CHICANO STUDIES II
CHICANO STUDIES II ChcLat 62
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Math Japom39ca 52 No 32000 469 512 469 FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES1 JANos ACZELquot JEANCLAUDE FALMAGNEquot AND RDUNCAN LUCEquot Received April 4 2000 ABSTRACT Functional equations are useful in the experimental sciences because they offer a tool for narrowing the possible models for a phenomenon A model can be formu lated by one or more not very restrictive equations which when paired with an empirical or logical constraint of a general character lead via functional equation techniques to precise quantitative relationships The article reviews various applications of functional equations in some areas of the behavioral sciences such as sensory psychology psy chophysics utility theory under uncertainty and aggregation of inputs and outputs in an economic or social context We also provide enough basic material on functional equations to make this review self contained 1 INTRODUCTION 11 A Behavioral Example Functional equations arise in the sciences because they allow researchers to formulate math ematical models in general terms via some not very restrictive equations that only stipulate basic properties of functions appearing in these equations without postulating the exact forms of such mctions However the data or the experimental situation itself may exhibit regularities or symmetries that can be captured by some other equations involving the same functions thereby constraining their forms and specifying the model We begin with an example dating back to the 19th century In a famous study the physicist Plateau2 gave a pair of painted disks one white one black to each of eight artists and asked them to return to their studios and paint a grey disk appearing midway between the two According to Plateau the eight resulting grey disks were virtually identical A possible formalization of such data is as follows cf Falmagne 1985 Label each disk by its luminance in conventional units Denote by M1 y the luminance of a disk appearing midway between the disks z and y with M in the same units as 1 and by Plateau s data can then be formalized by the homogeneity equation 11 Mry AM1y AL y gt 0 where the value of A re ects the differing conditions of illumination Realistically the domain of M should be restricted to a suitable positive region near the origin Here and also later in this paper we sometimes simplify the presentation and assume that the relevant functions are de ned on idealized domains such as 0 oo Usually such idealizations have no 2000 Mathematics Subject Hassi cation Primary 39B22 91A30 91B16 91E30 Secondary 26A24 26A48 26A51 39B12 39B72 91C05 Key wants and phmses Functional equations Cauchy and Pexider equations Invariance Weber s law and generalizations Fechner Thurstone model Gain control model Joint receipt segregation Separability Utility and value functions 1We thank Chris Doble for his comments on the rst draft of this paper 2The same Plateau 18011883 whose name was given to the classical problem of determining the minimal surface with a given twisted curve as its boundary 470 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE substantial impact on the results A conceivable mechanism for the operation performed by the artists in Plateau s experiment is that the grey disk results from some kind of mental averaging of the values of the two disks in the pair This averaging however is not necessarily carried out in the lux scale We can suppose for instance that there is some sensory scale u mapping the lux scale into the reals such that u a uy 12 ill9311 A function M satisfying this equation for some continuous strictly monotonic function u is called a quasiarithmetic mean As mentioned we assume that the scale u is defined on IR 0oo Combining 11 and 12 leads to the functional equation 13 u1 Au39l Aay gt 0 which has only two families of solutions for the function u We will solve a more general equation 220 in Section 221 Here we sketch the solution process for 13 to illustrate by way of introduction some functional equations methods in a simple case We write I ulR an open interval and de ne the functionj I x IL gt I with j s js by the equation 14 ms nun1a Note that j is continuous and nonconstant in 3 Applying u on both sides of 13 and rewriting the result in terms of j with s ua and t uy we get 15 j a Wjm steIgt0 For any xed A 15 is a Jensen equation the equality case of Jensen s inequality This equation is a particular case of Pexider s equation that will be solved in Section 214 The property expressed by 15 is immediately clear namely the midpoint of the two points sjs and tjt also es on the graph of 339 cf Fig 1 FIGURE 1 Jensen s equation 11 maniac a 1 t Iterating this observation for 9 5 for 31 t etc we understand that the graphs of continuous functions satisfying 15 for some xed A are straight lines segments Thus the FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 471 general continuous solution of 15 is J39At quot1Wt quotW m 7 0 With 14 we obtain ua mAux nA cf 223 Subtracting from this equation its particular case a 1 and de ning 16 lav uw 111 we get 17 Az mzx z m gt 0 There are two cases If mA 1 then we have cf 25 Cauchy s logarithmic equation 18 1a z la m gt 0 which has the general nonconstant continuous solution 19 la 71ml 1 gt 0 with 7 a 0 as we will see in Section 213 On the other hand if there exists a A0 with mo g 1 then from mAlz l lz mal la we get with A A0 a lAomAo 1 110 la ama 1 a a 0 because I is nonconstant Putting this into 17 gives 111 mAz mAmx Aa gt 0 This is Cauchy s power equation As we will see also in Section 213 its nonconstant continuous solutions are of the form 112 ma 25 a gt 0 a 0 In view of 16 19 110 112 and 12 we have shown that The general strictly monotonic solutions of the pair of functional equations 11 12 are given by 113 uw 71 6 Mwy xi1 and p p 13 114 ua ax 5 Mx y with arbitrary constants a a 0 a 0 7 a 0 and 6 For strictly increasing solutions we have gt0 anda gt0 This example illustrates how a functional equation or a system of such equations de rived in a particular situation can be solved by deducing from it some functional equations 472 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE whose solution is knOWn In the Plateau case 11 and 12 jointly led rst to 15 and then successively to 17 and to Cauchy s logarithmic equation 18 and Cauchy s power equation 111 which are two of the four fundamental Cauchy equations which will be reviewed in Section 2 together with a collection of basic functional equation results 12 Overview The purpose of this review is to describe in the style of the example in 11 some of the uses of functional equations in the behavioral and social sciences No pretense is made that the coverage is in any way exhaustive Rather we demonstrate a number of techniques using additional examples taken om areas in which the authors have done appreciable work Applications of functional equations in three empirical elds are discussed Section 3 gives examples in sensory psychology in particular psychophysics resembling the Plateau example but more complex and covering di erent situations Section 4 is devoted to indi vidual decision making under uncertainty utility theory and Section 5 covers consistent aggregation of inputs and outputs in a social or economic context Except for some proofs which are either omitted or abbreviated our discussion is self contained drawing on the mathematical material in Section 2 We start there with a summary of some classic types of functional equations solved long ago for details see Acz l 1987 that have proved use ful in the analysis of behavioral models from a functional equation viewpoint In all cases that we consider the functional equations are numerical ones generated by a mathematical model or a numerical representation of the phenomenon under consideration in which the modelling is incomplete and at least some of the functions are unspeci ed A substantial literature on representational measurement theory describes classes of qualitative systems that give rise to numerical representations Krantz Luce Suppes and Tversky 1971 Luce Krantz Suppes and Tversky 1990 and Suppes Krantz Luce and Tversly 1989 From this starting point functional equations enter the picture in at least three distinct ways One occurs when some invariance property holds An example of such an invariance is some type of homogeneity of one of the functions as illustrated by the function M in 11 and by several cases in Section 3 Other examples of invariance from 49 involve utility and weighting functions from utility theory The second arises when two independent measurement schemes give rise to distinct mea sures of the same underlying attribute An example in physics is mass It can be measured by concatenating objects on the pans of a pan balance in a vacuum which is used to determine the order of greater mass The resulting mass measure is additive over concate nation It can also be measured fundamentally by varying the container size and the choice of homogeneous substances within them on a pan balance This measure of mass exhibits a product structure with corresponding measures of volume and density Clearly because both mass measures represent the same ordering they are related by a strictly increasing function Any additional empirical law linking the two measurement structures imposes a constraint on that function in the sense that it must satisfy a functional equation In the physical case the link is a qualitative distribution law This approach with quite different linking laws is illustrated repeatedly in Section 4 on utility theory The third concerns what may be thought of as consistency principles One such principle is the commutative property used in modelling aggegation Suppose we have variables 12 that can be aggregated over either i or j and once done then over the remaining subscript An economic example concerns several types of inputs such as raw materials energy and labor to some production Often one wishes to speak of an aggregated result over each type of input and over the outputs The question is under what circumstances does the order of aggregation not matter This is discussed in Section 5 on consistent social aggregation FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 473 We keep notations consistent within sections but not necessarily accross them Achieving total consistency would have been at the cost of contravening well established notational conventions in the three empirical elds considered 2 SEVERAL BASIC FUNCTIONAL EQUATIONS 21 The Cauchy Family 211 The Four Equations The example discussed in Section 11 yielded two of four closely related functional equations that constitute the Cauchy family Later applications give rise to the other two So we treat all of them as a unit The rst known as Cauchy s fundamental equation or for short Cauchy s equation is 21 fs t fs ft s gt 0 t gt 0 It could be considered for all real st but here it is restricted to positive 9 t This restriction is easily removed other possible restrictions may not be so easily dropped if at all Define a new we could have introduced a new symbol for it to be the old f on 111 0oo f 0 O and 22 fv fv 0 lt 0 which is de ned because 39u gt O and the old f is defined at v It is easy to check that the new f satis es 23 fs t fs ft 8 e Rt 6 R where R stands for the set of real numbers Functions satisfying either version are called additive A second called Cauchy s exponential equation is 24 gs t gsgt s gt 0t gt 0 This functional equation is important because it formalizes the intuitive idea of a system having no memory of its past lack of memory property Indeed with s and t measuring disjoint contiguous time intervals it implies that the value of gs t gs is independent of 3 Provided g is positive it is reduced to the fundamental one by f 1n g Positivity will be shown to follow from 24 The third is Cauchy s logarithmic equation 25 lst 13 lt s gt 013 gt 0 and the fourth Cauchy s power equation 26 mst msmt s gt 0t gt 0 Functions m IR IR satisfying 26 are called multiplicative 474 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE By substitution 9 e t 6 equation 25 becomes 23 and 26 becomes the equation 27 91 w gvgw v E Rw E R The difference between 24 and 27 is that the latter is supposed to hold for all real arguments whereas the former is only for positive ones It is clear that it suf ces rst to show that g of 24 or similarly of 27 is positive and then to solve the fundamental equation 23 for the function f 212 The Four Smooth Solutions These equations have three types of solutions of which only one interests us 1 D ivial constant ones that are excluded simply by assuming the unknown function to be nonconstant which in the case of g in 24 or 27 excludes gt E 1 and gt E 0 2 Highly irregular solutions exist and are excluded in some fashion which we will do by assuming that on some closed subinterval the solution is bounded In particular we assume that g is bounded from above on a subinterval 11 of R of positive length no matter how small say by 8 3 Smooth ones that are our main interest We may list these in all cases c a 0 is a constant 28 ft ct t E R or t E R cf 21 or 23 29 gt e t E R or t E R cf 24 or 27 210 lt clnt t 6 R cf 25 211 mt t6 t 6 R cf 26 It is trivial to see that each solves its corresponding functional equation The only issue is to show that they are the unique nonconstant bounded ones 213 The Proofs We start with Equation 26 which arose in Section 11 as 111 We write it here in the form mzy mmmy 27 6 111 y E 1R The substitution 2 e y e and 91 2 me converts this into 92 w gvgw v E R w E R that is into equation 27 Restricting 27 to R we get 24 gs t gsgt s gt 0t gt 0 First we show that the nonconstancy condition and 24 imply that g is positive on R By 24 with s t 5 r gm 95 1 6 1a So 9 is nonnegative If there existed an so 6 R with gso 0 then 9 E 0 Indeed 97 95097 90 0 r gt so This does not yet exclude the possibility of gro gt 0 for some r0 lt so But that also leads to a contradiction Repeated application of 24 gives FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 475 212 gn u 91quot 390 E Rn 1 2 and for a sufficiently large positive integer n we have nro gt so whence 0 lt groquot 90211 0 a contradiction The proof is even simpler for 27 Thus 9 is positive and we can take logarithms on both sides of 24 to reduce the problem to solving 21 Because 9 was supposed to be bounded from above on an interval 11 by 399 so f lng is bounded from above on a b by 0 ln9 the positivity of g has no other consequence than that f can be so de ned Moreover by 22 the new f is bounded from below on b a by 0 But then f is bounded om both sides on every interval a b of length b a b 0 Indeed by 23 NUfUa05f 03fU b a f dl Ift E a39b then ta a 6 ab tb a39 E b a and so wfds nso fwdl Clearly fot ct gives solutions of 23 bounded on every nite interval We prove that this is the general such solution The function h f f0 formed from f0u cu and from an arbitrary solution f of 23 also satis es 23 We choose c in ht f t at so that hd 0 with d b a ie cf Fig 2 am mw 0 a n L FIGURE 2 Reduction of the Cauchy Equation ht f t f Ei t A ftgt Lsg Since this h satis es hs t hs ht and hd 0 we have 214 ht d ht hd ht which means that h is periodic with period 1 But with f bounded on 0d Fig 2 so is h f f0 remember that d b a and thus by the periodicity 214 it is bounded on the whole line IR If this h were not identically 0 then there would exist a to with hto 75 0 say hto lt 0 However hs t hs ht implies hnto nhto for all positive integers n Since hto lt O for large enough 12 the number nhto hnto could be as small as large a negative number as we want thus contradicting the boundedness of h on IR Therefore ht E 0 and we see from 213 0a aem as asserted and f is nonconstant iff c 90 0 Thus we have proved the following 476 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE Theorem 1 The general nonconstant solutions of 23 or 21 bounded from above on an interval are given by 28 REMARKS 1 The proof makes it clear that bounded from below may replace bounded from above Bounded from one side on an interval is also called locally bounded on one side 2 Obviously the only constant solution of 23 or 21 is f t E 0 Corollary 1 The result is unchanged if boundedness is replaced by either f is continuous at a point or is monotonic on an interval In the latter case f is strictly increasing or strictly decreasing if and only if c gt 0 or c lt 0 respectively but otherwise c is arbitrary Returning to 24 because f 1ng satis es 21 we have Corollary 2 The general nonconstant solutions of both 24 and 27 bounded on an interval from below by a positive constant or from above by any constant or monotonic on an interval or continuous at a point are given by 29 REMARKS 1 We need 9 bounded from below by a positive number if we want the f lng to be bounded from below at all 2 For the solutions f of 23 and g of 24 or 27 the assumptions that they are monotonic and nonconstant implies that they are strictly monotonic 3 The solution g will be strictly increasing or strictly decreasing according to whether c gt 0 or c lt 0 respectively As was noted earlier the substitution 3 e t e quot converts 25 to 23 and 26 to 24 Thus we have the following Corollary 3 The general nonconstant solutions of 25 bounded from one side on an interval are given by 210 and the general solutions of 26 locally bounded from below by a positive constant or locally bounded from above are given by 211 The solutions 1 and m are strictly increasing or strictly decreasing according to whether c gt 0 or c lt 0 respectively REMARK Some new things happen when either 24 or 25 is considered for nonnegative instead of positive variables In addition to 29 there is one and only one further locally bounded nonconstant solution of 24 for s 2 0 r 2 0 namely 1 if t 0 gt 0 iftgt0 Similarly as before a function is locally bounded if there exists a proper interval on which it is bounded Indeed s 0 in gs t gsgt s 2 0t Z 0 shows that either gO 1 or gt E 0 The former permits both gt 0 for t gt 0 and gt e for all t Z 0 On the other hand lay l1 l for all a 2 031 2 0 permits only the trivial solution 1a E 0 as can be seen by choosing y 0 214 The Pexider Equation We consider now the equation 215 fa y gx My FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 477 where 1y lie in an open region ie connected open set of R2 and one of f9 h say 9 is strictly monotonic This equation called the Pexider equation is similar to Cauchy s fundamental equation 23 except that it contains three unknown functions rather than one Yet as we shall see they are all determined a typical feature for functional equations but rather unusual for other equations Pexider equation has already arisen in Section 1 and will arise again in several applications Solving the Pexider equation is easily reduced to solving 23 If a Pexider equation holds on an open region in R2 then its validity can be extended to all 11111 E R2 see eg Acz l 1987 p 80 216 fv w 911 hw 11 E lRw 6 1R Substituting 11 0 and separately 11 0 we get respectively 217 91 NO 17 W fw a where a 90 b h0 Putting these back into 216 we see that Ft ft a b satisfies Cauchy s equation Fv w F1 Fw Because F is strictly monotonic Ft at c a 0 Thus see 217 we have the following Theorem 2 The general solution of the Pexider equation 215 on an open region R of 1R2 with at least one of f 9 h strictly monotonic is given by 218 ft ct a b 91 cu a hw cw b forvE 11 vw 6R we 11 1110 6 R t6 11w 1110 ER Here ca O a b are arbirary constants As with the Cauchy equation there are three additional variants involving multiplication as well as addition For example we will encounter f vw 91hw 11 E lRw 6 BL in Sections 223 and 443 The locally bounded nonconstant solutions are obviously 219 abtc 91 a116 hw bwc abc 74 0 22 The Invariance Equation 221 The Equation We solve the following functional equation an example of which arose in Section 11 see 13 and later we show applications to proposed invariance assumptions in the behavioral sciences 220 solf 1quotM1 MUM 1quot1 3511 E T G 111 where f r and p are real valued functions and T is an open cone with 1311 6 R2 ie a connected open set of pairs of positive reals closed under multiplication by any A gt 0 in the sense that if 1311 E T then Azy E T Clearly T e R1Az lt y lt B113 where B gt A gt 0 are constants The functions f 1390 are by supposition strictly monotonic and map their domains which are intervals onto intervals Thus they are continuous The domain of f and r is 0 oo Let their range set of function values be I and J respectively 47s JANos ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE By the above supposition these are open intervals open because of the strict monotonicity Thus cp is de ned onIJvw vEIwE J 222 Reduction to Cauchy Equations For all 11 E Iw E J there exist by the above a E lRy 6 111 such that v fa w ry Because 220 is supposed to hold for all 2 y in the open cone T which is an open region and because f 1 are continuous and strictly monotonic it follows that T 11110 I1 faw ry any 6 T is an open region Considering A to be a parameter we de ne 221 fxv f IV 10 MW TT1wl WU ltP1ltPtl yielding 90W w fxv U01 1111 E T a Pexider equation 215 Thus the solution is 222 WU mxt 0A bx fAv mw a TAU mxw bx Considering as a variable again we get in view of 221 223 f A50 mf39 00 COM 6 HM and similar equations for r and 0 As in Section 11 we get two distinct solutions 224 f 11 11150 31 m E IRquot011 7e 0 and 225 fa 011m 1 131 2 E 1Ra1 a 07 a 0 Because they satisfy 223 for m 1 or m X respectively and for all 11 a 01 y 0 and B1 the general solutions of 223 for strictly monotonic f are given by 224 and 225 Substitution yields a a1 ln or a B11 X1 respectively Similarly the equations for 739 and cp cf 218 220 and 221 rA11 cAmy b W108 COW 13 a b have as general solutions for strictly monotonic 1 and 90 226 79 02 11131 52 02 0 ltp1s a1 a2lns 3 11 012 0 and 227 731 01217 32 02 07 7e 0 cp 1s ozas7 31 H32 397 7e 0a3 7e 0 respectively The 0 1 expressions yield FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 479 228 PU exp 11632 a1 a2 7 0 17 229 W as e 07 e 0 223 The Result The following summarizes what we have just shown Theorem 3 The general strictly monotonic and continuous solutions f rltp of the func tional equation 220 where T is an open cone are given by 224 226 and 228 and by 225 227 and 229 with the constants restricted as indicated but otherwise arbitrary Corollary The general strictly monotonic and continuous solutions f of the functional equation 230 f391f fy f 1f fy 9331 6 TM 6 R where T is an open cone are given by f 0157 93 6 RH where a and 397 are arbitrary nonzero constants Equation 230 is of course the particular case of 220 where r f and 0 f 391 Substitution shows that 225 satis es 230 only with A 0 and 224 does not satisfy 230 with any 11 54 0 and 61 at all The equation 231 F 1Fvquot Fwquot F 1Fv Fw 910 A e in is transformed by v e w eVfa Fe into 230 Here the open cone T is replaced by R2 The same argument as above and as in Section 11 yields 232 fWJ mffv 93 E 1R E 131 Note that it is not restricted to x e As in 219 for E lR z 6 RH the general continuous strictly monotonic solution is given by mA X f a7 a7 a 0 On the other hand z 16 f 1 in 232 give f 6m 6A7 6 a 0 Because setting c 0 in 232 yields f 0 0 for strictly monotonic m we get 0127 for z 2 0 mquot 6rl397 for z lt 0 The function f is strictly monotonic only for positive 397 and negative 16 For Fp f 1n p we get 011n107 P 2 1 F 1 6llnpl 2 lt 1 with 7 gt 0 and 16 lt 0 as the general strictly monotonic and continuous solution of 231 Two behavioral examples of such invariance arguments are given in Sections 491 and 492 480 JANos ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE 3 PSYCHOPHYSICS Many applications of functional equations in psychophysics are in the style illustrated by the Plateau s situation discussed in Section 11 On the experimental side one starts with some homogeneity equation that seems to capture an important regularity in the data In the case of Plateau the function M in 11 is homogeneous of degree 1 Next one introduces some reasonable model involving one or more unknown functions formalizing a possible mechanism for explaining the behavior of the individuals in the study In Plateau s example such a model is obtained by assuming that the function M is a quasiarithmetic mean see 12 with the unknown function u Combining the homogeneity equation with the model results in a considerable speci cation of the possible forms of the unknown functions In our example u must be either a logarithmic or a power function cf 113 and 114 The cases reviewed in this section all involve a binary discrimination situation in which an individual must decide which of the two stimuli presented has more of some sensory attribute For instance the individual compares the loudness of pure tones differing only by their intensities which is measured in standard ratio scale units such as sound pressure level In a classic 19th century case involving E E Weber and G T Fechner the individual is presented with a pair of stimuli an 3 and the researcher s task is to estimate experimentally the probability Pc 3 that a stimulus of intensity 1 is judged as louder or brighter longer etc than a stimulus of intensity 3 In mathematical terminology the so called Weber Law states that the function P 118 0 1 is homogeneous of degree 0 31 Py Pay zy gt 0 As mentioned in Section 11 for the purpose of this review we indulge in some idealization and suppose that the variables take their values on sets such as R or Rh even though humans cannot realistically be presented with very intense stimuli eg the brightness of the sun In the case dealt with here such assumptions have no substantive impact on the results and can be generalized considerably see Falmagne 1985 There are however equations where the exact form of the results depends upon the domains of the functions cf Falmagne 1981 Fechner s contribution consisted in proposing that the individual s judgment resulted from a computation of the differences between the two sensations produced by 1 and 3 this difference being evaluated in terms of some unknown sensation scale Formally Fechner s idea is expressed by the equation 32 P390y F quot96 119 where the functions u and F are real valued strictly increasing and continuous but other wise not a priori speci ed In the context of Weber s Law 31 the possible forms of u and F in 32 are severely constrained Putting 32 into 31 we get the functional equation Fuz uy Fuz a gt 031 gt 0 A gt 0 which since F is strictly increasing is equivalent to u1 uy ua uy a gt 031 gt 0 gt 0 With 1y z 33 lz uz u1 the above equation gives lyz ly lz y gt 0 z gt 0 that is 25 whose strictly increasing solutions are given by 210 as FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 481 la clna c gt 0 With B u1 we see that all the strictly increasing continuous solutions u of the pair of functional equations 31 32 are given by 33 clna B a 6 EL while P and F are related by the equation 34 Ft Petc 1 t E R c E R B E R The other examples in this section are variations on this theme but require more work because we deal with cases in which each of the two arguments of the function P is a real vec tor Because each component of the two vectors may contribute separately to the sensation various models describing such contributions may be considered in the context of several possible forms of homogeneity As in the above example we take loudness discrimination as our experimental situation The next three sections summarize results by Falmagne and Iverson 1979 see also Falmagne Iverson amp Marcovici 1976 and Falmagne 1985 31 The Conjoint Weber Law Suppose that an individual wearing earphones is presented with a 1000 Hz tone delivered simultaneously with different intensities to the two auditory channels The impression created by such a stimulus is that of a single sensation the loudness of which depends upon the combination of two inputs The location of the sensation inside the individual s head also depends upon the combination of the two intensities but this aspect of the phenomenon is not relevant here We write aa for such a two dimensional stimulus where a and a stand for positive real numbers denoting the sound pressures in the left and right auditory channels respectively Extending our earlier notation we write Paa by for the probability that the individual judges the twodimensional stimulus a1 to be louder than the 2dimensional stimulus by Various concepts concerning the function P are gathered in the de nitions below 311 De nitions We suppose that P 1R1 gt0l is continuous with Paaby strictly increasing in a and strictly decreasing in b and strictly contramonotonic in a and 31 that is Paaby is either strictly increasing in a and strictly decreasing in y or vice versa We will call such a function a discrimination probability Note that the hypothesis that P is strictly co monotonic in the second and fourth variables is weaker than what is suggested by the binaural loudness summation example but makes sense because the formalism is then also applicable to other empirical cases For instance a pair a 1 could represent a stimulus of intensity 1 presented over a noisy backgound of intensity 1 cf Section 33 A discrimination probability P satis es the Conjoint Weber Law if it is homogeneous of degree zero that is 35 Pa Ax Ab Ag Paar by Aaarby gt 0 It is easily shown that 35 holds if and only if there is a function Q R1 0 1 such that 36 Paw 1121 Qawyby with Q strictly increasing in the rst variable and strictly decreasing in the third variable It is tempting to conceptualize the combination of the loudnesses of the two tones as an addition which is somehow performed by the auditory system in terms of some sensory 482 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE scales In the literature this phenomenon is in fact referred to as binaural loudness sum mation Falmagne and Iverson 1979 assumed that the function P satis es the equation 37 Paby Hfa 7197 fb 7 yl where f r and H are continuous functions with f strictly increasing r strictly monotonic and H strictly increasing in the rst variable and strictly decreasing in the second vari able When 37 holds for some functions f r and H we say that P satis es component additivity Falmagne and Iverson 1979 also considered the generalization of 37 represented by the equation of simple scalability 38 Pax 1721 H ga 2 gby with H as above and 9 continuous strictly increasing in the rst variable and strictly monotonic in the second variable When a function P satis es 38 for some functions g and H satisfying the stated conditions we say that P is simply scalable and we call gH a simple scale representation of P The Conjoint Weber Law puts stringent constraints on the functions entering the component additivity and even the simple scalability equations 312 Results We begin with a preparatory result which is of some intrinsic interest Lemma 1 Suppose that a discrimination probability P 1R1 gt01 has a simple scale representation gH cf 38 Then the following two conditions are equivalent i The discrimination probability P satis es the Conjoint Weber Law 35 ii Either the function g in 38 is homogeneous of degree 0 or there exists a constant 7E 0 a continuous function h Ri l 11L homogeneous of degree 3 strictly increas ing in the rst variable strictly monotonic in the second variable and a continuous strictly increasing function G such that 39 Paab y G acby gt 0 Sketch of proof Clearly each of the two cases of ii implies Assume that Pamby H ga zgb 21 for some simple scale representation gH Notice that ga a is de ned for any a 6 111 For the moment assume that ga a is either constant or strictly monotonic in a If ga a is constant we obtain successively Paa Ml Hgaagby H gAa a gb Ay Conjoint Weber Law H ga agb MD and by the strict monotonicity of H in the second argument we derive gAb Ag gby for all A by gt 0 In other words g is homogeneous of degree 0 and the rst case of ii follows FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 483 Suppose on the other hand that ga a is strictly monotonic By an argument of continuity which is omitted here one shows that there must exist a solution qaa gt 0 to the equation 310 gqaaqaa gaa a gt 0 a gt 0 where the function q is continuous and nonconstant on 1R1 With a qaa and a qb y this yields Paaby Hgaagby Hgaaga a Paa a a Applying the Conjoint Weber Law we obtain with 1 a Paaby Paa a a Pa A01 Aa Aa mg 311 M OI0139 where M s Pss 1 1 is strictly monotonic since gs s is strictly monotonic We have thus applying once more the Conjoint Weber Law wacgzgWeiss which implies 10116 IOWA 5 qby qb Ayl With m qx xq11 we get 10mm mMa a3915 with the function in nonconstant continuous on lR and satisfying mxX m mX which is Cauchy s power equation cf Section 211 26 According to 211 m A for some a 0 Thus q is homogeneous of degree a 0 If gtt is strictly increasing then M is also strictly increasing and the second case of ii obtains with h q G M and a 0 In the other case we de ne h lq and Gs Mls To complete the proof it remains to show that if gt t is nonconstant it must be strictly monotonic This fact also results from the continuity of g and the Conjoint Weber Law but we do not include the argument here D REMARK The result in Lemma 1 also holds under weaker conditions concerning the domain of the function P for instance Paaby is de ned for all aa b y E D where D is a nonempty positive convex open cone see Falmagne and Iverson 1976 Lemma 1 is instrumental for establishing the next two results Theorem 4 Suppose that P IR 01 is a simply scalable discrimination probability satisfying the Conjoint Weber Law 35 Then one of the two following possibilities holds a There are some real valued continuous strictly monotonic functions T gt 0 and M satisfying Taaa 311 Pazby M azby gt 0 484 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE b There is a continuous function Q0 strictly increasing in the rst argument strictly decreasing in the second argument such that 312 Paxby Qoaaby Proof Clearly Case b which describes homogeneity of degree O is compatible with the hypotheses of the theorem Let h H be the simple scale representation of P obtained in Lemma 1ii Thus 9 or h is homogeneous of degree Assume that Case b does not hold It is easily veri ed that we must have 6 79 0 Thus haa P 111 G aixabv gt 0 21 MM y gt for some continuous strictly increasing function G with haz 15 ha z 1 De ning Ts hs11 Ms C35 gives 311 Thus Case a is implied by the hypotheses of the theorem when b does not hold El Theorem 5 Suppose that P IR gtO1 is a discrimination probability satisfying the Conjoint Weber Law 35 together with the component additiuity equation 37 Pax 1w Hfa r073 fb ry malty gt 0 where f r and H are continuous functions with f strictly increasing r strictly monotonic and H strictly increasing in the rst vari able and strictly decreasing in the second variable Then one of the three equations ala 61 Pa 3b3y G a a 3 7 Paxby G a b Paazab1y Q0 51 i must hold for axby gt O In 313 and 314 G is continuous and strictly increasing and 5 gt O 6 79 O and 7 7e 0 are constants In 315 Q0 is continuous strictly increasing in its rst argument and strictly decreasing in its second argument Thus the function P is necessarily decreasing in its second argument and increasing in its fourth argument Proof The hypotheses of the theorem imply those of Theorem 4 Accordingly either 315 holds which with f r 1n and H s t Q0e et is clearly of the additive form 37 or else Case a of Theorem 4 holds that is 316 Pazby M H fa re M ry with T and M continuous and strictly monotonic Holding b b0 y yo constant and letting Km T yoM1Ht fbo ram FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 485 we get from the last equation in 316 a Kfa ra T 51 Thus for all gt 0 Klfa 7096 AKlfa 1116 which is the invariance equation 220 Accordingly cf 224 226 and 225 227 we have the following two solutions 317 fa Ala B1 rc Azm 32 where A1Bl A2 32 and are constants ALB gt 0 A2 79 0 318 fa A1 Ina Bl ra A2 Ina 32 where A1Bl A2 32 are constants A1 gt 0 A2 79 0 If 317 holds then Pa 2 b y HA1a B1 Agx 32A1b 31 Agy 32 H1a 6x b 633 with 6 142141 and H1st HA15 31 BzA1t 81 32 So the two functions h ac H a 61 and H form a simple scale representation of the discrimination probability P Notice that h is homogenous of degree 3 a 0 and that P satisfies the Conjoint Weber Law The conditions of Case ii of Lemma 1 are thus satisfied and there must exist a strictly increasing function G satisfying 313 A similar argument leads to 314 in the case of the solution 318 CI 32 The Conjoint Weber Inequality Empirically the Conjoint Weber Law may be satisfied only approximately For instance it may fail for small values of the stimuli In such a case the following Conjoint Weber Inequality may give a more adequate representation of the data 319 Paab y g Paz Aby A Z 1a 2 b gt 01 2 y gt 0 It is then natural to ask whether the type of expressions obtained for the discrimination probability P in Theorems 1 4 and 5 could hold asymptotically that is for large A in 319 An example of what can be obtained is outlined here without proof Suppose that P is a discrimination probability thus P is a continuous function mapping R into 01 We also suppose that Paaby is strictly increasing in a c strictly decreasing in b y and satisfies 39 of Lemma 1 that is Maw aix b ygt0 for a continuous strictly increasing function G and a function h gt 0 strictly increasing in both variables and satisfying continuity and other regularity conditions in particular for large a we have ha a 5 39ya for some constants 711 gt 0 Suppose moreover that the Conjoint Weber Inequality holds Then Taxx 11m P Aax Ab M a 139b gt 0 Hoe 3 Tbyy y l 486 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE The proof is similar to that of Theorem 4 but uses some new ideas cf Falmagne 1977 for details see Falmagne and Iverson 1979 This type of asymptotic result is sometimes referred to as a stability result cf the survey by Forti 1995 33 The Strong Conjoint Weber Law In some situations a stronger version of homogeneity can hold for a discrimination proba bility P As before suppose that two pairs of stimuli aa and by are presented to an individual who has to decide which pair seems louder In this case however the numbers a and y represent the intensity of a noisy background The relative loudness of a1 with respect to by may not change if on one hand a and b are multiplied by the same positive constant and on the other hand both 1 and y are multiplied by a possibly different positive constant This idea leads to the following homogeneity condition 320 Paaby Pa39rcb39ry a1by739 gt 0 We refer to 320 as the Strong Conjoint Weber Lawl Obviously this condition implies the Conjoint Weber Law So the results of the previous sections are applicable with possibly stronger consequences The following theorem is one of the results Theorem 6 Suppose that P is a simply scalable discrimination probability which is strictly increasing in the second variable and strictly decreasing in the fourth variable Then the following two conditions are equivalent i The Strong Conjoint Weber Law 320 holds ii Equation 311 holds with Ts 3quot for some constant B 0 lt H lt 1 Accordingly H 1p 321 Pa1by M acby gt 0 where M is as in 311 Proof Clearly ii implies The conditions of the theorem together with Condition imply the hypotheses of Theorem 4 However Case b of Theorem 4 cannot hold because here Panby is strictly increasing in 1 by hypothesis Thus Case a of Theorem 4 holds and we must have 311 Tamx P a1by M T1my for some real valued continuous function T gt 0 Note that we can assume without loss of generality that T1 1 Applying the Strong Conjoint Weber Law we get after simpli cation Tam TAarm T Wu TAbTy Setting by 1 s am t A r we obtain TsTt Tst st gt 0 a Cauchy power equation cf 26 Accordingly cf Corollary 3 to Theorem 1 we have Ts s for some constant Thus 321 must hold with 0 lt 3 lt 1 because Pa9 by is strictly increasing in both a and 13 D a 13b y Anquot gt 0 1In Falmagne and Iverson 1979 a related condition was also investigated represented by the equation Pa cc b y PAala139b39ry FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 48 34 The NearMisstoWeber s Law We turn now to some recent functional equation results and generalize Weber s Law con siderably in terms of a monomial equation see 328 to which for historical reasons we refer as the NearMisstoWeber s Law In the experimental literature of psychophysics a onedimensional version of the monomial equation 328 is referred to by that name cf Falmagne 1985 In Section 341 which summarizes results from Acz l and Falmagne 1999 we consider a situation in which an individual is presented with a pair xn ym of stimuli where x7 and ym denote real positive vectors ie xn E R3 ym 6 1R with n 2 2 For example xn is a stimulus described by a number of components ym denotes a noisy background which may also be described by components and Pxy is the probability that the individual detects stimulus xn in the backgound ym We suppose that the probabilities P satisfy a Fechner Thurstone difference representation 322 Pxn1ym Fuxn 9ym xn 6 R24 Ym 6 RT where u and 9 denote real valued functions The more general Gain Control representation 323 Pwy F x 6 111 y e m M16 is examined in Section 342 which describes results from Falmagne and Lundberg 2000 Thus the functional equations 322 and 323 play here a role similar to that of 37 and 38 in Section 31 while a monomial equation see 328 and 337 replaces the Conjoint Weber Law 341 FechnerThurstone Model We begin by simplifying the notation For reasons that soon will be apparent we single out the last component of X7 in 322 and write x7 xI with x E KKK 12 xquot E R ym y E To avoid multiplying the parentheses Pxcy denotes probability of detecting stimulus x x in the background y The function P is assumed to be continuous strictly increasing in its rst n variables and strictly decreasing in its last m variables Close attention is paid to the domains of variations of all the variables involved For 1 g 239 S n and l lt j S m let aa and bjb3 be 11 m real open intervals with 0 lt a lt l lt a and 0 lt bj lt l lt b Singling out the interval a a we de ne the Cartesian products 324 An1 a1a 1 X X an1a1 n gt 1 325 B 1b1ba x x 1bb39 m 2 1 We also suppose that the probabilities P satisfy a Fechner Thurstone difference represen tation 326 Pxxy F uxw 900 x E fin 1x Elamazt y 6 Bm which is a special case of 323 with u g and F continuous real valued and strictly increasing in all relevant variables Researchers are typically much more interested in the forms of the functions u and 9 than in that of F For that reason they routinely study the 488 JANOS ACZE L JEANCLAUDE FALMAGNE AND RDUNCAN LUCE phenomenon represented in 326 by estimating empirically a so that Pxzy p for some values of p and for many values of the variables involved in x and x In other terms they study the function g xy p H xyp satisfying 327 xyp 5 gt Pxzy p thus Pxxypy p Notice that for any xed x in An1 and y in Bm the function a H Pxz y is strictly increasing and continuous on a a Accordingly its range 3xy Pxlan city must be an open interval and so xy p is de ned for all points x E An1 y E Bm and p 6 Sky A simple model for the function g is offered by the product n l m 328 xyp 1 1 zquot gt11y jlt gtcp i1 j1 x 21 zn1 e An1 y y1 ym E Bm p 6 SK This monomial representation has the form of the laws of classical physics and is a natural one to consider here if each of the components is measured on a ratio scale as is the case in the empirical example mentioned at the beginning of Section 3 Acz l and Falmagne 1999 investigated the compatibility of the representations 326 and 328 which are linked by the equivalence 327 They shoWed that under reasonable background assumptions concerning the domains for the variables x and yj in 328 the equations 326 and 328 force all functions 1 in 328 to be constant Moreover either all functions 139 must also be constant and C expoF 391 where F 1 is the inverse of the function F in 326 or if at least one of the Cj s is nonconstant then all of them must have the form jp Hj expidF 1p for some constants 0 gt 0 1 S j S m and 6 a 0 None of these results hinges on the assumption that the function P is measuring a probability ie is bounded above by 1 and below by 0 It certainly need not be a discrimination probability in the sense used in Section 311 This can be achieved just by choosing the otherwise arbitrary continuous and strictly increasing function F so that its value lie between 0 and 1 Acz l and Falmagne s main result is reproduced below Notice that F turns out to be arbitrary as in the solution of 31 32 Theorem 7 Suppose that P is a pair of functions related by the equivalence 327 The following three conditions are then equivalent i The function P satis es 326 for some functions u g and F strictly increasing and continuous in all arguments but otherwise arbitrary Moreover 5 satis es 328 for some positive functions C 1 1 S i S n 1 and j 1 g j g m all de ned on J PA1gtlt aa me with at least one of the J nonconstant ii The function P satis es 326 with F strictly increasing and continuous and with u g speci ed by 329 uxz glnln i1 xi 330 gy lnln i1 J FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 489 for some constants a gt 0 1 S i S n 1 Bj gt 0 1 S j S m 7 6 A and B the latter four satisfying either Case a or Case b below Case a 6 lt 0 lt 397 A Z a a2 gt 1 B 2 H bgaj i1 j1 71 1 71 Caseb 6gt0gt7 0ltASanHa 0ltBSHbgj Accordingly the function P takes the form 1 A 7 1 B 7 331 Pxay F 1n1n 1nln 6 3 l l3911 9 6 szl y Fllnln w T rn llnln TB 239 gt0 a m 839 F lnln WU 1quot 1n1n ng in lt 0 iii The function 5 satis es 328 for some positive functions C n and 1 all de ned on J PA1gtlt aa n me with constant 1 a 1 S i S n 1 and nonconstant j 1 S j S m Moreover there exist constants 6 01 gt 0 1 S j S m A B satisfying either Case a or Case b above so that for all p e J 332 99 9139 exp60p 1 s j S m 333 Cp B quotPMGWM where G is a strictly increasing and continuous but otherwise arbitrary function on J Consequently 328 takes the form expi6G0l 111 in 334 xy p A H 9317 quot 15 H 9 i1 j1 The proof of this result contained in Acz l and Falmagne 1999 is quite long and we only summarize it here They begin by establishing the implication i gt iii and prove that when one of the 9 s in 328 is nonconstant then they all must be nonconstant and of the form speci ed by 332 with G F quot1 They then prove 333 Finally they show that all 1 must be constant if one of the s is nonconstant Equation obtains The representations 331 and 334 follow easily from each other with G F quot1 and 7 arbitrarily positive or negative in Case a or b respectively We have thus i gt iii ltgt ii It remains to establish ii and iii i which is readily obtained by observing that 331 has the form 326 with u and 9 de ned by 329 and 330 and that 334 has the form 328 with the n constant and the C and C de ned by 332 and 333 respectively 342 Gain Control Model The motivation here is similar to that of Acz l and Falmagne s work reviewed in the last section The two major differences are 1 only the case n m 1 of the monomial equation 328 is considered 2 the FechnerThurstone model 326 is replaced by the Gain Control model embodied in the equation 490 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE 335 Prcy F M a e an 1 e 11 b D Mfr where h gt 0 aa and bb are real open positive intervals and the function P is continuous strictly increasing in the rst variable and strictly decreasing in the second variable To be speci c for any a 6 aa let II P1bb g 01 stand for the range of the function y H Pay and de ne the function E by the equivalence 336 Ewp y ltgt Pay p a 6 laa p e Ix This is the equivalence 327 for m n 1 A simple model for the function C is offered by the power law 337 139 p 011207 with C and 1 real valued positive continuous functions In the style of Theorem 7 Falmagne and Lundberg 2000 derive the consequences of 335 and 337 holding jointly on the form of the functions u g h C and 10 They prove the following Theorem 8 Suppose that the functions P and E are linked by the equivalence 336 Sup pose moreover that the following two conditions are satis ed 1 the function P satis es 335 for some continuous and strictly increasing functions u g h and F with h gt 0 ii the function C satis es 337 for some continuous positive functions C and w both of them de ned on RP and strictly decreasing Then there exist ve positive constants a 39y6 and 1 and two constants 0 and c such that 338 u1 allins 6W c 339 931 v71ny 639quot c 340 hz 1111 6W with 341 lna620 71nb 020 Accordingly 335 takes the particular form 342 Pzy F a MY lnz 6 Moreover the functions C and 11 in 337 can be written as 343 up 17 a F1ltp 1 344 W exp 17 6a Few 0 The proof is based on the solution of the functional equation 345 gotvs ws s kt ft for real variables 9 and t which we state here as a lemma It is assumed that the domain of 345 is a region open connected D Q 1R2 FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 491 Lemma 2 Suppose that 345 is satis ed by the real valued continuous functions Lp w v k and f on a region D g 1R2 Suppose further that k and v are nonconstant and have no zeroes Then there are constants a 6 c and n such that 346 ws avs c 347 ft 6kt n and constants 8 u and 7 where 371 94 0 such that 348 123 Vs 6 1 349 kt rt 041 For a proof see Falmagne and Lundberg 1999 Sketch of a proof of Theorem 8 Applying the inverse F 1 of F on both sides of 335 we get with Pc y p and after rearranging yy WE F 1phw Solving for y z p and using also 337 w p 9quot1u F 1phm CP1 P Taking logarithms in the last equation both sides are positive yields 350 1n y1u F1Phl 1 11193 1n P With 351 Lp lnog l s lnzv ws uequot t F1p 352 03 Mes 7613 C 0 F t ft 111W 0 F t 350 leads to 345 Moreover it can be veri ed that all conditions of Lemma 2 are satis ed here Thus the functions to f v and k of 345 have the form given by 346 347 348 and 349 for the relevant constants Since vs he by 352 v is strictly increasing thus 9 6 in 348 cannot change Sign It can in fact be shown that s 6 lt 0 for all values of 3 leads to a contradiction Thus we can drop the abslolute value symbols in 348 and write 123 13 6W A similar argument of sign preserving permits to rewrite 349 as kt t 113 Using the expressions obtained for w f v and I together with 351 and 352 leads to the forms of the functions u g and h given in Theorem 8 and thus to 342 343 and El 4 UTILITY THEORY 41 General Background Binary Gambles A key notion in classical utility theory in the presence of uncertainty is that of a binary gamble It is formalized in terms of two primitive concepts 1 a set C of valued goods and bads generically called consequences 2 an algebra E of events on a set 1quot U8 N o probability measure is postulated in the following developments A binary rst order gamble is a triple wCy fezEC 065 492 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE The empirical interpretation is that an individual who chooses gamble 2 Cy over other gambles receives consequence 1 if the event C is realized in an experiment and consequence y otherwise It is understood that some more or less precise information regarding the chances of the event C occuring is available to the individual In many cases no actual experimental trial takes place and gambles are compared by the individual using thought experiments Whereas in many situations the gambles compared by an individual may arise from different sample spaces we only consider here the case of gambles de ned with respect to a xed sample space I and algebra of events 8 For a full presentation of the general theory involving n ary gambles and different samples see Luce 2000a We also deal here with compound binary gambIes such as 41 zCyDz maul 6 C C D 6 8 denoting a situation where the individual holding this gamble gets gamble 2C y if the event D is realized in an experiment In this case a new experiment is performed and the individual receives 1 if C occurs and y otherwise We may also consider cases in which in 41 2 itself may be a gamble All theoretically feasible sets of compound gambles of any level can be de ned recursively with 1 30 C 2 for n 2 0 Bn1 8 U 2Cy1y E B C E 8 The set of all compound gambles of any level is then de ned by B U an In practice however it is not realistic to suppose that individuals can conceptualize compound gambles of levels higher than 2 Accordingly in the rest of this section we only consider the case of gambles in 82 A preference order over 82 is assumed to exist which is transitive and connected thus t is a weak order Denote the converse order by j indifference by 25 5 and strict preference by gt f Within C there is assumed to be a unique element 6 which may be thought of as no change from the status quo neither a gain nor a loss Any element 1 E 32 is called a gain or good if 1 f e and a loss or bad if 1 lt 6 Much of utility theory has studied behavioral conditions axioms that describe interlocks between the gambling structure and the preference order In many cases one is able to show that the preference order can be represented numerically In all such situations the representations are order preserving in the sense that there is a utility function U 82 IR satisfying the following two formulas 42 a 3 y ltgt Wit 2 W any 6 B2 43 Ue 0 Conditions for the existence of such an orderpreserving representation are well known namely i must indeed be a weak order and 132 must include a countable orderdense subset eg Krantz Luce Suppes and Tversky 1971 Ch 2 What makes utility theory interesting is the investigation of possible decompositions of U1 C in terms of compo nents U1 U y and a weighting function W C r gt WC mapping the algebra of events 8 into 01 42 Basic Behavioral Assumptions The following assumptions which all seem a priori rational and for which there is some empirical support underlie the work They hold for any 2yz 6 81 and C E 8 with C I C Certainty 2I y 2 FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 493 Idempotence 5 Ca 1 Complementarity 17 C y yC 17 Consequence Monotonicity on First Component If 0 C C C I then m t 2 gt 9502 b 9621 Order Independence of Events If zy gt 6 then 5506 zDe ltgt 2106 yDe StatusQua Event Commutativity If 1 6 Bo and z t e then 44 ICeDe wDeCe where the successive gambles are from independent realizations of the chance experiment or phenomenon Note that in 44 a is the consequence resulting from the occurence of both C and D in independently run experiments regardless of the order in which they occur in all other cases the result is e In the rest of this section except when otherwise indicated the variables z y 2 etc ap pearing in the notation of gambles or alone denote elements in 131 43 Bilinear Representations One goal of the theory has been to understand the circumstances that give rise to the class of bilinear representations 45 U 56 C 1 U IWC U y80WC 1 where go is some stn39ctly decreasing function om 0 1 to 0 1 In doing so we assume that the range of U is a halfopen interval 0 k where k 00 is a possibility although for simplicity here we assume k lt co and the range of W is the closed interval 0 1 Conditions are known that are sufficient to ensure these ranges for U and W For simplicity in the sequel we use the abbreviation U 6 C y to mean UzC Two classes of bilinear models have received a great deal of attention those giving rise to the rankindependent utility RIU representation in which there are no constraints on the zy pairs and those that give rise to the rankdependent utility RDU ones in which we require 7 z y and assume that the other case is covered by the behavioral indifference of complementarity stated above We will take up RIU and RDU in that order 44 Axiomatization of Binary RIU Representations 441 Two Tradeoffs Luce 1998 explored the fact that 45 consists of two important kinds of tradeoffs The one is the additive tradeoff between the pairs 6 C and y C It has been axiomatized in the literature under the name additive conjoint measurement eg Krantz Luce Suppes and Tversky 1971 Ch6 and yields an additive form 46 UICy I1z0 I392yC where fori 12 11 C x gt0 k is surjective onto and I C 0 By certainty and 46 U39v Uzl y 1 1z1 II2yI Thus IlgyI is a constant which must be 0 because 112eI 0 The other tradeo in 45 arises when one sets 3 e yielding a multiplicatively sep arable trade off between 6 and C namely U1zW1 The key behavioral property un derlying the latter trade off called the Thomsen condition was shown by Luce 1996 to 494 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE derive from the above assumption of statusquo event commutativity 44 Moreover W1 preserves the order of event inclusion and so W10 0 and W1F 1 Because both U1W1 and In preserve the same order there is a strictly increasing and surjective f 0 k 0k such that for all a E C C E 8 47 I 1C flU139EW1C f0 0 Note that setting C F and using I12y 0 yields from 46 f U1 56 1 U1 W1 1 1112 F II11 1205111 Uz1 y U Under suitable structural conditions one can also justify the existence of certainty equiva lents CEaC y N zC y where CE 82 Bo and so we can write UzCy fU1CEzCy fU1zCy Thus from 46 48 flU1 DC 31 flU1 DW1Cl I 2yC where f0 0 and f is strictly increasing We explore three approaches one in this section and the others in 45 and 46 that are based on this equation 442 The RIU Functional Equation Setting a e in 48 which we can do because there is no constraint on the pair 6 y beyond being gains and using the fact that U1e 0 and complementarity I 2yaC flU1eW1Cl I 2yaC fU1eCy f U1y 5 6 flU1yW1al 2036 flU1yW1Ul Substituting this into 48 49 fU139v any fU1zW1C fU1yW1Ul Setting a y using idempotence and introducing the notations v U1 y w W1 C qw W1U qW1C we arrive at the functional equation 410 fv fvw f vqw 11 6 MW 6 01 where 411 f0 0 Since 410 is supposed to hold for all 11 E 0k necessarily vqw E 0 k and so qw E 0 1 for all w E 0 1 The derivation of 410 forces f to be a strictly increasing mapping of 0k onto 0k and q to be a strictly decreasing mapping of 0 1 onto 0 1 So they are continuous In the published proof Acz l Ger and Jarai 1999 however a much weaker assumption suf ced to solve 410 This assumption is that f is nonnegative on its domain 0k We omit the proof that except for two trivial solutions of 410 constant on 0 k cf 430 and FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 495 431 f is continuous and strictly increasing but we do show the equally surprising result that the problem can then be reduced to the case where f is continuously differentiable2 We show that under either of these assumptions the nontrivial solutions of 410 are 412 fv 1115 qw 1 w 1B v 0kw E0 1a gt 0 gt 0 If we de ne U U and W Wl then from this solution and 49 we obtain the rankindependent representation 45 with qWC 1 Note that this reduces to binary subjective expected utility if and only if W is nitely additive over disjoint unions 443 Solution of RIU Functional Equation The proof is divided into two major parts Reduction to the Di 39erentiable Case We introduce the integal mean of f 413 8u l u fvdv if u gt 0 and 90 0 u 0 Because f is continuous and positive on 0 k its integral mean exists and it is itself strictly monotonic positive and continuously differentiable on 0 k if we de ne 6 0 f 0 0 Surprisingly integrating 410 with respect to v dividing by u and setting 3 vw t vqw yields the same equation 410 this time with 9 in place of f 9a 50701 2 5 fvwdv 51 fvqwldv 1 W f d 1 quotM f gt 1 s s t t w 0 11 0 9W 9UltIwl for u gt 0 but with 90 0 also for u 0 Because 9 u fu a 0 9 1 is differentiable and so 414 m ge waw am is differentiable on 0 1 Observe that if we can show the differentiable solutions are those given by 412 then 6a cwquot fv ve v anquot Av v 6 MM gt m gt 0 and 414 yields the same expression as in 412 Solution in the Di erentiable Case So we now assume that f is strictly increasing and continuously differentiable on 0k and q is differentiable on 0 1 Differentiating 410 with respect to v and to we then get 415 f v wf vw qwf vqw 416 0 vf vw vq39wf vqw respectively Eliminating f vqw yields 417 q wf v wq39ltw qwlf vw 2The actual history of the proof went the opposite way It was rst solved on the assumption of di erentiability then continuity and strict increasing was shown to suf ce and nally that was established from the equation and nonnegativity 496 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE The expression in square brackets is nowhere 0 on 0 1 So 417 can be divided by wq w qw yielding the Pexider equation NW f v w 11 10kw 61010 where 39w q wwq w qw From 219 in Section 214 or Acz l 1987 pp 7374 we know its general continuous solution 1 is given by f v 1Wquot 11 10kl where a 0 because f v E 0 has been excluded Also b a 1 because for b 1 the resulting f v a lnv c is either negative or decreasing in the right neighborhood of 0 which is impossible So integration yields fv cw 397 v 0k Because f is strictly increasing continuous at 0 and f 0 0 we may conclude a gt 013 gt 07 0 Substituting this into 410 we obtain 412 as the general solution of 410 under the assumptions that f is strictly increasing and continuously differentiable on 0 k and q is differentiable on 0 1 Notice that q turned out to be continuously differentiable on 0 1 but not necessarily at 0 0r 1 and that q0 1 and q1 0 followed As was noted earlier the weaker assumption that f is nonnegative and nonconstant on 0 k is suf cient for the conclusion 45 RankDependent Utility Version 1 451 The Functional Equation In the rankdependent case we work with the constraint 1 t y and obtain the form for a lt y om the assumption of complementarity The argument is valid up to 48 La fU1Cy fU1wW1C I 2yC under the constraint 1 f y What does not work in the rankdependent case is setting 1 c this corrects Luce 1998 Luce and Marley 2000 explored three approaches to the ranked case The rst simply postulates without any real axiomatization that the y C terms also have a separable representation say U2 W2 S0 48 can be written 418 fU1 C 9 fU1W1C hU2yW2Clv In 418 let a y 12 U1y w W1C and de ne the function g by U2c gU1a gv and the function q by qw W2 Note that because of order preserving properties and the assumption of order independence of events 9 is strictly increasing and q is strictly decreasing with q0 1 and q1 0 Setting C 0 in 418 yields 419 f u 4901 and so 418 with z y implies 420 f v KW f 93919vqw 11 6 014 wqw 6 011 where we know that g 1 is de ned because 9 0k gt 0k is strictly monotonic and surjective As a consequence it is also continuous Although we know and will use the fact that f and q are strictly monotonic and contin uous the published proof supposes only that f is nonnegative and nonconstant on 0 k and derives strict monotonicity and continuity from that and the equation Equation 420 reduces of course to 410 if g identity We need however new ideas to solve 420 Because f and 9 both increase strictly and f0 90 0 it follows that q0 1 q1 0 FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 497 hold Ignoring these values for the time being we sketch the novel road map leading to all nontrivial solutions of 420 for more details see Acz l Maksa Ng and Pales 2000 452 Solution of Version 1 Functional Equation 420 Once again the proof is developed in two stages Linearizing the Equation and Convexity First we linearize the functional equation 420 with the aid of new functions defined as follows 421 28 1nqltesgt s 610oo1R and 422 Ft Nequot 000 411903quot Ht fly 1034 t E lnkoo Because the nontrivial solutions f of 420 are strictly increasing and g also increases strictly F and H both strictly decrease and G strictly increases With these functions and with s lnw t ln 1 420 is linearized as 423 Ft Ft s HGt Qs s e 0oot e lnkoo From this equation t H F t F t s is strictly decreasing because on the right hand side H decreases and G increases both strictly But then Ft Ft s gt Ft s Ft s s s gt 0 or with z t 23 t F t F F 32K 1 2 that is F is strictly Jensen midpoint convex Because F is also monotonic it is strictly convex see eg Roberts and Varberg 1973 p 219 This idea of Zsolt Pales and result proved to be very useful because we know a lot about convex functions For instance they are continuous they have everywhere leftside and rightside derivatives except maybe at the boundary of their interval of convexity moreover they are differentiable except for at most countably many points see Roberts and Varberg 1973 pp 47 Of course because G and H are monotonic they are almost everywhere differentiable see eg Riesz and SzokefalviNag 1990 pp 5 9 We write 423 as 424 H 1Ft Ft 3 Ct 23 3 E0oot e ln koo which we can do because H and thus H 1 is strictly monotonic decreasing and surjective Thus H 1 is almost everywhere differentiable on its domain Ft Ftss Em t6 lnkoo which because F is continuous and t H Ft Ft s is strictly decreasing is a non degenerate interval One can manipulate the two variables 3 and t and the differentiability of H almost everywhere and that of F up to at most countably many places so that the left hand side of 424 turns out to be everywhere differentiable in 3 Thus Q on the right and finally H 1 prove to be differentiable on the entire domain We do not yet know that F and G are everywhere differentiable but the rightside derivatives F and G exist everywhere as do the leftside ones the former because of the convexity of F and the latter because of 424 and the chain rule We differentiate 424 from the right with respect to s and t respectively and then eliminate H 1 Ft Ft 3 in order to get 425 Q39ltsgtF4lttsgt F4tgt1GtgtFlttsgt seiooote1 1nkoogt z gt t 498 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE Because F is strictly decreasing and strictly convex we know that F4 is negative everywhere and see eg Kuczma 1985 p 156 or Roberts and Varberg 1973 p 5 F t s F4t is sign preserving everywhere positive or everywhere negative Looking at 425 more thoroughly we see that also G and Q are sign preserving With the notation 1 G 1 426 LE MEN we obtain from 425 427 Ls t Lt MtNs s 6 0oo t 6 ln koo This is related to the equation 223 and is a particular case of several known functional equations see Acz l and Chung 1982 and Jarai 1984 By our previous results M and N preserve their signs and so 427 implies that L is strictly monotonic Accordingly the general solutions for s E 0 oo t 6 ln k 00 are given by the two sets of functions 42s Lt Ae B Mt Jed M3 gen 1 A 429 Lt At B Mt Ct Ns as The Solutions All we have to do now is play the tape in reverse through 426 422 and 421 and after passing some hurdles we arrive at the general solution of the functional equation 420 under the given conditions3 We recall also f0 90 q1 0 and q0 1 The nal result is the following theorem Theorem 9 If f is nonnegative and g is continuous and strictly monotonic then all so lutions of the functional equation 420 are given rst by the trivial degenerate solutions 430 fv 0 v E 0k q 0 1 01 g 0k 0k with g strictly increasing and surjective but otherwise both 9 and q arbitrary o ifv0 0 ifwe01 431 0k gtOk 6 ifv e01 W d if39w oa e01 g l l l l with g strictly increasing and surjective but otherwise arbitrary second by 432 fv av gv cl mum q39w 1 w 7 v E 0 k39w 6 0 1 and third by 433 f0 0 an will 91 5701 k Y lt10 1 100 v 6 07440 E 01 where a gt 06 gt 07 gt 0 and u gt k 3 are constants We have thus completely solved 420 under weak conditions Note that the solution 412 of 410 is the particular case 7 21 of 432 whereas 410 has no solution corre sponding to 433 3The hurdles arise by those conditions and by the unusual domain of validity of 427 which is why the theorem of Acz l and Chung 1982 was needed because it holds for arbitrary intervals At the point where we get F4 and G from 426 428 and 429 we see that these rightside derivatives are continuous thus F and G are everywhere differentiable see Kuczma 1985 p 156 and they as well as Q can be determined by integration FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 499 Utility representations With 432 the representation in utility terms becomes UxCy UWC Uy1 WC 95 t 2 which except for the condition 1 t y is the same as the RIU representation ie 45 with gz 1 z This is only half of the story To get what happens when a 4 y we invoke the assumption of complementarity ie 1C 3 315 1 yielding the full binary rankdependent representation WC Uy1 WC 95 t 2 U 434 U a C y Ux1 we Ultygtwltcgt a s 2 However the solution 433 yields after some algebraic manipulation a new representation of the utility of a binary gamble for 1 f y and p gt F1 UWC Uy1 WCl HUU9W0 which is called ratio rankdependent utility or more briefly RRDU Interestingly the par ticular case p 0 is just 434 whereas 432 is not a particular case of 433 although it is in a sense a limiting case Luce and Marley 2000 asked what behavioral properties force p 0 Three were found of which the most natural is mentioned in Section 481 Despite the nonsymmetric form of RRDU 435 it exhibits the important symmetry property called event commutativity 436 CyDy zDyCy which plays a role below Note that each side says that 1 is the consequence if in two independent realizations of the chance phenomenon C and D both occur and otherwise 3 is the consequence The difference is only in the order of occurrence and event commutativity assumes that does not matter to the decision maker Also note that the earlier assumed statusquo event commutativity used to insure separable representations is the special case of 436 with y e 46 RankDependent Utility Version 2 Luce and Marley 2000 took another approach that again began with 48 but then followed a more principled route Using the idempotence axiom and setting 1 y in 48 we get f U1y f U1yW1C I 2yC Substituting this back into 48 yields for 1 t y f 6 437 f U1 0 31 f U1W1Cl fU1y f U1yW1C This constraint on f seems inadequate for our purpose we must assume more A plausible experimentally supported added condition is event commutativity 436 which was just mentioned as a property of RRDU To see what that implies let u U1a v U1y w W1C t W1D k limsup U1 500 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE Then for all uvwt such that k gt u 2 39u 2 0 and wt E 0 1 418 yields fU1zCyaDzl fU1mC yW1Dl f U1y fU1yW1D flf 1flU1 C C yllW1Dl flU13l quot flU1yW1Dl f fquot1fU1W1Cl flU11l fU13W1CllW1D flU1yl flU1yW1Dl ffquot1fuw N fvwlt N fvt Thus event commutativity yields 438 f f 1fuw M fvwt fvt f f 1fut nu fvtw fvw where one has to show that f fv fvw is in the range of f Acz l and Maksa 2000 solved this equation Theorem 10 If the domain and the range of f are intervals f is strictly monotonic and twice di erentiable then for some constants aB39yn and p 439 fv 77 mmquot 7 440 fv nv 7 441 M nlnow s 1 If f is also supposed to be strictly increasing and f0 0 then there are just the following two solutions 442 fv nlnow 1 u gt 7 m gt 0 3 gt o 443 M 7wquot n gt 0 3 gt 0 From 442 one derives again the ratio rankdependent form 435 with the added restriction a 79 0 And 443 yields RDU ie the case with a 0 For the cases where a j y we simply use complementarity and the representation with the roles of a and y interchanged We hope that future research will remove the differentiability assumption as was the case for RIU and the rst version of RDU Section 45 47 RankDependent Utility Version 3 Marley and Luce 2000 gave an axiomatization based on two major properties event com mutativity and the following assumption called gains partition There exists a permutation M on 6 that inverts the order induced by f on 539 so that for zzlyy39 E C with a f y c39 t y39 and CC e 6 if 3636 5056 and yMCe y39MC39e then xCy C y39 At rst blush gains partition may seem trivial however it need not hold in a represen tation where the pieces U c WC U y and WM play separate roles not just as U xWC and U Representations of these more complex types have arisen for example in Luce s 1997 20003 treatment of binary mixed gains and losses which we do not go into here FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 501 A rather intricate argument which we do not repeat here gives rise to the functional equation Z 444 59 129010 f 1fz 1Pl 21 El01l Acz l Maksa and Pales 2000 proved the following Theorem 11 The general solutions fq O1 gt0oo both strictly monotonic and g 01 0oo strictly decreasing and surjecti39ue of the functional equation 444 are given by gm9 p qppc fltzA1 z c Menu zl where A gt 05 gt 00 lt 0 are otherwise arbitrary constants Equation 444 leads to an equation that is similar to 425 in Section 452 but here there is no solution corresponding to 432 Marley and Luce 2000 used Theorem 11 to show that the RDU representation holds with WMC 1 WC 48 Utility of Joint Receipt 481 Joint Receipt Luce 1991 and Luce and Fishbum 1991 were the rst to study theoretically the concept of a binary operation 63 over gambles called joint receipt Earlier empirical work had invoked the concept which in one study was called a duplex lottery It is the very natural idea of having or receiving two or more as it turns out things at once So if a and y are two gambles including pure consequences as a special case then a 63 y means having or receiving both of them Several natural assumptions are made For all xy z E 81 JR Commutativity a 69 3 3 69 1 JR Monotonicity x f y gt a 69 2 f y 63 2 JR Identity 2 69 e N 2 A major scienti c question is How does joint receipt interlock with the gambling struc ture The answer proposed by these authors is the highly rational and empirically sup ported property called binary segregation 445 xCe By69yCy mic 218 065 Luce and Marley 2000 showed that segregation added to rational rankdependent utility 435 forces a 0 ie binary RDU 482 RDU and Segregation Given the concept of joint receipt 63 binary rankdependent utility and segregation Luce and Fishbum 1991 1995 noted that by applying the utility function U of binary RDU to this expression and using the separability assumption we obtain U Uquot1UWCl 69 U 1Uy U 99WC Uyll WCl With 446 u U2 v Uy w WC and Gu39v U U391u EB U 1v the last equation becomes 447 Guw39v Guvw v1 w 0 lt 39u S u lt k0 lt w lt 1 502 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE where by the monotonicity of 63 G is strictly increasing in each argument which fact however is not used in what follows We prove that the solution is 448 Guv avu v where a O k 0 k is an arbitrary function If we de ne 449 guu Guv 1 then 447 can be written as guw wgu By xing u c close to k and setting 11 gucc we get t gt 2900 avt 0 lt t lt c This can be extended to all t 0 k From 449 we then get 448 Conversely 447 is satis ed by 448 for arbitrary a If we also require as follows from JR commutativity and 447 450 Cu vw Cu vw u1 11 then we get similarly Guv 6uv u Comparison with 448 gives avu v 6uv u or rewriting av 1 6u 1 6 constant 1 u Thus 451 Guv u v 61w is the general solution of 447 and 450 where 6 is an arbitrary constant Rewriting 451 in terms of the function U and using 446 we nd 452 U60 63 y Uw Uy 6UUy which is called a p additive representation because it is the only polynomial form with U e 0 that can be transformed into an additive representation Thus the assumptions of RDU JR idempotence JR commutativity and segregation imply JR Associativityn 1 69 y 63 2 1 e y 63 z ayz E 81 483 pAdditive and Separable Utility Of course we also know from the assumption of RDU that gambles of the form 2 C e where e is the identity of e have the multiplicative separable orderpreserving represen tation U From separability and p additivity with a common utility function and segregation Luce and Fishburn 1991 showed that RDU follows So that is a fourth way to axiomatize the RDU representation This has the virtue of making very transparent the source of rank dependence namely segregation But it also has a substantial weakness It is easy to axiomatize separability as noted above It is also easy to axiomatize the existence of an additive representation Krantz Luce Suppes and Tversky 1971 Ch 3 but it is far from clear that both can be done with the same utility function U Luce 1996 exPlored the question A necessary and sufficient condition was found namely the structure is said to be jointreceipt decomposable if for each a 6 81 and C 6 639 there exists an event D DzC 6 8 such that for all y 6 81 453 a e yCe 2 Ce e yDe FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 503 It is not very dif cult to show that this property follows when both U is padditive and U W forms a separable representation In the process one shows that D satis es 1 6U 13 1 6UvWC The deeper question concerns the other direction Suppose we have a p additive repre sentation U1 and a separable one U2W2 each of which we know how to axiomatize Does 453 mean that there exists a U that is both padditive and with some W also U W is a separable representation The answer is that for some gt 0 U 2 U23 is p additive and U W where W W2 is separable To show this one proceeds as follows Let f be de ned by U1 f U2 Since they both preserve the same order f is strictly increasing Then applying these assumptions to joint receipt decomposability and de ning p Ua q w Ppww one is led to the functional equation 454 WD WC 4 55 HOD 11 HWaqPQ aw Paq E 02 liiw E 0 1 where P maps 0 X01 into 0 1 and 456 HO 4 F 1Fp Fq FFql with F 6 f so that F maps 0 1 onto 0 1 and is strictly increasing First we simplify 456 to 457 H p q 1lt1gtplt1gtql WI 6 010 by introducing 458 MP 1 F P Clearly ltIgt 2 0 l 0 1 is a surjection and is strictly decreasing and so continuous By 457 H is strictly increasing and continuous in each variable and thus is also continuous as a function of two variables Fimctional equation for P Repeated application of 455 gives H Wm qPPwsl H P 1108 Hlpw qPPwls H PW9 1131 wPPw 8 Because H is strictly increasing in the second variable P satis es 459 131 w PpwPpw8 P E 01ws 6 01 So our solution plan is rst solve the functional equation 459 next plug that solution into 455 and then try to use that functional equation to nd the general solution H and so by 456 nd F Note that it would be easy to solve 459 if we could substitute p 1 but we cannot and as we will see in 473 below Il i mlPpw 0 for which 459 reduces to O 0 Solution of equation 459 First one can show that 460 P0w 0 and for p E0 1 461 Ppw 0 iff w 0 After excluding p 0 w 0 and s 0 from 459 we do the next best thing to substituting p 1 namely we substitute 1 c lt 1 and to counterbalance we choose w rc Then with the notation 1 c739 W 739 El0Cl 504 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE we get MT Acrs One easily veri es that Ac is determined up to a multiplicative constant So for c lt c lt 1 we can choose Ac de ned on 0 c so that Aclc Acc and thus Aczs Acs for all s 0c Thus these Ac s are restrictions to 0c of a function A 0 l gt0 oo such that Prs 139 Ocs E01 MT P 01 1 4 62 r 3 Mrs 139 E s 0 This 460 and 461 satisfy 459 and so we have solved that functional equation One can show that A 01 gt0oo is strictly decreasing and continuous so P is strictly increasing in its second variable and continuous First approach to 455 Up to here we summarized work in Acz l Luce and Maksa 1996 That paper accomplished the second step of the solution plan determining H G and F from 460 461 462 455 457 and 458 only under the assumption of di 39erentiability of both F and F 1 or equivalently Igt and Q Several attempts were made to eliminate this condition which certainly is not inherent to the original utility problem Partial results were achieved eg N g 1998 replaced it by assuming differentiability of P in one variable It is easy to see the dif culties when one substitutes as planned 462 into 455 with 457 MP 463 dgt1ltIgtltpgtqgtltqgt1wltr1ltIgtwltrgt pqeio1weo11gt which is a rather intimidating equation Second approach to 455 One idea that pushed the solution forward was replacing 463 by a simpler limiting case obtained by taking the limit as q 1 in 463 which it turns out does not add any new solutions Acz l Maksa and Pales 1999 Notice that ltIgt 0 1 gt0 1 is decreasing surjective and continuous therefore in view of 455 and 457 the following limit equations hold 43964 01 0a HQ q 1 qgtw Observe that we did not substitute 1 we took limits in two of the functions as q f l but de nitely not of Ppw as p 1 Putting 462 into 464 we get an equation much simpler than 463 465 5amp3 ltIgt p 610 1w 6 01 Now we are ready to execute the somewhat modi ed second step of our solution plan of determining ltIgt and thus H and f from 465 Determining ltIgt from 465 As with 420 earlier we linearize the functional equation 465 to 466 Kv Kuv KZu uv by introducing u lnpv lnw and de ning 467 Ku ln Igte u ln Ae39 FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 505 Because 13 and are continuous and strictly decreasing so too are K I and as well the function 11 I gt K u u Thus Kltv m v Klan u v 2 K u u Kuv K2uv So K and similarly f are strictly convex the onesided derivatives K1 K If I exist and are strictly increasing everywhere The convexity method of Section 452 works again and even more simply One differentiates 466 from the right or left with respect to u or 11 respectively and eventually gets for L lKf and for some function A the equation cf 427 Lu v Lv LuA39v with the strictly decreasing solutions 1 1 468 m a and accordingly g and 4 69 K39 a 1 C and according Z39 6 39 Lu 1eb y 39l39 65 139 But these are continuous and if the right derivatives of the convex functions K and Z are continuous then they are differentiable everywhere Integration gives K and f and the equations 467 yield lt1 and A Substitution into 465 and taking into account the fact that the strictly decreasing 1 maps 0 1 onto 0 1 does three things It eliminates 468 it restricts the constants in 469 to b gt 0 and it eliminates the constant of integration in lt1 Finally one gets gp dpb 11b b gt 0 and for p 0 1w 0 1 1 pb b a Continuity extends the validity of 470 to p 0 w 0 and w 1 Equations 457 and 458 nally leads to 471 Hp q pb qb pbqb1b Fp 1 1 pb a gt 012 gt 0 Equations 455 and 456 are satis ed by 470 and 471 This concludes the sketch of the proof of the following Theorem 12 Suppose that the strictly increasing function F maps 0 1 onto 0 1 and P maps 01gtlt0 1 into 0 1 Then 455 and 456 are satis ed by 472 F02 1 1 pb 2 6 01 b 1b 473 Ppw w la 15m p e 01w 6 01 for arbitrary positive constants a and b and there are no other solutions For more detail see Acz l Luce and Maksa 1996 and Acz l Maksa and Pales 1999 Using Theorem 12 Luce 1996 showed that if there is a padditive representation U1 a separable representation U2W2 and the structure is jointreceipt decomposable then there exists a B gt 0 such that U U is p additive and for W W that U W is a separable representation 506 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCB 49 Invariance Assumptions In applying utility theory to data having completely open ended monotonic utility and weighting functions leaves the models insuf ciently speci ed So one is led to ask whether the forms of these functions can be seriously constrained We know of such constraints only when the consequences are money and the chance events can be characterized in terms of probabilities Further work needs to be done for at least weights over events not just probabilities 491 Utility of money Consider money consequences 2 and y a gt 0y gt 0 Although it may seem plausible that 9 EB 31 1 3 some empirical evidence suggests otherwise A rather weaker assumption is that money and joint receipt 6B are related by the a multiplicative invariance condition of the form of 11 474 AmSAyAa By a gt0ygt 0gt0 If as arises under the conditions at the end of Section 482 63 has an additive representation which we denote by V then Luce 2000a pointed out that Vquot1Vm V040 AV1V D V00 which is 230 with V f and so for some gt 0 V1 c1176 and a 69y 1 l y51 492 Form of Weighting Functions For known probabilities in gambles of the form 1 p 0 p E 0 1 consider a separable representation U W where both U and W are strictly increasing in their arguments Prelec 1998 imposed an invariance condition which he showed to be equivalent to the following form for the weighting function 475 Wp exp 1 lnpquot p 6 01117 gt 0 Luce 2000b showed the same result using the following simpler invariance condition called reduction invariance For all a 6 6 1 qr pp q E 0 1 and integers N 23 476 wp0q0 120 gt wpN0qN0 rN0 Note that 1p0q0 means that a arises with probability pq and 1139 0 means that a arises with probability r therefore a person understanding probability theory will have r pq in which case reduction invariance is automatically satis ed and in 475 we have n 1 and Wp 17 p 6 01 The proof involves rst showing that reduction invariance holds with N replaced by an arbitrary real number A which follows by induction and a limiting process Using that and separability we are led to the functional equation 477 W391WPWq 7 W391WPWqlA PM 6 01 By simple logarithmic transformations this reduces to 231 and the solution follows with a little algebra FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 507 5 CONSISTENT AGGREGATION We present this topic by an example involving n different kinds of inputs contributing to the outputs of m producers Other examples could pertain to investments employment consumption etc The jth producer s output or maximal output by some measure de pends upon the inputs raj1 chn to that producer through possibly producerspeci c microeconomic production functions Fj j 1 m We ask whether there exist 71 1 aggregator functions G and Gk k 1 n so that the aggregated output depends only upon the n aggregated inputs through a macroeconomic function F that is GF1 L 11 1n Fm l m1 Imn FG1 L3911113m1 Gn l 1n l mn This is the generalized bisymmetry4 functional equation in m x n variables Table 1 below may help understanding the situation For convenience we use in this table the vector notation xj 2711 xjk Ejn 1 O m xok Z1kjk Imk k 1 n yZ1yjym ZZ1zkzn TABLE 1 Consistent Aggregation of Inputs and Outputs Inputs commodities and services Maximal Outputs Producers 1 k n production functions 1 9311 11 171 21 F1x1 j 111 17139 len yj FJxj m 23ml zmk zmn ym Fmxm 7 Aggregators 21 G1x1 2 G1xk zquot Glxn Gy Fz Two questions arise naturally what are the domains and ranges of the functions F1 Fm G1 Gn F and G and which of these function should be regarded as given and which as unknown We will consider both real and quite general domains and ranges As to the second question it is often argued that if outputs can be measured by their monetary value then they can be aggregated by addition that is 52 Cyiymy1mym and if the kinds of outputs are completely separated even 53 Gka1amzlzm k1n can be assumed In this case 51 leads to a Pexider equation cf 215 Section 214 and can easily be solved under reasonable regularity conditions F1 Fm and F will be 4Bisymmetry is the m 2 n 2 F1 F2 2 F G 2 G1 G2 case which is important in measurement theory cf Krantz Luce Suppes and Tversky 1971 508 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE af ne linear functions However the production functions most often used in practice such as the CD CobbDouglas function 54 Fzl 2quot azf zf z quot with 61 cma gt 0 and the CES Constant Elasticity of Substitution function 55 Fzl zn aclztf62z3 cnz1b with b 96 O are not linear except for b 1 or af ne This problem can be solved if we do not require that the G s be additions We show here that 54 and 55 are incompatible with 52 and 53 in the general framework of 51 On the other hand there is no compelling reason for assuming that inputs outputs etc have to be measured by money or other real valued indices or even that these variables can be so measured Accordingly it makes sense to solve 51 as we do in the sequel on quite general sets without imposing any order or topological properties cf von Stengel 1991 Acz l and Maksa 1996 Taylor 1999 In particular the sets could be discrete for instance consisting of all possible collections of inputs or outputs This may give a more realistic approximation of the empirical situation We shall see that a cornerstone of the solution is an abelian structure In addition to surjective onto functions which we used before we use the standard notions of injective 1to1 bijective 1to 1 and onto functions f from X into onto Y We denote by f X the image of X under f ie the range of f Note that if f is an injection then it is a bijection of X onto fX When in a function F 21 x x Zn S in n variables 21 2 we x all the variables except 2 we get a partial function F39 Z1 S really a family of such partial functions one for each possible xing of the values of 21 z1z1 2quot For arbitrary sets we have the following Acz l and Maksa 1996 Theorem 13 Let in Z 2 n 2 2 be integers For 1 5 j S m and 1 S k 5 it let Xjk Zk and S be nonempty sets with the functions EZX XXXjn lj szXlkxmek gtZk FZ1xgtltZ gtS GY1xYm S Then the two following conditions are equivalent i The partial functions F Xjk and 01 Xjk gt Zk are su39rjections Fquot Zk S and Gj gt S are injections and 51 is satis ed for all mi 6 XJk 1 S j lt m 1 S k S n ii There exists an abelian group T o T g S surjections fjk X jk T and Injections gjYj gtT andhkZk gtTsuch that 56 F21 znh1zloohnzn zkEZk15kSn 57 Gy1 ymglyl gmym yj EYjl Sjsm 58 Fjij 13quot 9171 fjl11 fjnjn 59 Gk1k fcmk h1f1kf 1k fmkmk Ivjk EXjk1 SJ Sm1 5k 5 Others results on consistent aggregation have been reported before and after this one For earlier results see Acz l s 1997 survey As to more recent ones Taylor 1999 has weakened the surjectivity conditions of Theorem 13 from supposing the surjectivity of the partial functions for all choices of the remaining variables to just one choice for each partial FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 509 function Accordingly T 0 becomes a cancellative a 01 a o y gt 9 y abelian semigroup closed under the commutative and associative operation 0 with unit element rather than an abelian group This result is more appropriate for applications to real intervals For example the conditions of Theorem 13 exclude the set of all nonnegative reals equipped with addition but this result does not At the same time Maksa 1999 proved Theorem 14 below for arbitrary real intervals without assuming any surjectivity We do not give here the details of the rather intricate proofs of either of these three results beyond mentioning that they rely on a double induction and on an adjustment of the solution on different domains In the theorem below we use CM for the class of all real valued functions de ned on a subset of 1Rl for some positive integer l which are continuous and also monotonic in all variables We also write fW1 gtlt x M for the image of the Cartesian product W1 x x W by the function f This is sometimes denoted by fW1a WIH Theorem 14 Let m gt 2 n gt 2 be integers and let Xjk 1 S j S m 1 S k S n be real intervals with szX xgtltX gt1R Gerlkxgtltka R FjX1 gtlt ijn Jj GkX1k X XX Ik FIlgtltxI R GJ1gtltgtltJm IR and FjGkFG 6 CM Suppose also that 51 holds for all 23 1 S j S m 1 g k S n Then there exist a real interval I and functions tJII HR akzIk HR VjZJj9R BijXjk9lR with ak jk 6 CM for1 5339s m 1 5kg n such that 510 F1 xn liakzk 21Zn IlgtltgtltIn kl Cyla39 aym yla ym 6 J1 X X Jm j 512 Fjj1a 112 7171 jkjk 1 513 Gk1k xmk a i jkjk DJ E Xjk1 S lt m 1 S k S n j1 In other words For consistent aggregation under the conditions of Theorems 13 and 14 the production functions are given either by 56 and 58 or by 510 and 512 and the aggregation functions either by 57 and 59 or by 511 and 513 on general sets or on real intervals respectively We can now answer the question raised earlier concerning the incompatibility of CD functions 54 and CBS functions 55 with an aggregation by addition as in 53 If even one aggregation function is a sum say 010171 xma1rm then no production function 512 can be CD Indeed putting 514 into 513 gives 515 11131m61118m1m 510 JANOS ACZEL JEANCLAUDE FALMAGNE AND RDUNCAN LUCE This is a Pexider equation A result proved in Acz l 1987 p 80 applies if 515 is satis ed for asj on real intervals Jj l S j 5 m and yields that all continuous solutions are given by cf 26 in Section 214 13113 1 9 31 1 lt j g m m 13 1 9 23 i1 where 1 gt O 31 3quot are constants so from 512 with wk 211 1 S k S n 516 Fjw1 wn 71717 101 3 Z jkwk k2 Another application of a Pexider equation shows that 516 cannot equal a CD function awf 10quot cf 54 The situation for CBS functions is similar and they too are incom patible with aggregation by addition We leave the details to the reader This pair of results does not say that inputs or outputs cannot actually be additively aggegated Rather it says that such an additive aggregation is not consistent with common and realistic produc tion functions As mentioned earlier it is consistent only with af ne or linear production functions The following question is often asked if all of the Fj functions are in some sense of the same form does that dictate that the macroeconomic function F must also be of the same form This is called the representativeness problem The answer of course depends upon exactly what is meant by of the same form In a trivial sense the answer is Yes because the F and Fj s in the above 510 and 512 solutions have the same structure A somewhat more sophisticated answer is that it need not be so For example if all the are CD functions then in 512 719 111 g 311423 611113 J or if they are all CES functions then in 512 7jy 11quot 31743 0km Neither conclusion restricts 1 and ak in 510 in any way Thus one may choose F to be a CD function a CBS function or some other function of the form 510 The general conclusion from solving the aggregation problem is that consistent or even representative aggregation is feasible only for appropriately chosen functions In general neither is possible if the aggregating functions are pre selected the wrong way The implications for the possibility of macroeconomic models are considerable Results of this kind have applications elsewhere For instance they put stringent con straints on models having legitimate application to data aggegated over trials of an exper iment or over individuals tested For some results concerning aggregation of probabilistic models of choice which among others characterize Luce s 1959 choice model as a special case see Acz l Maksa Marley and Moszner 1997 6 CONCLUSIONS As demonstrated in this article the solution of many functional equation problems consists in reducing a functional equation to another one which belongs to a running list of all those already solved Over time the list grows extending the reach of the techniques This process is supplemented by general results concerning broad classes of equations such as uniqueness FUNCTIONAL EQUATIONS IN THE BEHAVIORAL SCIENCES 511 theorems and theorems strengthening the regularity of functions eg from integrability to di erentiability of arbitrary order Uniqueness considerations are implicit in Sections 11 22 and 3 and a method of elevating regularity from continuity to differentiablity was explicitly used in Section 443 The examples of functional equations techniques described in this paper were taken from three quite different areas of behavioral sciences sensory psychology micro and macroe conomics and utility theory This diverse choice was deliberate and intended to suggest that such methods have potentially wide ranging applicability not only in the behavioral sciences obviously but in all the sciences In spirit these methods resemble those used in dimensional analysis of eg Krantz Luce Suppes and Tversky 1971 but they are con siderably more general in scope the expressions in the equations need not be monomials and the goal is to nail down the forms of some functions and not simply to uncover the values of some exponents In many cases they can be used to convince oneself and others that the functions specifying a model are the only feasible ones Within a given framework REFERENCES 1 Acz l J 1987 A Short Course on Emctional Equations Based on Applications to the Social and Behavioral Scienca DordrechtBostonLancasterTokyo ReidelKluwer 2 Acz l J 1997 Bisymmetry and consistent aggregation Historical review and recent results In A A J Marley Choice Decision and Measurement Essays in Honor of R Duncan Luce Mahwah NJ Lawrence Erlbaum Associates 225233 3 Acz l J amp Chung J K 1982 Integrable solutions of functional equations of a general type Studia Sci Math Hungar 17 5167 4 Acz l J amp F almagne J Cl 1999 C 39 t of 39 and quot391 representations of func tions arising from empirical phenomena J Math Anal Appl 234 632659 5 Acz l J Ger R amp Jarai A 1999 Solution ofa functional equation arising from utility that is both separable and additive Proc Amer Math Soc 127 29232929 6 Acz l J Luce R D amp Maksa G 1996 Solutions to three functional equations arising from different ways of measuring utility J NfathAnaJApp1 204 451 471 7 Acz l J Maksa G 1996 Solution of the rectangular m x n generalized bisymmetry equation and the problem of consistent aggregation J MathAnalApp1 203 104126 8 Acz l J Maksa G A functional equation generated by event commutativity in separable and additive utility theory Aequation Math In press 9 Acz l J Maksa G Marley AAJ amp Moszner Z 1997 Consistent aggregation of scale families of selection probabilities Math Soc Sci 333 227230 10 Acz l J Maksa G Ng C T amp Pa les Z 2000 A functional equation arising from ranked additive and separable utility Proc Amer Math SOC In press 11 Acz l J Maksa G amp Pales Z 1999 Solution of a functional equation arising from different ways of measuring utility J Math Anal Appl 233 740748 2 Acz l J Maksa G amp Pales Z Solution of a functional equation arising in an axiomatization of the utility of binary gambles Proc Amer Math Soc In press 13 Falmagne JC1 1977 Note Weber s inequality and Fechner s Problem J Math Psychol 16 267 271 14 Falmagne JCl 1981 On a recurrent misuse ofa classic functional equation result J Math Psychol 23 190193 15 F almagne J C1 1985 Elements of Psychophysical Theory New York Oxford University Press 16 Falmagne J C1 amp Iverson G 1979 Conjoint Weber s laws and additivity J Math Psychol 20 164183 17 Falmagne J Cl Iverson G amp Marcovici S 1979 Binaural loudness summation Probabilistic theory and data Psycho Review 86 25 43 18 Falmagne J Cl amp Lundberg A 1999 Compatibility of gain control and power law representations A Janos Acz l connection Aequationas Math 58 110 19 Forti GL 1995 Hyers Ulam stability of functional equations in several variables Aequationes Math 50 143 190 20 Jarai A 1984 A remark to a paper of JAcz l and JK Chung Studia Sci Math Hungar 19 273274
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