Weighing Risk and Uncertainty

Transcription

1 Pychological Review 1995, Vol. 12, No. 2, Copyright 1995 by the American Pychological Aociation, Inc. OOM-295X/95/S3. Weighing Rik an Uncertainty Amo Tverky an Craig R. Fox Stanfor Univerity Deciion theory itinguihe between riky propect, where the probabilitie aociate with the poible outcome are aume to be known, an uncertain propect, where thee probabilitie are not aume to be known. Stuie of choice between riky propect have uggete a nonlinear tranformation of the probability cale that overweight low probabilitie an unerweight moerate an high probabilitie. The preent article exten thi notion from rik to uncertainty by invoking the principle of boune ubaitivity: An event ha greater impact when it turn impoibility into poibility, or poibility into certainty, than when it merely make a poibility more or le likely. A erie of tuie provie upport for thi principle in eciion uner both rik an uncertainty an how that people are le enitive to uncertainty than to rik. Finally, the article icue the relationhip between probability jugment an eciion weight an itinguihe relative enitivity from ambiguity averion. Deciion are generally mae without efinite knowlege of their conequence. The eciion to invet in the tock market, to unergo a meical operation, or to go to court are generally mae without knowing in avance whether the market will go up, the operation will be ucceful, or the court will ecie in one' favor. Deciion uner uncertainty, therefore, call for an evaluation of two attribute: the eirability of poible outcome an their likelihoo of occurrence. Inee, much of the tuy of eciion making i concerne with the aement of thee value an the manner in which they are or houl be combine. In the claical theory of eciion uner rik, the utility of each outcome i weighte by it probability of occurrence. Conier a imple propect of the form (x, p) that offer a probability p to win $jc an a probability 1 p to win nothing. The expecte utility of thi propect i given by pu(x) + (1 p)w(o), where u i the utility function for money. Expecte utility theory ha been evelope to explain attitue towar rik, namely, rik averion an rik eeking. Rik averion i enne a a preference for a ure outcome over a propect with an equal or greater expecte value. Thu, chooing a ure $1 over an even chance to win $2 or nothing i an expreion of rik averion. Rik eeking i exhibite if a propect i preferre to a ure outcome with equal or greater expecte value. It i commonly aume that people are rik avere, which i explaine in expecte utility theory by a concave utility function. The experimental tuy of eciion uner rik ha hown that people often violate both the expecte utility moel an the Amo Tverky an Craig R. Fox, Department of Pychology, Stanfor Univerity. Thi reearch wa upporte by National Science Founation Grant SES an SBR The article benefitte from icuion with Daniel Kahneman an Peter Wakker. Correponence concerning thi article houl be aree to Amo Tverky, Department of Pychology, Stanfor Univerity, Stanfor, California principle of rik averion that unerlie much economic analyi. Table 1 illutrate a common pattern of rik eeking an rik averion oberve in choice between imple propect (aapte from Tverky & Kahneman, 1992), where C(x, p) i the meian certainty equivalent of the propect (x, p). Thu, the upper left-han entry in the table how that the meian participant i inifferent between receiving $14 for ure an a 5% chance of receiving $1. Becaue the expecte value of thi propect i only $5, thi obervation reflect rik eeking. Table 1 illutrate a fourfol pattern of rik attitue: rik eeking for gain an rik averion for loe of low probability, couple with rik averion for gain an rik eeking for loe of high probability. Choice conitent with thi pattern have been oberve in everal tuie, with an without monetary incentive' (Cohen, Jaffray, & Sai, 1987; Fihburn & Kochenberger, 1979; Herhey & Schoemaker, 198; Kahneman & Tverky, 1979; Payne, Laughhunn, & Crum, 1981; Wehrung, 1989). Rik eeking for low-probability gain may contribute to the popularity of gambling, wherea rik eeking for highprobability loe i conitent with the tenency to unertake rik in orer to avoi a ure lo. Becaue the fourfol pattern i oberve for a wie range of payoff, it cannot be explaine by the hape of the utility function a propoe earlier by Frieman an Savage (1948) an by Markowitz (1952). Intea, it ugget a nonlinear tranformation of the probability cale, firt propoe by Preton an Baratta (1948) an further icue by Ewar (1962) an other. Thi notion i one of the cornertone of propect theory (Kahneman & Tverky, 1979; Tverky & Kahneman, 1992), which provie the theoretical framework ue in the preent article. Accoring to thi theory, the value of a imple propect that offer a probability p to win $x (an probability 1 p 1 Rik eeking for long hot wa reporte by Kachelmeier an Shehata (1992) in an experiment conucte in China with real payoff that were conierably higher than the normal monthly income of the participant. 269

2 27 AMOS TVERSKY AND CRAIG R. FOX to win nothing) i given by w(p)v(x), where v meaure the ubjective value of the outcome x, an w meaure the impact of p on the eirability of the propect. The value of w are calle eciion weight; they are normalize o that w() =, an w( 1) = 1. It i important to note that w houl not be interprete a a meaure of egree of belief. A eciion maker may believe that the probability of hea on a to of a coin i onehalf but give thi event a lower weight in the evaluation of a propect. Accoring to propect theory, the value function v an the weighting function w exhibit iminihing enitivity: marginal impact iminihe with itance from a reference point. For monetary outcome, the tatu quo generally erve a the reference point that itinguihe gain from loe. Thu, iminihing enitivity give rie to an S-hape value function, with v() =, that i concave for gain an convex for loe. For probability, there are two natural reference point certainty an impoibility that correpon to the enpoint of the cale. Therefore, iminihing enitivity implie that increaing the probability of winning a prize by. 1 ha more impact when it change the probability of winning from.9 to 1. or from to.1 than when it change the probability from, ay,.3 to.4 or from.6 to.7. Thi give rie to a weighting function that i concave near zero an convex near one. Figure 1 epict the weighting function for gain an for loe, etimate from the meian ata of Tverky an Kahneman (1992). 2 Such a function overweight mall probabilitie an unerweight moerate an high probabilitie, which explain the fourfol pattern of rik attitue illutrate in Table 1. It alo account for the wellknown certainty effect icovere by Allai (1953). For example, wherea mot people prefer a ure $3 to an 8% chance of winning $45, mot people alo prefer a 2% chance of winning $45 to a 25% chance of winning $3, contrary to the ubtitution axiom of expecte utility theory (Tverky & Kahneman, 1986). Thi obervation i conitent with an S-hape weighting function atifying w(.2)/w(.25) a w(.8)/w( 1.). Such a function appear to provie a unifie account of a wie range of empirical fining (ee Camerer & Ho, 1994). A choice moel that i bae on a nonlinear tranformation of the probability cale aume that the eciion maker know the probabilitie aociate with the poible outcome. With the notable exception of game of chance, however, thee probabilitie are unknown, or at leat not pecifie in avance. People generally o not know the probabilitie aociate with event uch a the guilt of a efenant, the outcome of a football game, or the future price of oil. Following Knight (1921), eci- Table 1 The Fourfol Pattern of Rik Attitue Probability Note. Low High Gain C($1,.5) = $14 (rik eeking) G($ 1,.95) = $78 (rik averion) Lo C(-$1,.5) = -$8 (rik averion) C(-$1,.95) = -$84 (rik eeking) C i the meian certainty equivalent of the propect in quetion. ion theorit itinguih between riky (or chance) propect where the probabilitie aociate with outcome are aume to be known, an uncertain propect where thee probabilitie are not aume to be known. To ecribe iniviual choice between uncertain propect, we nee to generalize the weighting function from rik to uncertainty. When the probabilitie are unknown, however, we cannot ecribe eciion weight a a imple tranformation of the probability cale. Thu, we cannot plot the weighting function a we i in Figure 1, nor can we peak about the overweighting of low probabilitie an unerweighting of high probabilitie. Thi article exten the preceing analyi from rik to uncertainty. To accomplih thi, we firt generalize the weighting function an introuce the principle of boune ubaitivity. We next ecribe a erie of tuie that emontrate thi principle for both rik an uncertainty, an we how that it i more pronounce for uncertainty than for rik. Finally, we icu the relationhip between eciion weight an juge probabilitie, an the role of ambiguity in choice uner uncertainty. An axiomatic treatment of thee concept i preente in Tverky an Wakker( in pre). Theory Let S be a et whoe element are interprete a tate of the worl. Subet of S are calle event. Thu, S correpon to the certain event, an <f> i the null event. A weighting function W (on S) i a mapping that aign to each event in S a number between an 1 uch that W(<t>) =, W( S) = 1, an W( A) > W( B) if A D B. Such a function i alo calle a capacity, or a nonaitive probability. A in the cae of rik, we focu on imple propect of the form (x,.a), which offer $x if an uncertain event A occur an nothing if A oe not occur. Accoring to propect theory, the value of uch a propect i W(A)v(x), where W 7 i the eciion weight aociate with the uncertain event A. (We ue Wfor uncertainty an w for rik.) Becaue the preent treatment i confine to imple propect with a ingle poitive outcome, it i conitent with both the original an the cumulative verion of propect theory (Tverky & Kahneman, 1992). It i conitent with expecte utility theory if an only if Wi aitive, that i, W(A\JB) = W(A) + W(B) whenever^ O5 =. 3 Propect theory aume that W atifie two conition. (i) Lower ubaitivity: W(A)^W(A\J B)- W(B), provie A an B are ijoint an W( A U B) i boune away from one. 4 Thi inequality capture the poibility effect: The impact of an event A i greater when it i ae to the null event than when it i ae to ome nonnull event B. (ii) Upper ubaitivity: W(S) - W(S- A) > W(A U B) W( B), provie A an B are ijoint an W( B) i boune 2 Figure 1 correct a minor error in the original rawing. 3 For other icuion of eciion weight for uncertain event, ee Hogarth an Einhorn (199), Vicui (1989), an Wakker (1994).. 4 The bounary conition are neee to enure that we alway compare an interval that inclue an enpoint to an interval that i boune away from the other enpoint (ee Tverky & Wakker, in pre, for a more rigorou formulation).

3 WEIGHING RISK AND UNCERTAINTY cg 'tn 8 Q State Probability (p) Figure 1. Weighting function for gain (w+) an loe (w-). away from zero. 5 Thi inequality capture the certainty effect: The impact of an event A i greater when it i ubtracte from the certain event S than when it i ubtracte from ome uncertain event A\J B. A weighting function W atifie boune ubaitivity, or ubaitivity (SA) for hort, if it atifie both (i) an (ii) above. Accoring to uch a weighting function, an event ha greater impact when it turn impoibility into poibility or poibility into certainty than when it merely make a poibility more or le likely. To illutrate, conier the poible outcome of a football game. Let H enote the event that the home team win the game, V enote the event that the viiting team win, an T enote a tie. Hence, S = H U V U T. Lower SA implie that W(T) excee W(H\JT)- W(H), wherea upper SA implie that W(H\JV\JT)-W(HW) excee W( H U T) - W( H). Thu, aing the event T(a tie) to <t> ha more impact than aing Tlo H, an ubtracting Tfrom 5ha more impact than ubtracting T from HUT. Thee conition exten to uncertainty the principle that increaing the probability of winning a prize from to p ha more impact than increaing the probability of winning from q to q + p, an ecreaing the probability of winning from 1 to 1 p ha more irrlpact than ecreaing the probability of winning from q + p to q. To invetigate thee propertie empirically, conier four imple propect, each of which offer a fixe prize if a particular event (H, T,H(J V,orH U T) occur an nothing if it oe not. By aking people to price thee propect, we can etimate the eciion weight aociate with the repective event an tet both lower an upper SA, provie the value function i cale inepenently. Several comment concerning thi analyi are in orer. Firt, rik can be viewe a a pecial cae of uncertainty where probability i efine through a tanar chance evice o that the probabilitie of outcome are known. Uner thi interpretation, the S-hape weighting function of Figure 1 atifie both lower an upper SA. Secon, we have efine thee propertie in term of the weighting function Wihat i not irectly obervable but can be erive from preference (ee Wakker & Tverky, 1993). Neceary an ufficient conition for boune SA in term of the oberve preference orer are preente by Tverky an Wakker (in pre) in the context of cumulative propect theory. Thir, the concept of boune SA i more general than the property of iminihing enitivity, which give rie to a weighting function that i concave for relatively unlikely event an convex for relatively likely event. Finally, there i evience to ugget that the eciion weight for complementary event typically um to le than one, that i, W( A) + W( S - A) < 1 or equivalently, W(A) <; W(S)- W(S- A). Thi property, 5 The upper ubaitivity of W'K equivalent to the lower ubaitivity of the ual function W(A)= l-w(s-a).

4 272 AMOS TVERSKY AND CRAIG R. FOX Table 2 A Demontration ofsubaitivity in Betting on the Outcome of a Stanfor-Berkeley Football Game Problem Option fi Si f2 g2 fj g3 $25 $1 $25 $1 Event $1 $1 $1 $1 D $1 $25 $25 $1 Preference Note. A = Stanfor win by 7 or more point; B = Stanfor win by le than 7 point; C = Berkeley tie or win by le than 7 point; D = Berkeley win by 7 or more point. Preference = percentage of reponent (N = 112) that choe each option. calle ubcertainty (Kahneman & Tverky, 1979), can alo be interprete a evience that upper SA ha more impact than lower SA; in other wor, the certainty effect i more pronounce than the poibility effect. Some ata conitent with thi property are preente below. An Illutration We next preent an illutration of SA that yiel a new violation of expecte utility theory. We ake 1 12 Stanfor tuent to chooe between propect enne by the outcome of an upcoming football game between Stanfor an the Univerity of California at Berkeley. Each participant wa preente with three pair of propect, iplaye in Table 2. The percentage of reponent who choe each propect appear on the right. Half of the participant receive the problem in the orer preente in the table; the other half receive the problem in the oppoite orer. Becaue we foun no ignificant orer effect, the ata were poole. Participant were promie that 1% of all reponent, electe at ranom, woul be pai accoring to one of their choice. Table 2 how that, overall, fi wa choen over g!, f 2 over g 2, an g 3 over f 3. Furthermore, the triple (f,, f 2, g 3 ) wa the ingle mot common pattern, electe by 36% of the reponent. Thi pattern violate expecte utility theory, which implie that a peron who chooe f] over gi an f 2 over g 2 houl alo chooe f 3 over g 3. However, 64% of the 55 participant who choe fi an f 2 in Problem 1 an 2 choe g 3 in Problem 3, contrary to expecte utility theory. Thi pattern, however, i conitent with the preent account. To emontrate, we apply propect theory to the moal choice in Table 2. The choice off] overgi in Problem 1 implie that Similarly, the choice of f 2 over g 2 in Problem 2 implie that v(25)w(d)>v(lo)w(aub). Aing the two inequalitie an rearranging term yiel W(A)+ W(D) u(lo) W(A\JB) + W(CUD) > v(25) ' On the other han, the choice of g 3 over f 3 in Problem 3 implie that v(\)w(a\jb\jcl)d)>v(25)w(a(jd), D(1) u(25) Conequently, the moal choice imply W(AUD) W(AUBUCUD) 5 f (A)+ W(D) W(AUD) W(AUB) + W(C\JD) > W(AUBUCUD)' It can be hown that thi inequality i conitent with a ubaitive weighting function. Moreover, the inequality follow from uch a weighting function, provie that ubcertainty hol. To emontrate, note that accoring to lower SA, W{ A) + W( D) S: W(A U D). Furthermore, it follow from ubcertainty that (j W(AUB) + W(CVD)< W(AUBUCUD)^ I. Thu, the left-han ratio excee the right-han ratio, in accor with the moal choice. Note that uner expecte utility theory Wi an aitive probability meaure, hence the left-han ratio an the right-han ratio mut be equal. Relative Senitivity A note earlier, propect theory aume SA for both rik an uncertainty. We next propoe that thi effect i tronger for uncertainty than for rik. In other wor, both lower an upper SA are amplifie when outcome probabilitie are not pecifie. To tet thi hypothei, we nee a metho for comparing ifferent omain or ource of uncertainty (e.g., the outcome of a football game or the pin of a roulette wheel ). Conier two ource, A an B, an uppoe that the eciion weight for both ource atify boune ubaitivity. We ay that the eciion maker i le enitive to B than to A if the following two conition hol for all ijoint event A\, A 2 in A, an BI, B 2 in B, provie all value of Ware boune away from an 1. 7 lfw(b { )= W(A t )an W(S-B 2 ) = W(S-A 2 ), W(S-[A 1 UA 2 }). (2) The firt conition ay that the union of ijoint event from B "loe" more than the union of matche event from A. The econ conition impoe the analogou requirement on the ual function. Thu, a peron i le enitive 6 to B than to A if B prouce more lower SA an more upper SA than oe A. Thi efinition can be reaily tate in term of preference. 6 Relative enitivity i cloely relate to the concept of relative curvature for ubjective imenion introuce by Krantz an Tverky (1975).

5 WEIGHING RISK AND UNCERTAINTY 273 Table 3 Outline of Stuie Participant Source NBA fan (N=21) Stuy 1 Stuy 2 Stuy 3 Chance NBA playoff San Francico temperature NFL fan (# = 4) Chance Super Bowl Dow-Jone Note. NBA = National Baketball Aociation; NFL = National Football League. Pychology tuent (N=45) Chance San Francico temperature Beijing temperature To illutrate, conier a comparion between uncertainty an chance. 7 Suppoe B\ an B 2 are ijoint uncertain event (e.g., the home team win or the home team tie a particular football game). Let A \ an A 2 enote ijoint chance event (e.g., a roulette wheel laning re or laning green). The hypothei that people are le enitive to the uncertain ource B than to the chance ource A implie the following preference conition. If one i inifferent between receiving $5 if the home team win the game or if a roulette wheel lan re (p = 18/38), an if one i alo inifferent between receiving $5 if the home team tie the game or if a roulette wheel lan green (i.e., zero or ouble zero, p = 2/38), then one houl prefer receiving $5 if a roulette wheel lan either green or re (p = 2/38) to receiving $5 if the home team either win or tie the game. The following tuie tet the two hypothee icue above. Firt, eciion maker exhibit boune ubaitivity uner both rik an uncertainty. Secon, eciion maker are generally le enitive to uncertainty than to rik. Experimental Tet We conucte three tuie uing a common experimental paraigm. On each trial, participant choe between an uncertain (or riky) propect an variou cah amount. Thee ata were ue to etimate the certainty equivalent of each propect (i.e., the ure amount that the participant conier a attractive a the propect) an to erive eciion weight. The baic feature of the tuie are outline in Table 3. Metho Participant. The participant in the firt tuy were 27 male Stanfor tuent (meian age = 21) who repone to avertiement calling for baketball fan to take part in a tuy of eciion making. Participant receive $15 for participating in two 1-hour eion, pace a few ay apart. The participant in the econ tuy were 4 male football fan (meian age = 21), recruite in a imilar manner. They were promie that in aition to receiving $ 15 for their participation in two 1-hour eion, ome of them woul be electe at ranom to play one of their choice for real money. The participant in the thir tuy were 45 Stanfor tuent enrolle in an introuctory pychology coure (28 men, 17 women, meian age = 2) who took part in a 1-hour eion for coure creit. The repone of a few aitional participant (one from Stuy 1, four from Stuy 2, an three from Stuy 3) were exclue from the analyi becaue they exhibite a great eal of internal inconitency. We alo exclue a very mall number of repone that were completely out of line with an iniviual' other repone. Proceure. The experiment wa run uing a computer. Each trial involve a erie of choice between a propect that offere a prize contingent on chance or an uncertain event (e.g., a 25% chance to win a prize of $15) an a ecening erie of ure payment (e.g., receive $4 for ure). In Stuy 1, the prize wa alway $75 for half the reponent an $15 for the other half; in Stuie 2 an 3, the prize for all reponent wa $15. Certainty equivalent were inferre from two roun of uch choice. The firt roun conite of ix choice between the propect an ure payment, pace roughly evenly between $ an the prize amount. After completing the firt roun of choice, a new et of even ure payment wa preente, panning the narrower range between the lowet payment that the reponent ha accepte an highet payment that the reponent ha rejecte. The program enforce internal conitency. For example, no reponent wa allowe to prefer $3 for ure over a propect an alo prefer the ame propect over a ure $4. The program allowe reponent to backtrack if they felt they ha mae a mitake in the previou roun of choice. The certainty equivalent of each propect wa etermine by a linear interpolation between the lowet value accepte an the highet value rejecte in the econ roun of choice. Thi interpolation yiele a margin of error of ±$2.5 for the $15 propect an ±$1.25 for the $75 propect. We wih to emphaize that although our analyi i bae on certainty equivalent, the ata conite of a erie of choice between a given propect an ure outcome. Thu, reponent were not ake to generate certainty equivalent; intea, thee value were inferre from choice. Each eion began with etaile intruction an practice. In Stuy 1, the firt eion conite of chance propect followe by baketball propect; the econ eion replicate the chance propect followe by propect efine by a future temperature in San Francico. In Stuy 2, the firt eion conite of chance propect followe by Super Bowl propect; the econ eion replicate the chance propect followe by propect efine by a future value of the Dow-Jone inex. Stuy 3 conite of a ingle eion in which the chance propect were followe by propect efine by a future temperature in San Francico an Beijing; the orer of the latter two ource wa counterbalance. The orer of the propect within each ource wa ranomize. Source of uncertainty. Chance propect were ecribe in term of a ranom raw of a ingle poker chip from an urn containing 1 chip numbere conecutively from 1 to 1. Nineteen propect of the form (x, p) were contructe where p varie from.5 to.95 in multiple of.5. For example, a typical chance propect woul pay $15 if the number of the poker chip i between 1 an 25, an nothing otherwie. Thi eign yiel 9 tet of lower SA an 9 tet of upper SA for each participant. 7 Although probabilitie coul be generate by variou chance evice, we o not itinguih between them here, an treat rik or chance a a ingle ource of uncertainty.

6 274 AMOS TVERSKY AND CRAIG R. FOX Utah Win Portlan Win Figure 2. Event pace for propect enne by the reult of the Utah-Portlan baketball game. The horizontal axi refer to the point prea in that game. Each row enote a target event that efine a propect ue in Stuy 1. Segment that exten up to the arrowhea repreent unboune interval. Each interval inclue the more extreme enpoint relative to, but not the le extreme enpoint. Baketball propect were enne by the reult of the firt game of the 1991 National Baketball Aociation (NBA) quarter final erie between the Portlan Trailblazer an the Utah Jazz. For example, a typical propect woul pay $ 15 if Portlan beat Utah by more than 6 point. The event pace i epicte in Figure 2. Each of the 32 row in the figure repreent a target event A that efine an uncertain propect (x, A). For example, the top row in Figure 2, which conit of two egment, repreent the event "the margin of victory excee 6 point." Thi eign yiel 28 tet of lower SA an 12 tet of upper SA. For example, one tet of lower SA i obtaine by comparing the eciion weight for the event "Utah win" to the um of the eciion weight for the two event "Utah win by up to 12 point" an "Utah win by more than 12 point." Super Bowl propect were efine by the reult of the 1992 Super Bowl game between the Buffalo Bill an the Wahington Rekin. The event pace i epicte in Figure 3. It inclue 28 target event yieling 3 tet of lower SA an 17 tet of upper SA. Dow-Jone propect were efine by the change in the Dow-Jone Inutrial Average over the ubequent week. For example, a typical propect woul pay $15 if the Dow-Jone goe up by more than 5 point over the next even ay. The event pace ha the ame tructure a that of the Super Bowl (Figure 3). San Francico temperature propect were efine by the aytime high temperature in San Francico on a given future ate. The 2 target event ue in Stuie 1 an 3 are epicte in Figure 4. Thi eign yiel 3 tet of lower SA an 1 tet of upper SA. For example, a typical propect woul pay $75 if the aytime high temperature in owntown San Francico on April 1, 1992, i between 65 an 8. Similarly, Beijing temperature propect were efine by the aytime high temperature in Beijing on a given future ay. The event pace i ientical to the San Francico temperature in Stuy 3, a epicte in Figure 4. Reult To tet lower an upper SA, the eciion weight for each reponent were erive a follow. Uing the choice ata, we firt etimate the certainty equivalent C of each propect by linear interpolation, a ecribe earlier. Accoring to propect theory, if C(x,,4)=j;, then i>(}>) = W(A)v(x)anW(A) = v(y)/ v(x). The eciion weight aociate with an uncertain event A, therefore, can be compute if the value function v for gain i known. Previou tuie (e.g., Tverky, 1967) have inicate that the value function for gain can be approximate by a power function of the form v(x) x", a < 1. Thi form i characterize by the aumption that multiplying the prize of a propect by a poitive contant multiplie it certainty equivalent by the ame contant. 8 Thi preiction wa tete uing the ata from Stuy 1 in which each event wa paire both with a prize of $75 an with a prize of $15. Conitent with a power value function, we foun no ignificant ifference between C( 15, A) an 2C( 75, A) for any of the ource. Although the preent ata are conitent with a power function, the value of the exponent cannot be etimate from imple propect becaue the exponent a can be aborbe into W. To etimate the exponent for gain, we nee propect with two poitive outcome. Such propect were invetigate by Tverky an Kahneman (1992), uing the ame experimental proceure an a imilar ubject population. They foun that etimate of the exponent i not vary markely acro reponent an the meian etimate of the exponent wa.88. In the analyi that follow, we firt aume a power value function with an exponent of.88 an tet lower an upper SA uing thi function. We then how that the tet of SA i robut with repect to ubtantial variation in the exponent. Further analye are 8 Thi follow from the fact that for t >, the value of the propect (tx, A) i W(A) (tx) ; hence, C(tx, A) = JV(A)"" (x, which equal tc(x,a).'

7 WEIGHING RISK AND UNCERTAINTY 275 Buffalo win Wahington win Dow-Jone own Dow-Jone up Figure 3. Event pace for propect enne by the reult of the Super Bowl game between Wahington an Buffalo (an for the Dow-Jone propect). The horizontal axi refer to the point prea in the Super Bowl (an the change in the Dow-Jone in the next week). Each row enote a target event that efine a propect ue in Stuy 2. Segment that exten up to the arrowhea repreent unboune interval. Each interval inclue the more extreme enpoint relative to, but not the le extreme enpoint. bae on an orinal metho that make no aumption about the functional form oft). Uing the etimate Wfor each ource of uncertainty, we efine meaure of the egree of lower an upper SA a follow. Recall that lower SA require that W(A) > W(A U B) - W(B),forAr\B = <t>. Hence, the ifference between the two ie of the inequality, )^ W(A)+ W(B)- W(AUB), provie a meaure of the egree of lower SA. Similarly, recall that upper SA require that 1 - W(S - A) ;> W(A U B) - W( B), for A n B = <t>. Hence, the ifference between the two ie of the inequality, D'(A,B)= 1 - W(S-A)- W(A\JB)+ W(B), provie a meaure of the egree of upper SA. Table 4 preent the overall proportion of tet, acro participant, that trictly atify lower an upper SA (i.e., D>,D'> ) for each ource of uncertainty. Note that if ffwere aitive (a implie by expecte utility theory), then both D an D' are Stuy 1 6' 65' 7' 75' 8' Stuy 3 4 < 5' 6' Figure 4. Event pace for propect efine by future temperature in San Francico an Beijing. The horizontal axi refer to the aytime high temperature on a given ate. Each row enote a target event that efine a propect ue in Stuie 1 an 3. Segment that exten up to the arrowhea repreent unboune interval. Each interval inclue the left enpoint but not the right enpoint. 7' 8'

8 276 AMOS TVERSKY AND CRAIG R. FOX Table 4 Proportion of Tet That Strictly Satify Lower an Upper Subaitivity (SA) Source Chance Baketball Super Bowl Dow-Jone S.F. temp. Beijing temp. Stuy 1 Stuy 2 Stuy 3 Lower Upper Lower Upper Lower Upper SA SA SA SA SA SA Note. S.F. = San Francico; temp. = temperature expecte to be zero; hence, all entrie in Table 4 houl be cloe to one-half. However, each entry in Table 4 i ignificantly greater than one-half (p <.1, by a binomial tet), a implie bysa. To obtain global meaure of lower an upper SA, let an ', repectively, be the mean value of D an D' for a given reponent. Beie erving a ummary tatitic, thee inexe have a imple geometric interpretation if the riky weighting function i roughly linear except near the enpoint. It i eay to verify that within the linear portion of the graph, D an D' o not epen on A an B, an the ummary meaure an ' correpon to the "lower" an "upper" intercept of the weighting function (ee Figure 5). It lope, j = 1 ', can then be interprete a a meaure of enitivity to probability change. State Probability (p) Figure 5. A weighting function that i linear except near the enpoint ( = "lower" intercept of the weighting function; ' = "upper" intercept of the weighting function; = lope). For uncertainty, we cannot plot an ' a in Figure 5. However, an ' have an analogou interpretation a a "poibility gap" an "certainty gap," repectively, if W i roughly linear except near the enpoint. 9 Note that uner expecte utility theory, = ' = an 5=1, wherea propect theory implie ^,'>, an 5 = 1. Thu, propect theory implie le enitivity to change in uncertainty than i require by expecte utility theory. To tet thee preiction, we compute the value of, ', an, eparately for each reponent. Table 5 preent the meian value of thee inexe, acro reponent for each ource of uncertainty. In accor with SA, each value of an ' in Table 5 i ignificantly greater than zero (p <.5). Furthermore, both inexe are larger for uncertainty than for chance: The mean value of an of ' for the uncertain ource are ignificantly greater than each of the correponing inexe for chance (p <.1, eparately for each tuy). Finally, conitent with ubcertainty, ' ten to excee, though thi ifference i tatitically ignificant only in Stuie 2 an 3 (p <.5). Recall that all participant evaluate the ame et of riky propect, an that reponent in each of the three tuie evaluate two ifferent type of uncertain propect (ee Table 3). Figure 6 plot, for each reponent, the average enitivity meaure for the two uncertain ource againt for the riky ource. (One reponent who prouce a negative wa exclue from thi analyi.) Thee ata may be ummarize a follow. Firt, all value of for the uncertain propect an all but two value of for the riky propect were le than or equal to one a implie by SA. Secon, the value of are conierably higher for rik (mean =.74) than for uncertainty (mean 5 =.53), a emontrate by the fact that 94 out of 111 point lie below the ientity line (p <.1 by a ign tet). Thir, the ata reveal a ignificant correlation between the enitivity meaure for rik an for uncertainty (r =.37, p <.1). The average correlation between the uncertain ource i.4. If we retrict the analyi to Stuie 1 an 2 that yiele more table ata (in part becaue the riky propect were replicate), the correlation between enitivity for rik an for uncertainty increae to.51, an the mean correlation between the uncertain ource increae to.54. Thee correlation inicate the preence of conitent iniviual ifference in SA an ugget that enitivity to uncertainty i an important attribute that itinguihe among eciion maker. An axiomatic analyi of the conition uner which one iniviual i conitently more SA than another i preente in Tverky an Wakker (in pre). Robutne. The preceing analyi ummarize in Table 5 aume a power value function with an exponent a =.88. To invetigate whether the above concluion epen on the particular choice of the exponent, we reanalyze the ata uing ifferent value of a varying from one-half to one. To appreciate the impact of thi ifference, conier the propect that offer a one-thir chance to win $1. [We chooe one-thir becaue, accoring to Figure 1, w( 1 / 3) i approximately one-thir.] The certainty equivalent of thi propect i $33.33 if a = 1, but it i 9 More formally, thi hol when W(A U B) - W(A) oe not epen on A, for all A n B = <t>, provie W(A) i not too cloe to zero an W( A U B) i not too cloe to 1.

9 WEIGHING RISK AND UNCERTAINTY r 1..8 '«.6 I (rik) Figure 6. Joint itribution for all reponent of the enitivity meaure for rik an uncertainty. only $11.11 if a =.5. Table 6 how that a a ecreae (inicating greater curvature), increae an ' ecreae. More important, however, both an ' are poitive throughout the range for all ource, an the value of are ignificantly maller than one (p <.1) in all cae. SA, therefore, hol for a fairly wie range of variation in the curvature of the value function. Orinal analyi. The preceing analyi confirme our hypothei that people are le enitive to uncertainty than to chance uing the enitivity meaure. We next turn to an ori- Table 5 Meian Value of, ', an, Acro Reponent, Meauring the Degree of Lower an Upper Subaitivity (SA) an Global Senitivity, Repectively Source Chance Baketball Super Bowl Dow-Jone S.F. temp. Beijing temp Stuy : 1 ' Stuy 2 ' Note. S.F. = San Francico; temp. = temperature Stuy 3 ' nal tet of thi hypothei that make no aumption about the value function. Let B l, B^ enote ijoint uncertain event, an let A!, A 2 enote ijoint chance event. We earche among the repone of each participant for pattern atifying C(x, BI) ;> C(x, A t )an C(x, B 2 ) ^ C(x, A 2 ) or butc(x,b i \JB 2 )<C(x,A l UA 2 ), (3) anc(x,s-b 2 )^C(x,S-A 2 ) butc(x,s-[b t \JB 2 ])>C(x,S-[Ai\JA 2 ]). (4) A repone pattern that atifie either conition 3 or 4 provie upport for the hypothei that the reponent i le enitive to uncertainty (B) than to chance (A). Several comment regaring thi tet are in orer. Firt, note that if we replace the weak inequalitie in conition 3 an 4 with equalitie, then thee conition reuce to the efinition of relative enitivity (ee Equation 1 an 2). The above conition are better uite for the preent experimental eign becaue participant were not ake to "match" interval from ifferent ource. Secon, the preent analyi i confine to contiguou interval; conition 3 an 4 may not hol when comparing contiguou to noncontiguou interval (ee Tverky

10 278 AMOS TVERSKY AND CRAIG R. FOX & Koehler, 1994). Thir, becaue of meaurement error, the above conition are not expecte to hol for all comparion; however, the conition inicating le enitivity to uncertainty than to chance are expecte to be atifie more frequently than the oppoite conition. Let M(E, A) be the number of repone pattern that atify conition 3 above (i.e., le enitivity to uncertainty than to chance). Let M( A, B) be the number of repone pattern that atify 3 with the A an B interchange (i.e., le enitivity to chance than to uncertainty). The ratio m(b, A) = M(B, A)/ (Af(B,A) + M(A,B)) provie a meaure of the egree to which a reponent i le enitive to uncertainty than to chance, in the ene of conition 3. We efine M'( B, A), M' (A, B), an m'(b, A) imilarly for preference pattern that atify conition 4. If the reponent i invariably le enitive to B than to A, then the ratio m(b, A) an m'(b, A) houl be cloe to one. On the other han, if the reponent i not more enitive to one ource than to another, thee ratio houl be cloe to one-half. Table 7 preent the meian ratio, acro reponent, comparing each of the five uncertain ource to chance. A preicte, all entrie in the table are ignificantly greater than one-half (p <.5, by t tet), inicating that people are generally le enitive to uncertainty than to chance. We conclue thi ection with a brief methoological icuion. We have attribute the fining of boune ubaitivity an lower enitivity for uncertainty than for rik to baic pychological attitue towar rik an uncertainty capture by the weighting function. Alternatively, one might be tempte to account for thee fining by a tatitical moel that aume that the aement of certainty equivalent, an hence the etimation of eciion weight, i ubject to ranom error that i boune by the enpoint of the outcome cale, becaue C(x, A) mut lie between an x. Although boune error coul contribute to SA, thi moel cannot aequately account for the oberve pattern of reult. Firt, it cannot explain the ubaitivity oberve in imple choice experiment that o not involve (irect or inirect) aement of certainty equivalent, uch a the Stanfor-Berkeley problem preente in Table 2. More extenive evience for both lower an upper SA in imple choice between riky propect i reporte by Wu an Gonzalez (1994), who alo foun ome upport for the tronger hypothei that w i concave for low probabilitie an convex for moerate an high probabilitie. Secon, a tatitical moel cannot reaily account for the reult of the orinal analyi reporte above that reponent were le enitive to uncertainty than to chance. Thir, becaue a ranom error moel implie a bia towar one-half, it cannot explain the obervation that the eciion weight of an event that i a likely a not to occur i generally le than one-half (ee Figure 7, 8, an 9 below). Finally, it houl be note that ubaitivity an ifferential enitivity play an important role in the pricing of riky an uncertain propect, regarle of whether thee phenomena are riven primarily by pychological or by tatitical factor. Dicuion The final ection of thi article aree three topic. Firt, we explore the relationhip between eciion weight an Table 6 Meian Value of, ', an Acro Reponent, Meauring the Degree of Lower Subaitivity, Upper Subaitivity, an Global Senitivity, Repectively, for Several Value of a Between.5 an I Source an inex Chance (Stuy 1) ' Chance (Stuy 2) ' Chance (Stuy 3) ' Baketball 1 Super Bowl ' Dow-Jone ' SF temp (Stuy 1) ' SF temp (Stuy 3) ' Beijing temp ' ! a Note. SF = San Francico; temp = temperature juge probabilitie. Secon, we invetigate the preence of preference for betting on particular ource of uncertainty. Finally, we icu ecriptive an normative implication of the preent reult. Preference an Belief The preent account itinguihe between eciion weight erive from preference an egree of belief expree by probability jugment. What i the relation between the juge probability, P(A), of an uncertain event, A, an it aociate eciion weight W(A)'? To invetigate thi problem, we ake reponent, after they complete the choice tak, to ae the probabilitie of all target event. Following the analyi of eciion weight, we efine meaure of the egree of lower an upper SA in probability jugment a follow:

11 WEIGHING RISK AND UNCERTAINTY 279 Table8 Meian Value of, ', an, Acro Reponent, That Meaure the Degree of Lower an Upper Subaitivity, SA, an Global Senitivity, Repectively, for Juge Probability Source Baketball Super Bowl Dow-Jone S.F. temp. Beijing temp Stuy 1 ' Stuy 2 ' Stuy 3 Note. SA = Subaitivity; S.F. = San Francico; temp. = temperature; 5 = egree of global enitivity. ' D(A, B) = P(A ) + P(B)- P(A U B), D'(A,B)= 1 -P(S-A) + P(B)-P(AUB), provie A fl B = (j>. Clearly, P i aitive if an only if D = D' = for all ijoint A, B in 5. A before, let an ' be the mean value of D an D', repectively, an let i = 1 '. Table 8, which i the analog of Table 5, preent the meian value of, ', an, acro reponent, for each of the five uncertain ource. All value of an ' in Table 8 are ignificantly greater than zero (p <.5), emontrating both lower an upper SA for probability jugment. Comparing Table 8 an Table 5 reveal that the value of for juge probabilitie (overall mean.7) are greater than the correponing uncertain eciion weight (overall mean.55). Thu, probability jugment exhibit le SA than o uncertain eciion weight. Thi fining i conitent with a two-tage proce in which the eciion maker firt aee the probability P of an uncertain event A, then tranform thi value by the riky weighting function w. Thu, W( A) may be approximate by w[p(a)]. We illutrate thi moel uing the meian riky an uncertain eciion weight erive from Stuy 2 (auming a =.88). In Figure 7 we plot eciion weight for chance propect a a function of tate (objective) probabilitie. In Figure 8 an 9, repectively, we plot eciion weight for Super Bowl propect an for Dow-Jone propect a function of (meian) juge Table 7 Orinal Analyi of Differential Senitivity Source comparion Baketball v. chance Super Bowl v. chance Dow- Jone v. chance S.F. temp. v. chance Beijing temp. v. chance Stuy 1 m m' Stuy 2 Stuy 3 m m' m rri Note. Each entry correpon to the meian value, acro reponent, of m an m 1 meauring the egree to which reponent are le enitive to uncertainty than to chance. S.F. = San Francico; temp. = temperature. probabilitie. The comparion of thee figure reveal that the ata in Figure 8 an 9 are le orerly than thoe in Figure 7. Thi i not urpriing becaue juge probability (unlike tate probability) i meaure with error, an becaue the uncertain eciion weight exhibit greater variability (both within an between ubject) than riky eciion weight. However, the unerlying relation between probability an eciion weight i nearly ientical in the three figure. 1 Thi i exactly what we woul expect if the uncertain weighting function H^i obtaine by applying the riky weighting function w to juge probabilitie. The Subaitivity of probability jugment reporte in Table 8 i conitent with upport theory" (Tverky & Koehler, 1994), accoring to which P(A) + P(B)^P(A\J B). The combination of the two-tage moel (which i bae on propect theory) with an analyi of probability jugment (which i bae on upport theory) can therefore explain our main fining that eciion weight are more ubaitive for uncertainty than for chance. Thi moel alo implie that the eciion weight aociate with an uncertain event (e.g., an airplane accient) increae when it ecription i unpacke into it contituent (e.g., an airplane accient caue by mechanical failure, terrorim, human error, or act of Go; ee Johnon, Herhey, Mezaro, & Kunreuther, 1993). Furthermore, thi moel preict greater Subaitivity, ceteri paribu, when A U B i a contiguou interval (e.g., future temperature between 6 an 8 ) than when A U B i not a contiguou interval (e.g., future temperature le than 6 or more than 8 ). A more etaile treatment of thi moel will be preente elewhere. Source Preference The fining that people are le enitive to uncertainty than to rik houl be itinguihe from the obervation of ambigu- 1 The mooth curve in Figure 5 an 6 were obtaine by fitting the parametric form w(p) = &p~ l /(&p' 1 + [1 - p] T ), ue by Lattimore, Baker, an Witte (1992). It aume that the relation between w an p i linear in a log o metric. The etimate value of the parameter in Figure 7, 8, an 9, repectively, are.69,.69, an.72 for y, an.77,.76, an.76 for 8. " In thi theory, P(A) + P(S-A) = 1; hence, the equation for lower an upper SA coincie.

12 28 AMOS TVERSKY AND CRAIG R. FOX CD.6 - O '«Q State Probability (p) Figure 7. Meian eciion weight for chance propect, from Stuy 2, plotte a a function of tate (objective) probabilitie. ity averion: People often prefer to bet on known rather than unknown probabilitie (Ellberg, 1961). For example, people generally prefer to bet on either ie of a fair coin than on either ie of a coin with an unknown bia. Thee preference violate expecte utility theory becaue they imply that the um of the ubjective probabilitie of hea an of tail i higher for the unbiae coin than for the coin with the unknown bia. Recent reearch ha ocumente ome ignificant exception to ambiguity averion. Heath an Tverky (1991) howe that people who were knowlegeable about port but not about politic preferre to bet on port event rather than on chance event that thee people ha juge equally probable. However, the ame people preferre to bet on chance event rather than on political event that they ha juge equally probable. Likewie, people who were knowlegeable about politic but not about port exhibite the revere pattern. Thee ata upport what Heath an Tverky call the competence hypothei: People prefer to bet on their belief in ituation where they feel competent or knowlegeable, an they prefer to bet on chance when they feel incompetent or ignorant. Thi account i conitent with the preference to bet on the fair rather than the biae coin, but it preict aitional preference that are at o with ambiguity averion. The preceing tuie allow u to tet the competence hypothei againt ambiguity averion. Recall that the participant in Stuie 1 an 2 were recruite for their knowlege of baketball an football, repectively. Ambiguity averion implie a preference for chance over uncertainty becaue the probabilitie aociate with the port event (e.g., Utah beating Portlan) are necearily vague or imprecie. In contrat, the competence hypothei preict that the port fan will prefer to bet on the game than on chance. To etablih ource preference, let A an B be two ifferent ource of uncertainty. A eciion maker i ai to prefer ource A to ource B if for any event A in A an B in B. W( A) = W( B) implie W( S - A) > W( S - B), or equivalently, C(x, A) = C(x, B) implie C(x, S-A)> C(x, S-B),x>. To tet for ource preference we earche among the repone of each participant for pattern that atify C(x, A) C(x, B) an C(x, S - A)> C(x, S - B). Thu, a eciion maker who prefer to bet on event A than to bet on event B, an alo prefer to bet againt A than to bet againt B exhibit a preference for ource A over ource B. The preference to bet on either ie of a fair coin rather than on either ie of a coin with an unknown bia illutrate uch a preference for chance over uncertainty. Let K( A, B) be the number of repone pattern inicating a preference for ource A over ource B, a efine above, an let K(B, A) be the number of repone pattern inicating the oppoite preference. For each pair of ource, we compute the ratio k(\, B) = K(\, B)/(tf(A, B) + K(B, A)), eparately for each reponent. Thi ratio provie a comparative inex of ource preference; it houl equal one-half if neither ource i

13 WEIGHING RISK AND UNCERTAINTY r.6 g 'to "o Juge Probability (P) Figure 8. Meian eciion weight for Super Bowl propect, from Stuy 2, plotte a a function of meian juge probabilitie. preferre to the other, an it houl be ubtantially greater than one-half if ource A i generally preferre to ource B. The preent ata reveal ignificant ource preference that are conitent with the competence hypothei but not with ambiguity averion. In all three tuie, participant preferre to bet on their uncertain belief in their area of competence rather than on known chance event. The baketball fan in Stuy 1 preferre betting on baketball than on chance (meian k =.76, p <.5 by t tet); the football fan in Stuy 2 preferre betting on the Super Bowl than on chance (meian k =.59, though thi effect i not tatitically ignificant); an the tuent in Stuy 3 (who live near San Francico) preferre betting on San Francico temperature than on chance (meian k =.76, p <.1). Two other comparion conitent with the competence hypothei are the preference for baketball over San Francico temperature in Stuy 1 (meian k =.76, p <.5), an the preference for San Francico temperature over Beijing temperature in Stuy 3 (meian k =.86, p <.1). For further icuion of ambiguity averion an ource preference, ee Camerer an Weber (1992), Fox an Tverky (in pre), an Frich an Baron (1988). Concluing Comment Several author (e.g., Ellberg, 1961; Fellner, 1961; Keyne, 1921; Knight, 1921), critical of expecte utility theory, itinguihe among uncertain propect accoring to the egree to which the uncertainty can be quantifie. At one extreme, uncertainty i characterize by a known probability itribution; thi i the omain of eciion uner rik. At the other extreme, eciion maker are unable to quantify their uncertainty; thi i the omain of eciion uner ignorance. Mot eciion uner uncertainty lie omewhere between thee two extreme: People typically o not know the exact probabilitie aociate with the relevant outcome, but they have ome vague notion about their likelihoo. The role of vaguene or ambiguity in eciion uner uncertainty ha been the ubject of much experimental an theoretical reearch. In the preent article we have invetigate thi iue uing the conceptual framework of propect theory. Accoring to thi theory, uncertainty i repreente by a weighting function that atifie boune ubaitivity. Thu, an event ha more impact when it turn impoibility into poibility, or poibility into certainty, than when it merely make a poibility more likely. Thi principle explain Allai' example (i.e., the certainty effect) a well a the fourfol pattern of rik attitue illutrate in Table 1. The experiment reporte in thi article emontrate SA for both rik an uncertainty. They alo how that thi effect i more pronounce for uncertainty than for rik. The latter fining ugget the more general hypothei that SA, an hence the eparture from expecte utility theory, i amplifie by

14 282 AMOS TVERSKY AND CRAIG R. FOX 1..8 CO '55 o " Juge Probability (P) Figure 9. Meian eciion weight for Dow-Jone propect, from Stuy 2, plotte a a function of meian juge probability. vaguene or ambiguity. Conequently, tuie of eciion uner rik are likely to uneretimate the egree of SA that characterize eciion involving real-worl uncertainty. 12 Subaitivity, therefore, emerge a a unifying principle of choice that i manifete to varying egree in eciion uner rik, uncertainty, an ignorance. The pychological bai of boune ubaitivity inclue both jugmental an preferential element. A note earlier, SA hol for jugment of probability (ee Table 8), but it i more pronounce for eciion weight (ee Table 5). Thi amplification may reflect people' affective repone to poitive an negative outcome. Imagine owning a lottery ticket that offer ome hope of winning a great fortune. Receiving a econ ticket to the ame lottery, we ugget, will increae one' hope of becoming rich but will not quite ouble it. The ame pattern appear to hol for negative outcome. Imagine waiting for the reult of a biopy. Receiving a preliminary inication that reuce the probability of malignancy by one-half, we ugget, will reuce fear by le than one-half. Thu, hope an fear eem to be ubaitive in outcome probability. To the extent that the experience of hope an fear i treate a a conequence of an action, ubaitivity may have ome normative bai. If lottery ticket are purchae primarily for entertaining a fantay, an protective action i unertaken largely to achieve peace of min, then it i not unreaonable to value the firt lottery ticket more than the econ, an to value the elimination of a hazar more than a comparable reuction in it likelihoo. 12 Evience for ubtantial SA in the eciion of profeional option traer i reporte by Fox, Roger, an Tverky (1995). Reference Allai, A. M. (1953). Le comportement e I'homme rationel evant le rique, critique e potulate et axiome e 1'ecole americaine. Econometrica, 21, Camerer, C. F., & Ho, T.-H. (1994). Violation of the betweenne axiom an nonlinearity in probability. Journal of Rik an Uncertainty, 8, Camerer, C. R, & Weber, M. (1992). Recent evelopment in moeling preference: Uncertainty an ambiguity. Journal of Rik an Uncertainty, 5, Cohen, M., Jaffray, J. Y., & Sai, T. (1987). Experimental comparion of iniviual behavior uner rik an uner uncertainty for gain an for loe. Organizational Behavior an Human Deciion Procee, 39, Ewar, W. (1962). Subjective probabilitie inferre from eciion. Pychological Review, 69, Ellberg, D. (1961). Rik, ambiguity, an the Savage axiom. Quarterly Journal of Economic, 75, Fellner, W. (1961). Ditortion of ubjective probabilitie a a reaction to uncertainty. Quarterly Journal of Economic, 75,

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