Prospect theory is currently the main descriptive theory of decision under uncertainty. It generalizes expected

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1 MANAGEMENT SCIENCE Vol. 55, No. 5, May 9, pp ssn 5-99 essn nforms do.87/mnsc INFORMS Addtve Utlty n Prospect Theory Han Blechrodt Erasmus School of Economcs, Erasmus Unversty, 3 DR Rotterdam, The Netherls, blechrodt@ese.eur.nl Ulrch Schmdt Kel Insttute for the World Economy Department of Economcs, Unversty of Kel, 498 Kel, Germany, uschmdt@bwl.un-kel.de Horst Zank Economcs, School of Socal Scences, Unversty of Manchester, Manchester M3 9PL, Unted Kngdom, horst.zank@manchester.ac.uk Prospect theory s currently the man descrptve theory of decson under uncertanty. It generalzes expected utlty by ntroducng nonlnear decson weghtng loss averson. A dffculty n the study of multattrbute utlty under prospect theory s to determne when an attrbute yelds a gan or a loss. One possblty, adopted n the theoretcal lterature on multattrbute utlty under prospect theory, s to assume that a decson maker determnes whether the complete outcome s a gan or a loss. In ths holstc evaluaton, decson weghtng loss averson are general attrbute-ndependent. Another possblty, more common n the emprcal lterature, s to assume that a decson maker has a reference pont for each attrbute. We gve preference foundatons for ths attrbute-specfc evaluaton where decson weghtng loss averson are dependng on the attrbutes. Key words: addtve utlty; prospect theory; decson weghtng; loss averson Hstory: Receved March 5, 8; accepted December, 8, by Davd E. Bell, decson analyss. Publshed onlne n Artcles n Advance February, 9.. Introducton Many decson stuatons nvolve outcomes that consst of several attrbutes. In appled decson analyses, t s useful to decompose the utlty functon over these multattrbute outcomes nto separate utlty functons over the dfferent attrbutes to reduce the number of preference elctatons. Ths s only justfed f the decson-maker s preferences satsfy partcular assumptons. Several authors have dentfed the preference condtons that allow decomposng multattrbute utlty functons nto addtve, multplcatve, related decompostons (e.g., Farquhar 975, Fshburn 965, Keeney Raffa 976). Most of these decomposton results have been derved under expected utlty. Abundant evdence exsts, however, that (subjectve) expected utlty s not vald as a descrptve theory of decson under (uncertanty) rsk. The descrptve defcences of expected utlty complcate the emprcal assessment of the preference condtons underlyng decompostons: t cannot be excluded that observed volatons of preference condtons are due to volatons of expected utlty rather than to volatons of a decomposton. To obtan robust tests of the approprateness of decompostons, t s desrable to derve condtons that are vald even when expected utlty s volated. In ths paper, we study multattrbute utlty theory under prospect theory (Kahneman Tversky 979, Tversky Kahneman 99). Prospect theory s currently the most nfluental theory of decson makng under uncertanty. It models two major devatons from expected utlty: nonlnear decson weghtng loss averson,.e., the tendency that people treat outcomes as devatons from a reference pont are more senstve to losses than to gans of the same magntude. Both nonlnear decson weghtng loss averson are wdely documented n the emprcal lterature. Despte ts popularty, some evdence has been accumulated recently revealng lmtatons of the theory (see the summares of Marley Luce 5 Brnbaum 8). Fshburn (984), Myamoto (988), Dyckerhoff (994), Myamoto Wakker (996) also studed multattrbute utlty under nonexpected utlty, but only consdered outcomes of the same sgn. Lke us, Zank () Blechrodt Myamoto (3) studed multattrbute utlty theory under prospect theory but ther approach was dfferent than the approach of ths paper, as we explan next. A central ssue n multattrbute prospect theory s to determne when an attrbute yelds a gan or a loss. Consder, for example, a research assocate 863

2 864 Management Scence 55(5), pp , 9 INFORMS (RA) who contemplates changng jobs. In evaluatng dfferent jobs, the RA has to consder several aspects, e.g., salary, commutng tme, cost of lvng, amount of research tme, etc. How does the RA determne whether a partcular job offer s an mprovement (a gan) compared wth her reference pont (presumably her current job)? One possblty s that she frst determnes whether the job offer, as a whole, s a gan or a loss compared to her reference pont, then apples the decomposton to determne how attractve the job offer s compared wth other offers. Ths holstc evaluaton was used by Zank () by Blechrodt Myamoto (3). Another approach, the focus of ths paper, s that the RA determnes a reference pont for each attrbute evaluates job offers as gans losses on each attrbute. Ths attrbute-specfc evaluaton seems plausble when the number of attrbutes s large the choce s complex. A decson context where the attrbute-specfc evaluaton s partcularly ntutve s welfare theory: there we are nterested n whether each ndvdual s welfare s above some reference level. The attrbute-specfc evaluaton s commonly assumed n emprcal studes on loss averson for trade-offs under certanty was found to be descrptvely accurate by some studes (Bateman et al. 997, Blechrodt Pnto, Tversky Kahneman 99). Also emprcal studes for decson under rsk reled on the attrbute-specfc evaluaton. See, for example, Payne et al. (984) who study managers choces among captal budget proposals nvolvng cash flows at two ponts n tme, Fscher et al. (986) who consder both rsky, multperod cash flows rsky job alternatves. Both studes use attrbute-specfc reference ponts. No preference foundaton of the attrbute-specfc evaluaton exsted untl now. Provdng such a foundaton s the topc of ths paper. The dfference between the holstc the attrbute-specfc evaluaton s that n the former, loss averson decson weghtng are attrbute-ndependent, whereas n the latter they depend on the attrbutes. As we show n 5, the holstc the attrbute-specfc evaluaton are n general equvalent only when people behave accordng to expected utlty,.e., when loss averson does not affect people s preferences there s no decson weghtng. An example to further clarfy the dfference between the holstc the attrbute-specfc evaluaton s n 3. Ths paper gves preference foundatons for addtve utlty under prospect theory the attrbutespecfc evaluaton. We restrct our attenton to the addtve decomposton for two reasons: Frst, t s commonly appled n many areas of economcs decson analyss. Second, other decompostons, such as multplcatve multlnear utlty, rase specal problems under the attrbute-specfc evaluaton. Solvng these problems s beyond the scope of ths paper. The remander of ths paper s organzed as follows. Secton gves notaton explans prospect theory for sngle-attrbute outcomes. In 3, we move to multattrbute utlty where we frst assume, for ease of exposton, that there are just two attrbutes, both numercal. Secton 4 gves preference foundatons for prospect theory wth addtve utlty under the attrbute-specfc evaluaton. As mentoned, weghtng functons are defned per attrbute they may dffer across attrbutes n the attrbute-specfc evaluaton. To force them to be equal across attrbutes requres addtonal condtons. We wll characterze ths specal case n 5. We extend our results to the case where there are more than two attrbutes n 6 to the case of nonnumerc outcomes n 7. Secton 8 concludes the paper wth some observatons on the emprcal measurement of addtve utlty n prospect theory under the attrbute-specfc evaluaton. All proofs are n the appendx.. Prospect Theory for Sngle-Attrbute Outcomes We consder a decson maker n a stuaton where there s a fnte number, n of dstnct states of nature, exactly one of whch obtans. S = n denotes the fnte set of states of nature. Subsets of S are called events. In a medcal decson problem, the states of nature can, for example, be mutually exclusve dseases, the decson maker has to choose between dfferent treatments before knowng what the actual dsease s. We consder decson under uncertanty where the probabltes for the states of nature may, but need not, be gven. The assumpton of a fnte number of states of nature s made for expostonal purposes. The results of ths paper can be extended to an nfnte state space usng tools from Wakker (993). The extenson to decson under rsk,.e., the case where probabltes are objectvely gven, s as n Köbberlng Wakker (3, 5.3). The decson-maker s problem s to choose between prospects. Each prospect s an n-tuple of outcomes, one for each state of nature. Formally, a prospect s a functon from the set of states of nature to the set of outcomes C. We denote the set of prospects as P = C n. We shall wrte f f n for the prospect f that gves f j f state j occurs. A constant prospect gves the same outcome for each state of nature. For ease of exposton, we assume n ths secton that outcomes are one-dmensonal. The set of outcomes C s a nondegenerate convex subset of. Outcomes are defned wth respect to a reference pont. The reference

3 Management Scence 55(5), pp , 9 INFORMS 865 pont s a constant prospect, that we wll denote as r. We assume that the reference pont s fxed,.e., we restrct attenton to preferences wth respect to one reference pont. Varatons n the reference pont are analyzed by Schmdt (3). Let denote a preference relaton on the set of prospects. As usual, denotes the asymmetrc part of (strct preference) denotes the symmetrc part of (ndfference), denote reversed preferences. We shall use the same notaton for the bnary relatons on C, derved through constant prospects. An outcome x r s a gan an outcome x r s a loss. A prospect f s rank-ordered f f f n. For each prospect, there exsts a permutaton, such that f f n. For each permutaton, let P = f P f f n. That s, P s the set of all prospects that are rank-ordered by. If two prospects can be rank-ordered by a common, then they are comonotonc. For each event A S, the set P A contans those prospects that yeld gans for states n A, no gans for states not n A. We defne the set P A as the ntersecton of P A P. Subsets of sets P A are sgncomonotonc. A real-valued functon V P represents on P f for all f, g P we have f g f only f (ff) V f V g. A functon V s a rato scale f t s unque up to unt,.e., f V can be replaced by U f only f U = V for postve. Aweghtng functon or capacty W s a functon on S, such that W =, W S =, for any two events A B, f B A then W B W A. W s strctly ncreasng f W B <W A whenever B s a proper subset of A. Prospect theory holds f there exsts a utlty functon U C wth U r = such that prospects f P A wth A = k for some k n are evaluated by wth PT f = k + j U f j + j= n j=k+ j U f j () + j = W + j W + j j =W j n W j + n (a) (b) choces preferences correspond wth ths evaluaton. PT f denotes the prospect theory value, orpt value for short, of f, W + W are weghtng functons for gans losses, respectvely. We wll assume throughout that U s strctly ncreasng,.e., for all x y C, U x U y ff x y, contnuous. If prospect theory holds, then utlty s a rato scale the weghtng functons are unquely determned. 3. Prospect Theory for Two-Attrbute Outcomes From now on C = C C s a product of two nondegenerate convex subsets of. Hence we now deal wth two product structures: the two-dmensonal structure of C the n-dmensonal structure C n. In what follows, the ndex wll refer to the attrbutes, the ndex j to the states of nature. Hence f j denotes the th attrbute of the outcome that s obtaned under state j Outcomes n C wll be denoted as x = x x or as x x for short. Note that although we assumed x x to be numercal, the notaton x x should not be nterpreted as multplcaton. Let P denote the set of prospects on C n P the set of prospects on C n For a fxed f P, we defne a preference relaton on P by f g ff f f g f. In 4 we mpose a condton that mples that the choce of f s mmateral. By restrctng attenton to constant prospects n P, we can defne a preference relaton on C In a smlar fashon we can defne on P on C. A functon U C s addtve f U x U x + U x where U s a real-valued functon on C, =. The functons U U are jont rato scales f U U can be replaced by V V f only f V = U, > That s, any common change n unt s allowed. In the holstc evaluaton, any outcome x that s ndfferent to r can also be nterpreted as a reference pont. Hence, t does not make sense to consder gans or losses on any separate dmenson n the holstc evaluaton. What matters n the holstc evaluaton s whether an outcome x s a gan or a loss compared to r (.e., whether x r or x r, respectvely). Under the holstc evaluaton, addtve decomposablty means that a prospect f P A wth A = k for some k n s evaluated as PT f = k + j U f j + U f j j= + n j=k+ j U f j + U f j (3) where the decson weghts are defned as n Equatons (a) (b). The unqueness results of prospect theory apply, whch mples that the attrbute utlty functons are jont rato scales the weghtng functon s unque. There s only one permutaton functon that apples to both attrbutes. In ths representaton, the decson weght that s assgned to a sngle-attrbute utlty functon U, = depends on whether the entre outcome s a gan or a loss. If an outcome x s a gan then the decson weght + s appled, f t s a loss then s appled. Preference foundatons for Equaton (3) were gven by Zank () Blechrodt Myamoto (3).

4 866 Management Scence 55(5), pp , 9 INFORMS The attrbute-specfc evaluaton assesses for each attrbute separately whether t yelds a gan or a loss the magntude of each. That s, the attrbutespecfc evaluaton nterprets reference-dependence for each attrbute separately. We wll denote the reference pont on the frst attrbute by r the reference pont on the second attrbute by r. x C s a gan f x r, a loss f x r, x C s a gan f x r, a loss f x r. We assume that preferences are monotonc n each attrbute. Then, unlke n the holstc evaluaton, the reference pont wll be unque. We further assume that r s an nteror pont of C that r s an nteror pont of C. Ths ensures that C C both contan outcomes that are gans outcomes that are losses, that genune tradeoffs between gans losses exst for both attrbutes. For each prospect f, there exst permutatons such that f f n f f n. Let P = f P f f n. That s, P s the set of all prospects where outcomes are rank-ordered by. P s defned smlarly. For each event A S, the set P A contans those prospects that yeld gans on the frst attrbute for states n A no gans on the frst attrbute for states not n A. Smlarly, P A contans those prospects that yeld gans on the second attrbute for states n A no gans on the second attrbute for states not n A. We defne P A = P A P P A = P A P. Subsets of P A are sad to be sgn-comonotonc on C subsets of P A are sad to be sgn-comonotonc on C. Under the attrbute-specfc evaluaton, a prospect f P A P A wth A = k A = k for some k k n s evaluated as k PT f = + j U f j + wth j= k + + j U f j + j= n j=k + n j=k + j U f j j U f j (4) + j = W + j W + j = (5a) j = W j n W j + n = (5b) preferences choces correspond wth ths evaluaton. The functons U U are strctly ncreasng contnuous satsfy U r = U r =. The decson weghts + are the decson weghts for gans losses for the frst attrbute, + are the decson weghts for gans losses for the second attrbute, W + W are the weghtng functons for gans losses for the frst attrbute, W + W are the weghtng functons for gans losses for the second attrbute. The utlty functons are jont rato scales the attrbute weghtng functons are unque. A comparson between Equatons (3) (4) reveals that the holstc evaluaton the attrbute-specfc evaluaton dffer both n loss averson n decson weghtng. An example may further clarfy the dfference between the holstc the attrbute-specfc evaluaton of addtve utlty. Suppose that the RA consders a job offer from a unversty n a dfferent town. The uncertanty she faces s whether her husb wll be able to fnd a sutable job n the new town. If he does, ther combned annual ncome wll be $8K but she wll only have 5 hours research tme per week because she wll have to take over some domestc actvtes from her husb (e.g., takng care of the chldren). If he does not fnd a job, ther combned annual ncome wll be $4K but she wll have 3 hours research tme per week because her husb wll take care of all domestc actvtes. In the example, there s only one source of uncertanty (whether or not the RA s husb fnds a sutable job). In real-lfe applcatons, there may be dfferent sources of uncertanty affectng the attrbutes separately. For example, the RA s research tme may not be affected by her husb fndng a job (because she can hre someone to take care of her domestc actvtes) but t s affected by whether or not she wll be able to fnd a sutable home near the unversty (f not, commutng wll negatvely affect the tme avalable for research). To model such stuatons we have to refne the state space (events are husb fnds job home near the unversty, husb fnds job but home far from unversty, etc.). For smplcty of exposton, we only consder one source of uncertanty. The RA s preferences are monotonc both n money (more money s preferred) n research tme (more research tme s preferred). Suppose that currently the RA her husb earn $5K per year she has hours research tme per week. Suppose also that $8K 5 h $5K h $4K 3 h The RA s reference pont s $5K h n the holstc evaluaton. In the attrbute-specfc evaluaton, the RA s reference pont for annual ncome s $5K for research tme t s hours per week. In the holstc evaluaton, where we determne frst the sgn of an outcome then apply the decompostons, we assume that the RA s utlty functon for gans s u x x u r r her utlty functon for losses s u x x u r r, where s a

5 Management Scence 55(5), pp , 9 INFORMS 867 coeffcent reflectng loss averson u s a basc utlty functon, expressng the RA s atttude toward outcomes, whch s reference ndependent (Tversky Kahneman 99, Köbberlng Wakker 5). We assume that the holstc basc utlty s addtve such that u x x = u x + u x. In the attrbute-specfc evaluaton, where frst the decomposton s appled then t s determned whether attrbutes yeld gans or losses, the utlty for gans s u x u r for losses t s u x u r ; =, where the are attrbute-specfc loss averson coeffcents. The RA does not care about job aspects other than wage rate avalable research tme. If event s, her husb fnds a job event s, her husb does not fnd a job, then, accordng to the holstc evaluaton (Equaton (3)), the PT value of the new job s equal to ( u 8 + u 5 u 5 + u ) + + ( u 4 + u 3 u 5 + u ) (6) accordng to the attrbute-specfc evaluaton (Equaton (4)), t s equal to ( + u 8 u 5 ) + ( u 5 u ) + ( u 4 u 5 ) + + ( u 3 u ) (7) A comparson between Equatons (6) (7) shows that both decson weghtng loss averson dffer between the two evaluatons. Loss averson decson weghtng are common for all ndvdual attrbutes n the holstc evaluaton; the attrbute-specfc evaluaton, n general, allows for dfferent degrees of loss averson dfferent weghtng functons for each of the ndvdual attrbutes. The possblty of attrbute-dependent weghtng functons can be realstc n applcatons. Rottenstrech Hsee () showed that decson weghtng depends on the outcome doman wth people devatng more from expected utlty for affect-rch outcomes, outcomes that arouse strong emotons. Examples of such outcomes are health states envronmental effects. For example, Smth Keeney (5) studed trade-offs between consumpton health. In such a settng t mght well be that people weght health rsks dfferently than consumpton rsks. Dyer et al. (998) compared dfferent alternatves for dsposng surplus weapons-grade plutonum. Here decson makers may weght rsks to the envronment dfferently from rsk regardng the costs of the alternatves. In 5, we characterze the specal case of the attrbute-specfc evaluaton where the weghtng functons are the same across dfferent attrbutes. There s no emprcal evdence to conclude that loss averson dffers across dfferent attrbutes, but ntutvely ths seems to make sense. 4. Preference Foundatons Ths secton develops preference foundatons for addtve prospect theory under the attrbute-specfc evaluaton,.e., Equaton (4). We contnue to assume that C = C C wth C C nondegenerate convex subsets of. 4.. General Preference Condtons Ths subsecton presents the stard preference condtons that are used n both the holstc attrbutespecfc approaches. The preference relaton on the set of prospects P s a weak order f t s complete (for all prospects f g f g or g f ) transtve. Any prospect f P yelds both a prospect f P a prospect f P, hence, each prospect f may be vewed as an element of the product P P. Hence, we can denote prospects as f f. Weak separablty holds when for all f g P for all f g P f f g f ff f g g g when for all f g P for all f g P f f f g ff g f g g. Weak separablty entals that the relatons on P on P are well-defned. Further stard propertes are monotoncty for outcomes contnuty: outcome monotoncty holds f for = f j g j for all j mples f g wth strct preference holdng f one of the antecedent nequaltes s strct; contnuty holds f for all prospects f, the sets g P g f g P g f are both closed n C n, = 4.. Trade-off Consstency To defne trade-off consstency we ntroduce some notaton. For x C, f P, = j S defne x j f = ( f f j x f j+ f n ) that s, x j f s the prospect f wth f j replaced by x. Let a b c d C. We wrte ab cd f () there exst f g P, f P a state j such that a j f f b j g f c j f f d j g f where a j f, b j g, c j f, d j g are sgn-comonotonc on C, or () there exst v w C, f P such that a f v f b f w f c f v f d f w f where a f, b f, c f, d f are rank-ordered prospects n P v f, w f are rank-ordered prospects n P. In the frst two ndfferences, the prospect on the second attrbute s kept fxed, n the fnal two

6 868 Management Scence 55(5), pp , 9 INFORMS ndfferences, everythng outsde state of nature s kept fxed. The relatonshp may be nterpreted as measurng strength of preference. For example, f a>b, from the ndfferences a j f f b j g f c j f f d j g f, we can see that ab cd means that, n the presence of f, a trade-off of a for b s an equally good mprovement as a trade-off of c for d: both exactly offset recevng f nstead of g for all other states of nature. A smlar nterpretaton can be assgned to the ndfferences a f v f b f w f c f v f d f w f. The relatons are defned entrely n terms of observed ndfferences no new prmtves beyond observed choce are assumed n ther defnton. Hence, we stay entrely wthn the revealed preference paradgm when usng the relatons. Let w x y z C We defne wx yz f () there exst f g P, f P a state j such that f w j f f x j g f y j f f z j g where w j f, x j g, y j f, z j g are sgn-comonotonc on C or () there exst a b C, f P such that a f w f b f x f a f y f b f z f where w f, x f, y f, z f are rank-ordered prospects n P a f, b f are rank-ordered prospects n P. We say that satsfes trade-off consstency on C f mprovng the frst attrbute of an outcome n any relatonshp breaks that relatonshp. That s, f ab cd a a then t cannot be that a b cd. Loosely speakng, trade-off consstency on C ensures that the relatonshp s well-behaved when nterpreted as a strength of preference relatonshp. If the strength of preference of a over b s equal to the strength of preference of c over d, then the strength of preference of a over b cannot be equal to the strength of preference of c over d, when a s strctly better than a. Smlarly, satsfes trade-off consstency on C f mprovng the second attrbute of an outcome n any relatonshp breaks that relatonshp. That s, f wx yz y y then t cannot be that wx y z. Trade-off consstency holds f trade-off consstency holds both on C on C. An mportant advantage of trade-off consstency as a preference condton s that t s closely related to measurements of utlty by the trade-off method (Wakker Deneffe 996). Ths makes t easy to test trade-off consstency emprcally. Emprcal studes that have used the trade-off method nclude Abdellaou (), Etchart-Vncent (4), Schunk Betsch (6), Abdellaou et al. (7) amongst others. Trade-off consstency s a powerful condton. It has two effects. Frst, t ensures that we can defne prospect theory functonals for both attrbutes second, t ensures that the overall evaluaton s addtve n these two prospect theory functonals Representaton for Two Attrbutes To derve the representng functonal for preferences, we need an addtonal assumpton. Solvablty holds f for any two prospects f g P there exsts outcomes such that ( f f g f f g. Solvablty mples that the attrbute utlty functons U U are unbounded. The next theorem characterzes Equaton (4). Theorem. The followng two statements are equvalent: () s represented by the functonal n Equaton (4) wth strctly ncreasng weghtng functons W +, W, W +, W contnuous, strctly ncreasng utlty functons U U. () satsfes () weak orderng, () contnuty, (3) weak separablty, (4) outcome monotoncty, (5) solvablty, (6) trade-off consstency. The unqueness results of prospect theory apply, that s, W, =, are unquely determned, the utlty functons U U are jont rato scales. the weghtng functons W + 5. Common Weghtng Functons In the attrbute-specfc evaluaton, the weghtng functons may dffer across the two attrbutes. In some cases, however, t mght be reasonable to take the weghtng functons ndependent of the attrbutes. Emprcal evdence suggests, for example, that decson weghts for money for health are close (Abdellaou compared wth Blechrodt Pnto ). Usng common weghtng functons facltates the use of prospect theory n practcal applcatons, because fewer elctatons are requred. In ths secton we wll gve a preference foundaton for the specal case of Equaton (4) where the weghtng functons do not depend on the attrbutes. By contnuty connectedness of C C, there exst gans x C x C losses y C y C, such that x r r x y r r y, hence, such that U x = U x U y = U y. Recall that r s the constant prospect that gves r r n every state of nature. For any event B, let x B f denote the prospect f wth f j replaced by x for all j n B. We can now defne a condton that ensures attrbute ndependence of the weghtng functons for gans for losses.

7 Management Scence 55(5), pp , 9 INFORMS 869 We say that satsfes attrbute-ndependence for states, f for all x C x C for whch x r r x for all events B, x r B r r x B r. Note that the condton holds for all x C x C, but x x must be ether both gans or both losses for otherwse the ndfference x r r x cannot obtan. We wll now explan the dea behnd the condton. As mentoned before, f x r r x then U x = U x. If Equaton (4) holds x x are both gans, the ndfference x r B r r x B r mples that W + B U x = W + B U x W + B = W + B follows from U x = U x. A smlar lne of argument shows that W B = W B whenever x x are losses. Because these equaltes hold for all events B, we obtan the followng result: Corollary. If we add attrbute-ndependence for states to statement () of Theorem, then the weghtng functons n statement () of Theorem are attrbutendependent,.e., W + = W + = W + W = W = W. If = A = A then Corollary also mples that the decson weghts + are attrbutendependent. Ths follows from the defnton of the decson weghts, Equatons (5a) (5b). Havng the weghtng functons ndependent of the attrbutes does not make the attrbute-specfc evaluaton equal to the holstc evaluaton. Ths s easly seen by referrng back to the example of the RA consderng the new job. Under the attrbute-specfc evaluaton wth common weghtng functons, Equaton (7) becomes + u 8 u 5 + u 5 u + u 4 u u 3 u 5 showng that the attrbute-specfc evaluaton clearly dffers from the holstc evaluaton, Equaton (6). Note that t s not only the presence of the loss averson parameter that dstngushes the holstc from the attrbute-specfc evaluaton. In general, the two evaluatons dffer even f a prospect yelds only gans or only losses. Consder agan the job offer example but suppose now that the RA s reference pont for annual earnngs s $3K for research tme t s hours per week. The preference $8K 5 h $4K 3 h stll holds. Let E denote the event husb fnds a job E the event husb does not fnd a job. Under the holstc evaluaton, the job s value s W + E ( u 8 +u 5 u 3 +u ) + ( W + E )( u 4 +u 3 u 3 +u ) under the attrbute-specfc evaluaton t s W + E ( u 8 u 3 ) + ( W + E )( u 5 u ) + ( W + E )( u 4 u 3 ) +W +( E u 3 u ) Equalty only holds f W + E = W + E,.e., f W + E + W + E =. Ths must hold for all events E E, for the attrbute-specfc evaluaton ths can only be the case f W + s a probablty measure. A smlar argument can be used to derve that W must be a probablty measure. Hence, for outcomes of the same sgn attrbute ndependent weghtng, the attrbute-specfc evaluaton agrees wth the holstc evaluaton only when both representatons reduce to subjectve expected utlty. 6. More Than Two Attrbutes We wll now extend our results to more than two attrbutes. Let C = C C m, m>. Each C s a nondegenerate convex subset of. The reference pont on the th attrbute s denoted r s assumed to be an nteror pont of C. We wll denote the set of prospects on C n as P wrte prospects as f f m. Let g f denote the prospect f P wth f replaced by g, let g h k f denote the prospect f P wth f replaced by g f k replaced by h k. Weak separablty s now defned as follows: for all m, f g P, f g P, f f g f ff f g g g. The defntons of outcome monotoncty, contnuty, solvablty easly generalze to the case of more than two attrbutes. For trade-off consstency we defne ab cd f () there exst f g P, f P, a state j such that a j f f b j g f c j f f d j g f where a j f, b j g, c j f, d j g are sgn-comonotonc on C, or () there exst v w C k, f P such that a f v f k k f b f w f k k f c f v f k k f d f w f k k f where a f, b f, c f, d f are rank-ordered prospects n P v f k, w f k are rank-ordered prospects n P k. Trade-off consstency holds f each -relatonshp satsfes trade-off consstency on C. We are now n a poston to extend Theorem to the case of more than two attrbutes. Theorem 3. The followng two statements are equvalent: () s represented by V = m = V f where the V are prospect theory functonals wth strctly ncreasng weghtng functons W + W contnuous, strctly ncreasng utlty functons U.

8 87 Management Scence 55(5), pp , 9 INFORMS () satsfes () weak orderng, () contnuty, (3) weak separablty, (4) outcome monotoncty, (5) solvablty, (6) trade-off consstency. The unqueness results of prospect theory apply, that s, the weghtng functons W + W are unquely determned, the utlty functons U are jont rato scales. Attrbute ndependence can easly be extended to the case of more than two attrbutes, so that the arguments precedng Corollary can stll be used to ensure that the weghtng functons are attrbutendependent. 7. General Outcomes For ease of exposton, we have assumed thus far that all attrbutes are numercal. In many real-world decsons, ths assumpton s too restrctve. An example s health, the area n whch decson analyss s most frequently appled (Keller Klenmuntz 998, Smth von Wnterfeldt 4). Health conssts of two dmensons, survval duraton health qualty, health qualty s a nonnumerc attrbute. The extenson of our analyss to nonnumerc attrbutes s as follows. Assume that the C are connected topologcal spaces. C = C C m s endowed wth the product topology so s C n. The reference ponts r are n the nteror of C for each. Redefne outcome monotoncty as follows: for all, ff j g j for all j then f g. The strct verson of outcome monotoncty s not necessary here as t follows from the verson wth weak preferences trade-off consstency (Köbberlng Wakker 3, Lemma 6). We can now state the extenson of our results to nonnumerc attrbutes. Corollary 4. If the C, = m, are connected topologcal spaces, then Theorems 3 stll hold f we drop n (), the requrement that the attrbute-wse utlty functons are strctly ncreasng. The proof of ths clam follows easly from the proofs of Theorems 3. Corollary can stll be used to ensure that the weghtng functons are attrbute-ndependent. 8. Emprcal Measurement A few comments concernng the emprcal mplementablty of addtve prospect theory under the attrbute-specfc evaluaton are worth mentonng. For emprcal purposes a frst step s obvously the verfcaton of the preference condtons that have been dentfed n ths paper. When these are satsfed, the elctaton of the attrbute-specfc evaluaton s smpler than that of the holstc evaluaton because we do not need to know the rankng of outcomes. Essentally, we can apply the known elctaton technques for sngle-dmensonal prospect theory to each of the attrbutes. When attrbute-ndependence for states holds, the weghtng functons have to be assessed only once. A procedure to measure utlty under prospect theory was recently proposed by Abdellaou et al. (7). Ther method uses varous elctaton technques (probablty equvalence, certanty equvalence, Wakker Deneffe s 996 trade-off method). A smpler procedure was proposed by Abdellaou et al. (8). Ther method only uses certanty equvalence questons but s less general than the procedure of Abdellaou et al. (7) n that t assumes that the utlty functons are power functons. The weghtng functons W + W, m can be measured ether through the nonchoce-based methods of Tversky Fox (995), Fox Tversky (998), Wu Gonzalez (999), or Klka Weber () or through the choce-based method of Abdellaou et al. (5). If probabltes are known then the methods of Abdellaou () or Blechrodt Pnto () can be appled. We can also use the representaton results for sngle-dmensonal prospect theory (Prelec 998, Wakker Tversky 993, Wakker Zank ) to restrct the functonal forms of the utlty functons the weghtng functons. If preferences do not change when we multply all levels of an attrbute by a common constant (whle holdng the other attrbute constant), then the attrbute utlty functon must be a power functon. If preferences are nvarant to addng a constant to all levels of an attrbute such that the sgn of the attrbute levels s preserved ( the other attrbute s held constant), then the attrbute utlty functon s exponental. When probabltes are known preferences satsfy Prelec s (998) compound nvarance condtons, then the weghtng functons must have the form w p = exp ln p, =+. Condtons for dervng exponental or power weghtng functons are presented n Decdue et al. (9). Acknowledgments Tony Marley, Peter P. Wakker, two anonymous revewers gave helpful comments. The frst author s research was made possble by a grant from the Netherls Organzaton for Scentfc Research (NWO). Appendx. Proofs Proof of Theorem. That () mples () s routne. Hence we assume () derve (). By weak order, weak separablty, outcome monotoncty, contnuty, on P can be represented by V V f V f wth V strctly ncreasng n V V. V represents V represents. By contnuty, V V are contnuous, by outcome monotoncty, they are strctly ncreasng. We wll now show that V V are prospect theory functonals. For a prospect f P, defne the prospect f +

9 Management Scence 55(5), pp , 9 INFORMS 87 by f + j = f j f f j r by f + j = r otherwse, the prospect f by fj = f j f f j r by fj = r otherwse. That s, f + s the postve part of f f s ts negatve part. In a smlar fashon, we defne f + f. Consder on P. Because satsfes outcome monotoncty C s nondegenerate, all states of nature are nonnull (a state s null f replacng any outcomes n that state does not affect the preference). Also, because r les n the nteror of C, s truly mxed ( s truly mxed f there exsts a prospect f such that f + r f + r, that s, genune trade-offs between gans losses occur). By Theorem n Köbberlng Wakker (3) there exsts a prospect theory representaton for wth U, the contnuous utlty functon over C, U r =, W +, W, the weghtng functons over gans losses on the frst attrbute, respectvely. Köbberlng Wakker s (3) weak monotoncty follows from outcome monotoncty sgn-comonotonc trade-off consstency follows from trade-off consstency on C. By Proposton 8. n Wakker Tversky (993), gan-loss consstency can be dropped from Köbberlng Wakker s (3) condtons when the number of states of nature exceeds two. Ths s the case n our analyss f we nterpret attrbutes as events (Sarn Wakker 998, Corollary B.3). U s strctly ncreasng because V s strctly ncreasng. W + W are strctly ncreasng by outcome monotoncty. By Observaton 3 n Köbberlng Wakker (3), U s a rato scale W + W are unque. By solvablty, U s unbounded. By a smlar lne of argument, there exsts a prospect theory representaton for, wth U the contnuous strctly ncreasng utlty functon on C, U r =, U a rato scale, W + W the unque strctly ncreasng weghtng functons over gans losses on the second attrbute, respectvely. By solvablty, U s unbounded. So far we have shown that V PT f PT f represents. It remans to show that V s addtve. We wll do so by showng that the rate of trade-offs between PT PT s everywhere constant. Take f P let f be a rank-ordered prospect n P. Take C such that f. Then f s a rank-ordered prospect n P. Let g be such that f j g j for all j wth at least one of these preferences strct. By solvablty there exsts an outcome such that (f f f g.by outcome monotoncty. Next we consder the prospect (f f. By solvablty we can fnd an outcome such that (f f f g Hence,. We proceed n ths manner to construct a stard sequence on the second attrbute for whch s s for all natural s. It s easly verfed that ths mples that PT s f PT s f = PT f PT f. Suppose wthout loss of generalty that PT f PT f =. Next we construct a stard sequence on the frst attrbute by elctng ndfferences t f f t f f, t = such that all prospects nvolved are rank-ordered. These ndfferences mply that t t for all natural t, thus that PT t f PT t f = PT f PT f. The ndfferences also defne a rate of trade-off between PT PT. Let PT f PT f = c. Then the rate of trade-off between PT PT s constant for all the ponts we have elcted thus far. Ths clam follows from trade-off consstency. By trade-off consstency, we must have f s f f s f for any s = Applyng trade-off consstency agan mples that we must have t f s f t f s f for any s = ; t = Hence the rate of trade-off between PT PT s everywhere c. Next we double the densty of the grd that we constructed above. By contnuty of U connectedness of C we can fnd an outcome / such that PT / f PT f = / Let = / construct a new stard sequence / / by elctng ndfferences (f / s f f / s g. It follows from outcome monotoncty that / =, hence, n general / s = s, s = We construct a new stard sequence / / on the frst attrbute by settng / = elctng ndfferences / t f / f / t f / f, t = We have to show that the rate of trade-off between PT PT n ths new grd / / / / s stll constant. For ths we have to show that / t = t,. Then / j = j for all j = follows from the constructon of the stard sequence. By the constructon of the stard sequence, / f / f / f / f. By trade-off t = We wll show that / = consstency / f / f / f / f = f f f f. By transtvty because = /, / f f f f. By outcome monotoncty / = Hence, the rate of trade-off between PT PT s stll constant when we double the densty of the grd. We contnue ths doublng of densty nfntely, creatng ncreasngly fne stard sequences m m m m m= On the resultng ncreasngly fne grds the rate of trade-off between PT PT remans constant by a smlar proof as for the case where the densty of the grd was doubled. Because U U are unbounded, for any natural m there can be no x C no x C such that x m t for all t or x m s for all s. There can also be no outcomes nfntely close to n the sense that there s always an outcome from the grd that les between an outcome x C, an outcome from the grd that les between an outcome x C when x s unequal to x s unequal to.ifx then U x U = c>, hence, there exsts a natural number m such that m <c. By constructon, there s an element m of the grd such that m x. A smlar argument shows that the grd nterferes everywhere. Let x y be two outcomes such that x y. Suppose that U x U y = d>. Then there exsts a natural number m such that m <d by constructon there s an element m s of the grd such that y m s x. Fnally, because U U are unbounded there cannot be elements x x that are so bad that they are never ncluded n any grd. Consder an outcome x. Then we can construct a prospect f wth f j = x for all j. Because U s unbounded, we can construct a prospect g such that f j g j for all j. Let = x construct a new grd by elctng ndfferences (f f

10 87 Management Scence 55(5), pp , 9 INFORMS f g f f f g ) etc. Ths produces a dense grd that ncludes x. By contnuty we can extend the dense grd to all outcomes. Hence, we have shown that on the whole doman the rate of trade-off between PT PT s constant for rank-ordered prospects. Hence, for rank-ordered prospects V PT f PT f s addtve: V f = PT f + PT f Because U U are contnuous unbounded C C are connected, we can for any prospects f f fnd rank-ordered prospects g g such that f g f g. We set V PT f PT f = PT g + PT g Fnally, we show that PT f + PT f represents. Suppose that f g There are rank-ordered prospects f g such that f f g g By transtvty, f g Hence, PT f + PT f = PT f + PT f PT g + PT g = PT g +PT g whch completes the proof of statement (). The unqueness results follow from the unqueness results for PT PT combned wth the fact that on each grd the rate of trade-off between PT PT must be constant. Ths completes the proof of Theorem. Proof of Theorem 3. That () mples () s routne. 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