theory of measurement applications to utility Naval Research Logistics Quarterly 6 28 3 294 Pfanzagl J 1968 Theory of Measurement New York Wiley 2nd ed published 1971 Ramsay l O 1976 Algebraic representation in the physical and be havioral sciences Synthese 33 419 453 Roberts F S 1979 Measurement theory Reading MA AddisonWesley Roberts F S amp Luce R D 1968 Axiomatic thermodynamics and extensive measurement Synthese 18 311 326 Savage L J 1954 Foundations of statistics New York Wiley 2nd revised ed New York Dover 1972 Scott D amp Suppes P 1958 Foundational aspects of theories of mea surement Journal of Symbolic Logic 23 113 128 Stevens S S 1946 On the theory of scales of measurement Science 103 677 680 Stevens S S 1951 Mathematics measurement and psychophysics In S S Stevens Ed Handbook of experimental psychology pp 1 49 New York Wiley Stevens S S 1959 Measurement psychophysics and utility In C W Churchman amp P Ratoosh Eds Measurement De nitions and theories pp 18 63 New York Wiley Stevens S S amp Davis H 1938 Hearing Its psychology and physiology New York Wiley Suppes P amp Zinnes J L 1963 Basic measurement theory In R D Luce R R Bush amp E Galanter Eds Handbook of mathematical psychology Vol 1 pp l 76 New York Wiley Von Neumann J amp Morgenstem 0 1947 Theory of games and eco nomic behavior Princeton NJ Princeton University Press Appendix 1 Some Structure Preserving Concepts 9G X So S S is said to be a relational structure if and only if X is a nonempty set and So SI S are relations or operations on X a is said to be a homomorphism of the relational structure 96 X if S is an operation If R is a subset of reals and R0 is the usual ordering z of the reals restricted to R then in measurement theory homo morphisms of 96 into 5 are called representations 1p is said to be an isomorphism of 96 onto it if and only if p is a S S 51 into the relational structure it R R0 R1 R homomorphism of 96 into it go is onto 32 and 1 isaone to one function if and only if p is a function from X into R for k 0 n St and Rh q is said to be an automorphism of 96 if and only if tp is an isomorphism have the same number it of arguments and for all x x in X of 96 onto itself Skoch 39 xik if and only if Rkl xl I 39 900k The set of automorphisms G of a relational structure 96 is closed under the operation of composing functions it a t B is de ned by lf 3quot 15 339 relauon and a a 3x a x It is easy to show that G at is a group G i5 PlSkx1 3 z Rkl POCi a 050 called the automorphism group of 96 Appendix 2 Some Concepts About Conjoint Structures Let 3 be a binary relation on the Cartesian product A X P and 69 A x P z t is said to be a conjoint structure if and only if the following two conditions are satis ed a Weak ordering is transitive and connected b Independence For all a b in A if for some p in P a p z b p then for all a in P a q z b q and for all p q in P if for some a in A a p z a q then for all bin A b p z b q Suppose C 7 is a conjoint structure Define 5 on A as follows For all a b in A a 2 b if and only if for somep in P a p g a 1 It is easy to show that A is a weak ordering of A Similarly a weak ordering 1 can be de ned on P 6 is said to satisfy unrestricted solvability if and only if for all a b in A and p q in P there exist c in A and r in P Such that c p b q and a r b a 6 is said to satisfy restricted solvability if and only if for all a a and binAandpqinP if 1217 b q a 0 p then for some a in A a p I q and for all a b in A and p p and q in P if a p 2 b q a a p then for some pin P a p I q Appendixes continue on next page


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