psychological review vol. 76, no. 1, intransitivity of

Psychological Review1969, Vol. 76, No. 1, 31-48

INTRANSITIVITY OF PREFERENCES

AMOS TVERSKY'

Hebrew University of Jerusalem, Jerusalem, Israel

It is shown that, under specified experimental conditions, consistent and pre-dictable intransitivities can be demonstrated. The conditions under whichintransitivities occur and their relationships to the structure of the alternativesand to processing strategies are investigated within the framework of a generaltheory of choice. Implications to the study of preference and the psychologyof choice are discussed.

Whenever we choose which car to buy,which job to take, or which bet to play weexhibit preference among alternatives.The alternatives are usually multidimen-sional in that they vary along several at-tributes or dimensions that are relevant tothe choice. In searching for the laws thatgovern such preferences, several decisionprinciples have been proposed and investi-gated. The simplest and probably themost basic principle of choice is the trans-itivity condition.

A preference-or-indifference relation,denoted >, is transitive if for all x, y, and 2

x > y and y > z imply x > z. [1]

Transitivity is of central importance to bothpsychology and economics. It is thecornerstone of normative and descriptivedecision theories (Edwards, 1954, 1961 ;Luce & Suppes, 1965; Samuelson, 1953),and it underlies measurement models ofsensation and value (Luce & Galanter,1963; Suppes & Zinnes, 1963). The essen-tial role of the transitivity assumption inmeasurement theories stems from the factthat it is a necessary condition for the ex-istence of an ordinal (utility) scale, «, suchthat for all x and y,

Transitivity is also a sufficient condition forthe existence of such a scale, provided thenumber of alternatives is finite, or countable.

Individuals, however, are not perfectlyconsistent in their choices. When facedwith repeated choices between x and y,people often choose * in some instances andy in others. Furthermore, such inconsist-encies are observed even in the absence ofsystematic changes in the decision maker'staste which might be due to learning orsequential effects. It seems, therefore,that the observed inconsistencies reflectinherent variability or momentary fluctu-ation in the evaluative process. This con-sideration suggests that preference shouldbe defined in a probabilistic fashion. Todo so, let P(x,y) be the probability ofchoosing x in a choice between x and y,where P(x,y) + P(y,x) = 1. Preferencecan now be defined by

x > y if and only if P(x,y) > -|. [3]

The inconsistency of the choices is thusincorporated into the preference relation asx is said to be preferred to y only when it ischosen over y more than half the time. Re-stating the transitivity axiom in terms ofthis definition yields

u(x) > u(y) if and only if x > y. [2] P(x,y) >1 This work was supported in part by United

States Public Health Service Grant MH-04236 andby National Science Foundation Grant GM-6782 tothe University of Michigan and in part by CarnegieCorporation of New York B-3233 to Harvard Uni-versity, Center for Cognitive Studies. The authorwishes to thank H. William Morrison for his helpfulcomments and David H. Krantz for his valuableassistance, including the suggestion of the likelihoodratio test.

and P(y,z) > Jimply P(x,z) > J. [4]

This condition, called weak stochastictransitivity, or WST, is the most generalprobabilistic version of transitivity. Vio-lations of this property cannot be attribut-able to inconsistency alone.

Despite the almost universal acceptanceof the transitivity axiom, in either algebraic

31

32 AMOS TVERSKY

or probabilistic form, one can think of sev-eral choice situations where it may beviolated. Consider, for example, a situa-tion in which three alternatives, x, y, and z,vary along two dimensions, I and II, andwhere their values on these dimensions aregiven by the following payoff matrix.

DimensionsI II

A Iternativesxyz

2e3e4e

6e4e2e

The alternatives may be job applicantsvarying in intelligence (I) and experience(II), where the entries are the candidates'scores on the corresponding scales or dimen-sions. Suppose the subject (S) uses the fol-lowing decision rule in choosing betweeneach pair of alternatives: if the differencebetween the alternatives on Dimension I is(strictly) greater than e, choose the alter-na^ive that has the higher value on Dimen-sion I. If the difference between thealternatives on Dimension I is less than orequal to e, choose the alternative that hasthe higher value on Dimension II. It iseasy to see that this seemingly reasonabledecision rule yields intransitive preferenceswhen applied to the above matrix. Sincethe differences between x and y and be-tween y and 2 on the first dimension are notgreater than t, the choice is made on thebasis of the second dimension and hence xis chosen over y and y is chosen over z. Butsince the difference between x and z on thefirst dimension is greater than e, z is chosenover x yielding an intransitive chain ofpreferences.

Formally, such a structure may be char-acterized as a lexicographic semiorder,abbreviated LS, where a semiorder (Luce,1956) or a just noticeable difference struc-ture is imposed on a lexicographic ordering.As an illustration, let us restate this rule interms of the selection of applicants. Anemployer, regarding intelligence as far moreimportant than experience, may choose thebrighter of any pair of candidates. Cogni-zant that intelligence scores are not per-fectly reliable, the employer may decide toregard one candidate as brighter than an-

other one only if the difference betweentheir IQ scores exceeds 3 points, for ex-ample. If the difference between theapplicants is less than 3 points, the em-ployer considers the applicants equallybright and chooses the more experiencedcandidate. Essentially the same examplewas discussed by Davidson, McKinsey, andSuppes (1955). Such a decision rule isparticularly appealing whenever the rele-vant dimension is noisy as a consequenceof imperfect discrimination or unreliabilityof available information. Where this de-cision rule is actually employed by indi-duals, WST must be rejected.

Other theoretical considerations proposedby Savage (1951), May (1954), Quandt(1956), and Morrison (1962) suggest thatWST may be violated under certain con-ditions. No conclusive violations of WST,however, have been demonstrated in studiesof preferences although Morrison (1962)provided some evidence for predictablein transitivities in judgments of relativenumerosity, and Shepard (1964) produced astriking circularity in judgments of relativepitch. Several preference experimentshave tested WST, for example, Edwards(1953), May (1954), Papandreou, Sauer-lander, Bownlee, Hurwicz, and Franklin(1957), Davis (1958), Davidson and Mar-schak (1959), Chipman (1960), and Gris-wold and Luce (1962). All these studiesfailed to detect any significant violation ofWST.

The present paper attempts to explorethe conditions under which transitivityholds or fails to hold. First, the LS de-scribed above is utilized to construct alter-natives which yield stochastically intransi-tive data. The conditions under whichWST is violated are studied within theframework of a general additive differencechoice model and their implications for thepsychology of choice are discussed.

EXPERIMENTS

General Considerations

The purpose of the following studies wasto create experimental situations in whichindividuals would reveal consistent pat-

INTRANSITIVITY OF PREFERENCES 33

terns of intransitive choices. The experi-ments are not addressed to the question ofwhether human preferences are, in general,transitive; but rather to the question ofwhether reliable intransitivities can beproduced, and under what conditions. Theconstruction of the alternatives was basedon the LS described in the introduction.The application of the LS to a specific ex-perimental situation, however, raises seriousidentification problems.

In the first place, the LS may be satisfiedby some, but not all, individuals. Onemust identify, therefore, the 5s that satisfythe model. This, however, is not an easytask since even if the LS is satisfied by allpeople, they may differ in the manner inwhich the alternatives are perceived or pro-cessed. Different individuals can char-acterize the same alternatives in terms ofdifferent sets of attributes. For example,one employer may evaluate job applicantsin terms of their intelligence and experiencewhereas another employer may evaluatethem in terms of their competence andsociability. Similarly, one S may con-ceptualize (two-outcome) gambles in termsof odds and stakes, while another mayview them in terms of their expectation,variance, and skewness. Since the predic-tions of the model are based on the dimen-sional structure of the alternatives, thisstructure has to be specified separately foreach 5. In order to induce 5s to use thesame dimensional framework, alternativesthat are defined and displayed in terms of agiven dimensional representation have beenemployed.

Then, even if all individuals satisfy theLS relative to the same dimensions, theymay still vary in their preference thresholdas well as in the relative importance thatthey attribute to the dimensions. A differ-ence between an IQ of 123 and an IQ of 127,for instance, may appear significant to somepeople and negligible to others.

These considerations suggest treatingeach 5 as a separate experiment and con-structing the alternatives according to thedimensions and the spacing he uses. Al-ternatively, one may select, for a criticaltest, those 5s who satisfy a specified cri-

terion relative to a given representation.(It should be noted that the preselection of5s or alternatives, on the basis of an inde-pendent criterion, is irrelevant to the ques-tion of whether WST is consistently vio-lated for any given 5.) Both methods areemployed in the following studies. The firststudy investigates choice between gambleswhile the second one is concerned with theselection of college applicants.

Experiment I

The present study investigates prefer-ences between simple gambles. All gam-bles were of the form (x,p,o) where onereceives a payoff of §x if a chance event poccurs, and nothing if p does not occur.The chance events were generated by spin-ning a spinner on a disc divided into a blackand a white sector. The probability ofwinning corresponded to the relative sizeof the black sector. The gambles employedin the study are described in Table 1.

Each gamble was displayed on a cardshowing the payoff and a disc with the cor-responding black and white sectors. Anillustration of a gamble card is given inFigure 1. Note that, unlike the outcomes,the probabilities were not displayed in anumerical form. Consequently, no exactcalculation of expected values was possible.The gambles were constructed so that theexpected value increased with probabilityand decreased with payoff.

Since the present design renders theevaluation of payoff differences easier thanthat of probability differences, it was hy-pothesized that at least some 5s wouldignore small probability differences, andchoose between adjacent gambles on the

TABLE 1THE GAMBLES EMPLOYED IN

EXPERIMENT I

Gamble

abcde

Probability ofwinning

7/248/249/24

10/2411/24

Payoff (in $)

S.OO4.754.504.254.00

Expected value(in*)

1.461.581.691.771.83

34 AMOS TVERSKY

$4.50

FIG. 1. An illustration of a gamble card.

basis of the payoffs. (Gambles are calledadjacent if they are a step apart along theprobability or the value scale.) Since ex-pected value, however, is negatively cor-related with payoff, it was further hypothe-sized that for gambles lying far apart in thechain, 5s would choose according to ex-pected value, or the probability of winning.Such a pattern of preference must violatetransitivity somewhere along the chain(from a to e).

In order to identify 5s who might exhibitthis preference pattern, 18 Harvard under-graduates were invited to a preliminarysession. The 5s were run individually.On each trial the experimenter presented5 with a pair of gamble cards and askedhim which of the gambles he would ratherplay. No indifference judgment was al-lowed. The 5s were told that a single trialwould be selected at the end of the sessionand that they would be able to play thegamble they had chosen on that trial.They were also told that the outcome of thisgamble would be their only payoff.

To minimize the memory of earlierchoices in order to allow independent re-plications within one session a set of five"irrelevant" gambles was constructed.These gambles were of the same generalform (x,p,o) but they differed from thecritical gambles in probabilities and payoffs.

In the preliminary session, all 5s werepresented with all pairs of adjacent gam-bles (a,b; b,c; c,d; d,e) as well as with thesingle pair of extreme gambles (a,e). In

addition, all 10 pair comparisons of the"irrelevant" gambles were presented.Each of the 15 pairs was replicated 3 times.The order of presentation was randomizedwithin each of the three blocks.

The following criterion was used toidentify the potentially intransitive 5s.On the majority of the adjacent pairs (i.e.,three out of the four) 5 had to prefer thealternative with the higher payoff, while onthe extreme pair, he had to prefer the onewith the higher expected value (i.e., choosee over a). A gamble was said to be pre-ferred over another one if it was chosen onat least two out of the three replications ofthat pair. Eight out of the 18 5s satisfiedthe above criterion and were invited toparticipate in the main experiment.

The experiment consisted of five testsessions, one session every week. In eachsession, all 10 pair comparisons of the testgambles along with all 10 pair comparisonsof the "irrelevant" gambles were presented.Each of the 20 pairs was replicated fourtimes in each session. The position of thegambles (right-left) and the order of the pairswere randomized within each block of 20pairs. The 5s were run individually underthe same procedure as in the preliminarysession. Each of the test sessions lastedapproximately f of an hour. The choiceprobabilities of all eight 5s between the fivegambles are shown in Table 2. Violationsof WST are marked by superscript x andviolations of the LS are marked by super-script y.

The data indicate that although two 5s(7 and 8) seemed to satisfy WST, it wasviolated by the rest of the 5s. Further-more, all violations were in the expecteddirection, and almost all of them were in thepredicted locations. That is, people chosebetween adjacent gambles according to thepayoff and between the more extreme gam-bles according to probability, or expectedvalue. This result is extremely unlikelyunder the hypothesis that the intransitivi-ties are due to random choices. Had thisbeen the case, one should have expected theviolations to be uniformly distributed withan equal number of violations in each of thetwo directions.


TABLE 2

PROPORTION OF TIMES THAT THE Row GAMBLE WASCHOSEN OVER THE COLUMN GAMBLE BY

EACH OF THE EIGHT SUBJECTS

Subject

1

2

3

4

5

6

7

8

Gamble

abcde

abcde

abcde

abcde

abcde

abcde

abcde

abcde

Gamblea b c

— .75 .70— .85

—

— .40' .65— .70

. — .

— .75 .70— .80

—

— .50 .45— .65*

—

— .75 .65— .80

—

— 1.00 .90— .80

—

— .45* .65— .60

—

— .60 .70— .65

—

d

.45'

.65

.80—

.50

.40%y

.75—

.60

.65

.95—

.20

.35

.70"—

.35"

.55

.65—

.65

.75

.90—

.60

.40**

.50—

.75

.75

.60—

e

.15"

.40*

.60

.85—

.25*

.35"

.55

.75• —

.25*

.40*

.801.00

—

.05

.10

.40

.85*—

.60^

.30"

.65

.70• —

.20*

.55

.65

.75—

.60*

.65

.75

.70—

.85*

.85

.80

.40^~

To further test the statistical significanceof the results, likelihood ratio tests of bothWST and the LS were conducted for each 5.This test compares a restrictive model (orhypothesis) denoted MI (such as WST orthe LS) where the parameter space is con-strained, with a nonrestrictive model, de-noted Ma, which is based on an uncon-strained parameter space. The test sta-tistic is the ratio of the maximum value ofthe likelihood function of the sample underthe restrictive model, denoted L*(Mi), tothe maximum value of the likelihood func-tion of the sample under the nonrestrictivemodel, denoted L*(M0). For a large sam-ple size, the quantity

= - 2 Ini*(Jlfo)

» Violations of WST.» Violations of the LS.

has a chi-square distribution with a numberof degrees of freedom that equals the num-ber of constrained parameters. Using thisdistribution, one can test the null hypothe-sis that the data were generated by the re-strictive model. For further details, seeMood (1950).

In the present study, L*(M0) is simplythe product of the binomial probabilities,while L*(Mi) is obtained from it by sub-stituting a value of one-half in the aboveproduct for those choice probabilities thatwere incompatible with the particular re-strictive model. The tested version of theLS was that in the (four) pairs of adjacentgambles, preferences are according to pay-off while in the most extreme pair of gam-bles, preferences are according to expectedvalue. The obtained chi-square valueswith the associated degrees of freedom andsignificance levels are displayed in Table 3.

The table shows that WST is rejected atthe .05 level for five Ss, while the LS is re-jected for one S only. It is important tonote that the test for rejecting WST is veryconservative in that it depends only on themagnitude of the violations and ignorestheir (predicted) location and direction.

The last column of Table 3 reports theQ values corresponding to the ratio of themaximum likelihoods of WST and the LS.Since both models are of the restrictivetype and the two chi-squares are not in-

36 AMOS TVERSKY

TABLE 3LIKELIHOOD RATIO TEST FOR ALL SUBJECTS UNDER WST AND THE LS

Subject

12345678

Q(WST,Mo)

11.827.846.02

15.94S.187.36.40.00

df

33232110

P<

.01

.05

.05

.01

.10

.01

.75—

Q(LS,M0)

.00

.00

.00

.00

.40

.001.80

11.62

df

00001032

P<

_

—

—.75—.50.01

Q(WST,LS)

11.827.846.02

15.944.787.36

-1.20-11.62

Note.— Q(Mi,M») - 2 In

dependent the distribution of this statisticis not known. Nevertheless, its values aresubstantially positive for six out of theeight 5s, suggesting that the LS accountsfor the data better than WST.

In a postexperimental interview, 54

described his behavior as follows: "Thereis a small difference between Gambles a andb or b and c etc., so I would pick the onewith the higher payoff. However, there isa big difference between Gambles a and eor b and e etc., so I would pick the one withthe higher probability." This is, in fact, agood description of his actual choices.When asked whether this type of behaviormight lead to in transitivities, he replied,"I do not think so, but I am not sure."The 5s did not remember for sure whetherany of the pairs were replicated during theexperiment, although they were sure thatmost gambles appeared in more than onepair in any one of the sessions. When thetransitivity assumption was explained tothe 5s, they reacted by saying that althoughthey did not pay special attention to it,they were almost certain that their prefer-ences were transitive.

The degree of intransitivity obtained inan experiment depends critically on thespacing of the alternatives and the selectionof the display. To study the effects ofchanges in the payoff or the probabilitystructure, three sets of gambles portrayedin Table 4 were compared.

Note that Set I is the one used in themain experiment. Set II was obtainedfrom it by increasing the probability differ-ences between adjacent gambles, and Set

III by decreasing the payoff differences be-tween them. All sets were constructed sothat the expected value increased with theprobability of winning and decreased withthe payoff.

To compare the three sets, 36 Harvardundergraduates (who did not participatein the earlier sessions) were invited for asingle session. Each 5 was presented withfive pairs of gambles from each one of thethree sets. These included the four pairsof adjacent gambles and the single pair ofextreme gambles from each set. Each ofthe 15 pairs was replicated three times, in arandomized presentation order. The 5swere run individually under the procedureemployed in the earlier sessions. Further-

TABLE 4

GAMBLE SETS I, II, AND III

Set

I

II

I I I

Probability

7/248/249/24

10/2411/24

8/2410/2412/2414/2416/24

7/248/249/24

10/2411/24

Payoff

5.004.754.504.254.00

5.004.754.504.254.00

3.703.603.503.403.30

Expected value

1.461.581.691.771.83

1.671.982.252.482.67

1.081.201.311.421.51


more, the same criterion for circularitywas investigated. That is, S had tochoose between most (three out of four)adjacent gambles according to payoff andbetween the extreme gamble according toprobability. The results showed that, outof the 36 5s, 13 satisfied the criterion inSet I, 6 satisfied the criterion in Set II and8 satisfied the criterion in Set III. Thesefindings indicate that the probability andthe payoff differences used in Set I yieldmore in transitivities than those used inSets II and III.

Experiment IIThe second experimental task is the selec-

tion of college applicants. Thirty-sixundergraduates were presented with pairsof hypothetical applicants and were askedto choose the one that they would ratheraccept. Each applicant was described bya profile portraying his percentile ranks onthree evaluative dimensions, labeled I, E,and S. The 5s were told that DimensionI reflects intellectual ability, Dimension Ereflects emotional stability, and DimensionS reflects social facility. An illustrativeprofile is shown in Figure 2.

The 5s were further told that the profileswere constructed by a selection committeeon the basis of high school grades, intel-

TABLE STHE 10 PROFILES USED IN THE PRE-LIMINARY SESSION OF EXPERIMENT II

100(I) (E) (s)

100

abcdefghij

Dimensions

I

63666972757881848790

E

96908478726660544842

S

958575655545352515S

FIG. 2. An illustrative applicant's profile.

Note.—I = intellectual ability, E = emotional stability,S = social facility.

ligence and personality tests, letters ofrecommendation, and a personal interview.Using this information, all applicants wereranked with respect to the three dimensionsand the three corresponding percentileranks constitute the applicant's profile.The 5s were then told that

The college selection committee is interested inlearning student opinion concerning the type of ap-plicants that should be admitted to the school.Therefore, you are asked to select which you wouldadmit from each of several pairs of applicants.Naturally, intellectual ability would be the mostimportant factor in your decision, but the otherfactors are of some value, too. Also, you shouldbear in mind that the scores are based on the com-mittee's ranking and so they may not be perfectlyreliable.

The study consisted of two parts: a pre-liminary session and a test session. Theprofiles used in the preliminary session aregiven in Table 5.

The profiles were constructed such thatthere was a perfect negative correlation be-tween the scores on Dimension I and thescores on Dimensions E and S. The (ab-solute) difference between a pair of profileson Dimension I is referred to as their Idifference. A choice between profiles issaid to be compatible with (or accordingto) Dimension I whenever the profile withthe higher score on that dimension is se-lected, and it is said to be incompatiblewith Dimension I whenever the profile

38 AMOS TVEESKY

with the lower score on that dimension isselected.

Since Dimension I is the most importantto the present task, and since the graphicaldisplay hinders the evaluation of small Idifference it was hypothesized that the LSwould be satisfied by some of the 5s. Forsmall I differences these 5s would chooseaccording to Dimensions E and S, but forlarge I differences they would choose ac-cording to Dimension I. The purpose ofthe preliminary session was to identify5s who behaved in that fashion and to col-lect preference data that could be employedin constructing new sets of profiles to beused in the test session.

The 5s were run individually. On eachtrial the experimenter presented 5 with apair of profiles and asked him to make achoice. Indifference judgment was not al-lowed. The 5s were presented with all 45pair comparisons of the 10 profiles in thesame randomized order.

The criterion for participation in the testsession was that at least six out of ninechoices between the adjacent profiles (a,b;b,c; c,d; d,e; e,f;f,g; g,h; h,i; i,j) were ac-cording to Dimensions E and S and at leastseven out of the 10 choices between the ex-treme profiles (a,j; a,i; a,h; a,g; b,j; b,i;b,h; c,j; c,i; d,j) were according to Dimen-sion I. Fifteen out of the 36 5s satisfiedthis criterion and were invited to the testsession.2

2 In some pilot studies in which 5s were run in agroup and only two dimensional profiles were used

Using the data obtained in the prelimi-nary session, the following procedure wasemployed to construct a special set of fiveprofiles for each 5. Let n(d) denote thenumber of choices (made by a given 5 inthe preliminary session) between profileswhose I difference was at most 5 and thatwere incompatible with Dimension I. Sim-ilarly, let m(8) denote the number of choicesbetween profiles whose I difference was atleast S and that were compatible with Di-mension I. Note that S = 3, 6, 9, • • •, 27and that, by the selection criterion em-ployed, w(3) > 6 and w(18) > 7 for all theselected 5s. The values of n(d) and m(d)were computed for each 5 and the value ofS' for which n(t>) + m(d) is maximized wasobtained.

To illustrate the procedure, the choicesmade by 5s in the preliminary session,along with the values of n(d), m(8), andn(8) + m(S) are shown in Table 6. Avalue of 1 in an entry indicates that theprofile in that row was selected over theprofile in that column. A value of 0 indicatesthe opposite.

Note that the diagonals of Table 6 rep-resent pairs of profiles that have equal Idifferences, and that these differences in-crease with the distance from the main(lowest) diagonal. Thus, pairs of adjacentprofiles are on the lowest diagonal whilepairs of extreme profiles are on the higherdiagonals. Inspection of Table 6 reveals

the proportion of 5s satisfying the above criterionwas considerably lower.

TABLE 6

CHOICES MADE BY Ss IN THE PRELIMINARY SESSION AND THE RESULTINGVALUES OF n(S), m(S), AND »(«) + m($)

Profile

abcdefghi3

a b c d e

1 1 0 01 1 1

1 01

f

00110

g

101011

h

0001111

i

01010100

j

000010011

5 «(5)

27 2324 2321 2318 2215 2112 199 176 133 7

»»(«) n(i)+m(J)

1 243 265 288 30

11 3215 3418 3520 3322 29

Note.—The preference of a row profile over a column profile is denoted by a 1, and the reverse preference is denoted by 0.


that most choices on the three lower diag-onals were incompatible with Dimension I,while most choices on the six upper diag-onals were compatible with Dimension I.The value of S which maximizes n (5) + m (5)is 9, which is taken as an estimate of thepreference threshold e. (It should benoted that e was originally defined as asubjective rather than objective difference.Consequently, it need not be independentof the location of the scores, and differentestimates of 5' may be obtained for differentparts of the scale. In the present study,however, only a single estimate of e wasobtained for each S.)

On the basis of these estimates, 5s weredivided into four groups, and a special setof profiles was constructed for each group.The new sets were constructed so that theintermediate I differences equaled the esti-mated threshold, e, for each 5. Morespecifically, the I differences in the fourpairs of adjacent profiles (a,b; b,c; c,d; d,e)were smaller than t, the I differences in thethree pairs of extreme profiles (a,e; a,d;b,e) were larger than e, and the I differencesin the three pairs of intermediate profiles(a,c; b,d; c,e) equaled e. The four sets ofprofiles, constructed for the test session, areshown in Table 7. Note that in each ofthe sets of Table 7 there is a perfect negativecorrelation between Dimension I and Di-mensions E and S, and that the profiles areequally spaced. The four sets differ fromeach other in the location and the spacingof the profiles. The differences betweenadjacent profiles on Dimensions I, E, and Srespectively are 3, 6, and 10 in Set I; 6, 10,and IS in Set II; 9, 12, and 20 in Set III;12, 16, and 23 in Set IV. Under the hy-pothesized model this construction was de-signed to yield preference patterns wherechoices between the four adjacent profilesare incompatible with Dimension I, whereaschoices between the three extreme profilesare compatible with Dimension I.

The test session took place approximately2 weeks after the preliminary session. The5s were reminded of the instructions andthe nature of the task. They were runindividually, and each one was presentedwith all 10 pair comparisons of the five new

TABLE 7FOUR SETS OP PROFILES CONSTRUCTED

FOR EXPERIMENT II

Set

I

11

I I I

IV

Profiles

abcde

abcde

abcde

abcde

Dimensions

I

6972757881

6672788490

5463728190

4254667890

E

8478726660

9080706050

9078665442

9680644832

S

7565554535

8570554025

9575553515

967350274

Note.—I = intellectualS = social facility.

ability, E = emotional stability,

profiles along with all 10 pair comparisonsof five "irrelevant" profiles introduced tominimize recall of the earlier decisions. Eachof the 20 pairs was replicated three timesduring the session. The order of presenta-tion was identical for all 5s and it was ran-domized within each block of 20 pairs. Thechoice frequencies of all 10 critical pairs ofprofiles are shown in Table 8 for each 5.Since only three replications of each paircomparison were obtained, the likelihoodratio test could not be properly applied tothese data. Instead, the maximum likeli-hood estimates of the choice probabilities,under both WST and the LS, were ob-tained for each S. The observed propor-tion of triples violating WST, denoted TT,was then compared with the expected pro-portions, based on the maximum likelihoodestimates under WST and the LS, denotedWST (IT) and LS(» respectively. Table 8shows that the observed values of IT exceedthe maximum likelihood estimates of TTunder WST for all but one 5 (p < .01 by a

40 AMOS TVERSKY

TABLE 8FREQUENCIES OF SELECTING THE FIRST ELEMENT OF EACH PAIR OVER THE

SECOND, TOTALED OVER THE THREE REPLICATIONS

Subject

123456789

101112131415

Total

Set

III

IIIIIIIIIIII

IIIIIIIIIIIIIVIV

Pair

a, b

323303333332323

b,c

312302332331222

c, d

223203333232233

d,e

132301113333333

a, c

221102221222212

b,d

112001112231102

c, e

223002113232022

a,d

100002002110112

b, e

000000001020110

a,e

000001000010000

7T

.4

.2

.4

.3

.0

.3

.2

.3

.4

.5

.4

.2

.3

.3

.4

.307

WSTM

.213

.1Q6

.262

.125

.000

.171

.197

.125

.237

.324

.238

.196

.228

.228

.238

.199

LSM

.316

.292

.303

.241

.300

.285

.295

.242

.281

.391

.366

.292

.275

.275

.372

.302

Note.—The values of T denote the observed proportions of intransitive triples, whereas the values of LS(TT) and WST (T) denotethe expected proportions under the two models, respectively.

sign test). Furthermore, the LS predictedthe observed proportions better than WSTfor 11 out of 15 5s. Finally, the overallproportion of intransitive triples (.307) issignificantly higher (p < .01) than thevalue expected under WST (.199), but it isnot significantly different from the valueexpected under the LS (.302), according toa chi-square test. Hence, WST is rejectedbecause both the overall proportion of in-transitive triples and the TT values of a sig-nificant majority of 5s exceed their expectedvalue under WST.

The 5s were interviewed at the end of thetest session. None of the 5s realized thathis preferences were intransitive. More-over, a few 5s denied this possibilityemphatically and asked to see the experi-menter's record. When faced with hisown in transitivities one 5 said "I must havemade a mistake somewhere." When theLS was explained to that 5, however, hecommented, "It is a reasonable way to makechoices. In fact, I have probably madesome decisions that way." The relationbetween the model and its logical conse-quences was obviously not apparent to our5.

THEORY

The empirical studies showed that, underappropriate experimental conditions, thebehavior of some people is intransitive.Moreover, the intransitivities are system-atic, consistent, and predictable. Whattype of choice theory is needed to explainintransitive preferences between multi-dimensional alternatives?

The lexicographic semiorder that wasemployed in the construction of the alter-natives for the experiments is one suchmodel. It is not, however, the only modelthat can account for the results. Further-more, despite its intuitive appeal, it isbased on a noncompensatory principle thatis likely to be too restrictive in many con-texts. In this section, two choice theoriesare introduced and their relationships tothe transitivity principle are studied.

Let A = AI X • • • X An be a set ofmultidimensional alternatives with ele-ments of the form x = (xi, • • • , # „ ) , y= (yi, • • •, y«). where #,• (» = ! , • • • , « )is the value of Alternative x on Dimensioni. Note that the components of x may benominal scale values rather than real num-


bers. A theory of choice between suchalternatives is essentially a decision rulewhich determines when x is preferred to y,or when p(x,y) > %. A more elaboratetheory may also provide an explicit formula

In examining the process of choice be-tween multidimensional alternatives, twodifferent methods of evaluation have beenconsidered (Morrison, 1962). The first isbased on independent evaluations. Ac-cording to this method, one evaluates thetwo alternatives, x and y, separately, andassigns scale values, u(x) and u(y), to eachof them. Alternative * is, then, preferredto Alternative y if and only if u(x) >u(y}.The scale value assigned to an alternative isa measure of its utility, or subjective value,which is assumed to depend on the sub-jective values of its components. Morespecifically, there are scales u\, • • • , «„ de-fined on A i, • • • , A n respectively such thatUi(xi) is the subjective value of the *th com-ponent of Alternative x. It is further as-sumed that the overall utility of an alter-native is expressable as a specified functionof the scale values of its components.Among the various possible functional rela-tions, the additive combination rule hasbeen most thoroughly investigated. Ac-cording to the additive (conjoint measure-ment) model, the subjective value of analternative is simply the sum of the sub-jective value of its components.

Stated formally, a preference structuresatisfies the additive model if there existreal-valued functions u, Ui, • • •, un suchthat

x > y if and only ifn n

«(*)= L «<(*<) ^ E«<Cy<) = «(?)• [5]*-l i-l

Axiomatic analyses of this model, whichare based on ordinal assumptions, havebeen provided by Debreu (1960), Luce andTukey (1964), Krantz (1964), and Luce(1966) under solvability conditions. Nec-essary and sufficient conditions for addi-tivity have been discussed by Adams andFagot (1959), Scott (1964), and Tversky(1967b). For some of the empirical appli-

cations of the model, see Shepard (1964)and Tversky (1967a). Note that thecommonly applied multiple-regressionmodel is a special case of the additivemodel where all the subjective scales arelinear.

The second method of evaluation isbased on comparisons of component-wisedifferences between the alternatives. Ac-cording to this method one considers quan-tities of the form 8,- = Ui(xi) — Ui(yt)which correspond to the difference betweenthe subjective values of x and y on the ithdimension. To each such quantity, oneapplies a difference function, <f>i, whichdetermines the contribution of the particu-lar subjective difference to the overallevaluation of the alternatives. The quan-tity $,-(5<) can be viewed, therefore, as the"advantage" or the "disadvantage" (de-pending on whether 3,- is positive or nega-tive) of x over y with respect to Dimensioni. With this interpretation in mind, it isnatural to require thai <fo( — 5) = — 0<(5).The obtained values of 0<(6<) are, then,summed over all dimensions, and x is pre-ferred over y whenever the resulting sum ispositive.

Stated formally, a preference structuresatisfies the additive difference model if thereexist real-valued functions MI, • • • , « » andincreasing continuous functions 0i, • • • , $„defined on some real intervals such that

x > y if and only if

£ *<[«<(*<) -t-i

> 0

where

^(- 5) =- <t>{(5) for all t.

An axiomatic analysis of the additivedifference model will be presented else-where. Essentially the same model wasproposed by Morrison (1962). A set ofordinal axioms yielding a (symmetric) ad-ditive difference model of similarity (ratherthan preference) judgments has been givenby Beals, Krantz, and Tversky (1968).

A comparison of the additive model(Equation 5) with the additive differencemodel (Equation 6) from a psychologicalviewpoint reveals that they suggest differ-

42 AMOS TVERSKY

ent ways of processing and evaluating thealternatives. A schematic illustration ofthe difference is given below.

X =

y =

' , Xi, • • -,Xn)

In the simple additive model, the alterna-tives are first processed "horizontally," byadding the scale values of the components,and the resulting sums are then compared todetermine the choice. In the additivedifference model, on the other hand, thealternatives are first processed "vertically,"by making intradimensional evaluations,and the results of these vertical compari-sons are then added to determine the choice.Although the two models suggest differentprocessing strategies, the additive model isformally a special case of the additivedifference model where all the differencefunctions are linear. To verify this fact,suppose 0j(8i) = ttSi for some positive tiand for all i. Consequently,

= £ tiUi(Xi) — £ tiUi(yi).»=i i-i

Thus, if we let »<(*<) = <<«<(*<) for all i,then Equation 6 can be written as x > y ifand only if

which is the additive model of Equation 5.Hence, if the difference functions are

linear the two models (but not necessarilythe processing strategies) coincide. Thevertical processing strategy is, thus, com-patible with the additive model if and onlyif the difference functions are linear.

The proposed processing strategies, aswell as the models associated with them, arecertainly affected by the way in which the

information is displayed. More specifi-cally, the additive model is more likely to beused when the alternatives are displayedsequentially (i.e., one at a time), while theadditive difference model is more likely tobe used when the dimensions are displayedsequentially. Two different types of polit-ical campaigns serve as a case in point.In one type of campaign, each candidateappears separately and presents his viewson all the relevant issues. In the secondtype the various issues are raised separatelyand each candidate presents his view on thatparticular issue. It is argued that the"horizontal" evaluation method, or thesimple additive model, is more likely to beused in the former situation, while the"vertical" evaluation method, or the addi-tive difference model, is more likely to beused in the latter situation.

Although different evaluation methodsmay be used in different situations, there areseveral general considerations which favorthe additive difference model. In the firstplace, it is considerably more general, andcan accommodate a wider variety of prefer-ence structures. The LS, for example, is alimiting case of this model where one (ormore) of the difference functions approachesa step function where 0(5) = 0 whenever8 < e. Second, intradimensional compari-son may simplify the evaluation task. Ifone alternative is slightly better than an-other one on all relevant dimensions, it willbe immediately apparent in a component-wise comparison and the choice will indeedbe easy. If the alternatives, however, areevaluated independently this dominancerelation between the alternatives may beobscured, which would certainly complicatethe choice process. But even if no suchdominance relation exists, it may still beeasier to use approximation methods whenthe evaluation is based on component-wisecomparisons. One common approximationprocedure is based on "canceling out"differences that are equal, or nearlyequal, thus reducing the number of dimen-sions that have to be considered. In decid-ing which of two houses to buy, for example,one may feel that the differences in styleand location cancel each other out and the


choice problem reduces to one of decidingwhether it is worth spending $x more for alarger house. It is considerably moredifficult to employ this procedure when thetwo alternatives are evaluated indepen-dently.

Finally, intradimensional evaluations aresimpler and more natural than interdimen-sional ones simply because the comparedquantities are expressed in terms of thesame units. It is a great deal simpler toevaluate the difference in intelligence be-tween two candidates than to evaluate thecombined effect of intelligence and emo-tional stability. In choosing between twow-dimensional alternatives, one makes 2«interdimensional evaluations when the al-ternatives are evaluated independently ac-cording to the additive model, but only ninterdimensional evaluations along with nintradimensional evaluations according tothe additive difference model.

Now that the two models have beendefined and compared, their relationshipsto the transitivity principle are investi-gated. It can be readily seen that thesimple additive model satisfies the transitiv-ity principle, for the assumptions that x ispreferred to y, and y is preferred to z implythatw(tf) > u(y)andu(y) > u(z). Hence,u(x) > u(z), which implies that x must bepreferred to z. Note that the argumentdoes not depend on the additivity assump-tion. Transitivity must, therefore, besatisfied by any model where a scale valueis assigned to each alternative and the pref-erences are compatible with Equations 2or 3.

Under what conditions does the additivedifference model satisfy the transitivityprinciple? The answer to this question isgiven by the following result, which de-pends on the dimensionality of thealternatives.3

Theorem: If the additive difference model(Equation 6) is satisfied then the followingassertions hold whenever the differencefunctions are defined.

8 In referring to the dimensionality of the alterna-tives, denoted n, only nontrivial dimensions havingmore than one value are considered. The fact thattransitivity holds whenever n = 2 and <j!>i = 0s hasbeen recognized by Morrison (1962, p. 19).

1. For n > 3, transitivity holds if andonly if all difference functions are linear.That is, <£,-(5) = tjS for some positive h andfor all i.

2. For n = 2, transitivity holds if andonly if <£i(5) = <£2(/5) for some positive t.

3. For n = 1, transitivity is alwayssatisfied.The proof is given in the appendix. Thetheorem shows that the transitivity as-sumption imposes extremely strong con-straints on the form of the differencefunctions. In the two-dimensional case,the difference functions applied to the twodimensions must be identical except for achange of unit of their domain. If thealternatives have three or more dimensions,then transitivity is both necessary andsufficient for the linearity of all the differ-ence functions. Recall that under thelinearity assumption, the additive differencemodel reduces to the simple additive model,which has already been shown to satisfytransitivity. The above theorem asserts,however, that this is the only case in whichthe transitivity assumption is compatiblewith the additive difference model. Putdifferently, if the additive difference modelis satisfied and if even one difference func-tion is nonlinear, as is likely to be the casein some situations, then transitivity mustbe violated somewhere in the system. Theexperimental identification of these intrans-itivities in the absence of knowledge of theform of the difference functions might bevery difficult indeed. The LS employedin the design of the experimental research isbased on one extreme form of nonlinearitywhere one of the difference functions is, orcan be approximated by, a step function.The above theorem suggests a new explana-tion of the intransitivity phenomenon, interms of the form of the difference functions,which may render it more plausible than itseemed before.

Most of the choice mechanisms that havebeen purported to yield in transitivities (in-cluding the LS) are based on the notion ofshifting attention, or switching dimensions,from one choice to another. Consequently,they assume that some relevant informationdescribing the alternatives is ignored or dis-

44 AMOS TVERSKY

carded on particular choices. In contrastto this notion, intransitivities can occur inthe additive difference model in a fully com-pensatory system where all the informationis utilized in the evaluation process.

Both the additive model and the additivedifference model can be extended in anatural way. To do so, let F be an in-creasing function and suppose that allchoice probabilities are neither 0 and 1.The (extended) additive model is said tobe satisfied whenever Equation 5 holds and

P(x,y) = / Z «<(*<) - £ Ui(y{) 1. [7]L <-i <=i J

Similarly, the (extended) additive differ-ence model is said to be satisfied wheneverEquation 6 holds and

P(x,y) = F( Z *«[>«(*,) - «<(?,)] V\ i=i /

[8]

Both models are closely related to theFechnerian or the strong utility model (seeLuce & Suppes, 1965). This model assertsthat there exists a function u and a distri-bution function F such that

= F[_u(x) - [9]

Note that Equation 7 is a special case ofEquation 9, where the utilities are additive,while Equation 8 is an additive generaliza-tion of Equation 9 to the multidimensionalcase. The two most developed probabilis-tic models of Thurstone (1927, Case V) andLuce (1959) can be obtained from the Fech-nerian model by letting F be the normal orthe logistic distribution function respec-tively.

It can be easily shown that Equation 9and, hence, Equation 7 satisfy not onlyWST, but also a stronger probabilisticversion of transitivity called strong sto-chastic transitivity, or SST. According tothis condition, if P(x,y) > | and P(y,z)> | then both P(x,z) > P(x,y) and P(x,z]

Clearly, SST implies WST but notconversely. However, if Equation 8 isvalid with n > 3, and if WST is satisfied,then according to the above theorem,Equation 8 reduces to Equation 7 whichsatisfies SST as well. Thus, we obtain the

somewhat surprising result that, under theextended additive difference model, withn > 3, WST and SST are equivalent.

DISCUSSION

In the introduction, a choice model (theLS) yielding intransitive preferences wasdescribed. This model was employed inthe design of two studies which showed that,under specified experimental conditions,consistent intransitivities can be obtained.The theoretical conditions under whichintransitivities occur were studied withinthe framework of a general additive differ-ence model. The results suggest that in theabsence of a model that guides the con-struction of the alternatives, one is unlikelyto detect consistent violations of WST.The absence of an appropriate model com-bined with the lack of sufficiently powerfulstatistical tests may account for the failureto reject WST in previous investigations.

Most previous tests of WST have beenbased on comparison between the observedproportion of intransitive triples, IT, and theexpected proportion under WST. As Mor-rison (1963) pointed out, however, thisapproach leads to difficulties arising fromthe fact that in a complete pair comparisondesign only a limited proportion of triplescan, in principle, be intransitive. Speci-fically, the expected value of T for an 5 whois diabolically (or maximally) intransitive

k + 1is 77; ^r where k is the number of alter-4(& — 2)natives. As k increases, this expression ap-proaches one-fourth, which is the expectedvalue of IT under the hypothesis of randomchoice (i.e., P(x,y) = J for all x,y). Mor-rison argued, therefore, that unless theintransitive triples can be identified in ad-vance, it is practically impossible (with alarge number of alternatives) to distinguishbetween the diabolically intransitive S andthe random 5 on the basis of the observedvalue of IT. These considerations suggestthat a more powerful test of WST can beobtained by using many replications of afew well-chosen alternatives rather than byusing a few replications of many alterna-tives. The latter approach, however, hasbeen employed in most studies of preference.


What are the implications of the presentresults for the analysis of choice behavior?

Casual observations, as well as the com-ments made by 5s, suggest that the LS (orsome other nonlinear version of the additivedifference model) is employed in some real-world decisions, and that the resulting in-transitivities can also be observed outsidethe laboratory. Consider, for example, aperson who is about to purchase a compactcar of a given make. His initial tendencyis to buy the simplest model for $2089.Nevertheless, when the saleman presentsthe optional accessories, he first decides toadd power steering, which brings the priceto $2167, feeling that the price difference isrelatively negligible. Then, following thesame reasoning, he is willing to add $47 fora good car radio, and then an additional$64 for power brakes. By repeating thisprocess several times, our consumer endsup with a $2593 car, equipped with all theavailable accessories. At this point, how-ever, he may prefer the simplest car overthe fancy one, realizing that he is not willingto spend $504 for all the added features,although each one of them alone seemedworth purchasing.

When interviewed after the experiment,the vast majority of 5s said that people areand should be transitive. Some 5s foundit very difficult to believe that they had ex-hibited consistent intransitivities. If in-transitivities of the type predicted by theadditive difference model, however, aremanifest in choice behavior why were 5s soconfident that their choices are transitive?

In the first place, transitivity is viewed,by college undergraduates at least, as alogical principle whose violation representsan error of judgment or reasoning. Con-sequently, people are not likely to admit theexistence of consistent intransitivities.Second, in the absence of replications, onecan always attribute intransitivities to achange in taste that took place betweenchoices. The circular preferences of thecar buyer, for example, may be explainedby the hypothesis that, during the choiceprocess, the consumer changed his mindwith regard to the value of the added ac-cessories. If this hypothesis is misapplied,

the presence of genuine intransitivities isobscured. Finally, most decisions aremade in a sequential fashion. Thus, hav-ing chosen y over x and then 2 over y, one istypically committed to z and may not evencompare it with x, which has already beeneliminated. Furthermore, in many choicesituations the eliminated alternative is nolonger available so there is no way of findingout whether our preferences are transitive ornot. These considerations suggest that inactual decisions, as well as in laboratoryexperiments, people are likely to overlooktheir own intransitivities.

Transitivity, however, is one of the basicand the most compelling principles of ra-tional behavior. For if one violates trans-itivity, it is a well-known conclusion thathe is acting, in effect, as a "money-pump."Suppose an individual prefers y to x, z to y,and x to z. It is reasonable to assume thathe is willing to pay a sum of money to re-place x by y. Similarly, he should bewilling to pay some amount of money toreplace y by z and still a third amount toreplace z by x. Thus, he ends up with thealternative he started with but with lessmoney. In the context of the selection ofapplicants, intransitivity implies that, if asingle candidate is to be selected in a seriesof pair comparisons, then the chosen candi-date is a function of the order in which thepairs are presented. Regardless of whetherthis is the case or not, it is certainly an un-desirable property of a decision rule.

As has already been mentioned, thenormative character of the transitivityassumption was recognized by 5s. In fact,some evidence (MacCrimmon, 1965) indi-cates that when people are faced with theirown intransitivities they tend to modifytheir choices according to the transitivityprinciple. Be this as it may, the fact re-mains that, under the appropriate experi-mental conditions, some people areintransitive and these intransitivities can-not be attributed to momentary fluctua-tions or random variability.

Is this behavior necessarily irrational?We tend to doubt it. It seems impossibleto reach any definite conclusion concerninghuman rationality in the absence of a de-

46 AMOS TVERSKY

tailed analysis of the sensitivity of thecriterion and the cost involved in evaluatingthe alternatives. When the difficulty (orthe cost) of the evaluations and the consist-ency (or the error) of the judgments aretaken into account, a model based on com-ponent-wise evaluation, for example, mayprove superior to a model based on inde-pendent evaluation despite the fact that theformer is not necessarily transitive whilethe latter is. When faced with complexmultidimensional alternatives, such as joboffers, gambles, or candidates, it is ex-tremely difficult to utilize properly all theavailable information. Instead, it is con-tended that people employ various ap-proximation methods that enable them toprocess the relevant information in makinga decision. The particular approximationscheme depends on the nature of the al-ternatives as well as on the ways in whichthey are presented or displayed. Thelexicographic semiorder is one such an ap-proximation. In general, these simplifica-tion procedures might be extremely usefulin that they can approximate one's "truepreference" very well. Like any approxi-mation, they are based on the assumptionthat the approximated quantity is inde-pendent of the approximation method.That is, in using such methods in makingdecisions we implicitly assume that theworld is not designed to take advantage ofour approximation methods. The presentexperiments, however, were designed withexactly that goal in mind. They attemptedto produce intransitivity by capitalizing ona particular approximation method. Thisapproximation may be very good in gen-eral, despite the fact that it yields intransi-tive choices in some specially constructedsituations. The main interest in the pres-ent results lies not so much in the fact thattransitivity can be violated but rather inwhat these violations reveal about thechoice mechanism and the approximationmethod that govern preference betweenmultidimensional alternatives.

REFERENCES

ADAMS, E., & FAGOT, R. A model of riskless choice.Behavioral Science, 1959, 4, 1-10.

BEALS, R., KRANTZ, D. H., & TVERSKY, A. Thefoundations of multidimensional scaling. Psy-chological Review, 1968, 75, 127-142.

CHIPMAN, J. S. Stochastic choice and subjectiveprobability. In D. Willner (Ed.), Decisions,values and groups. Vol. I. Oxford: PergamonPress, 1960.

DEBREU, G. Topological methods in cardinal util-ity theory. In K. J. Arrow, S. Karlin, & P.Suppes (Eds.), Mathematical methods in the socialsciences. Stanford: Stanford University Press,1960.

DAVIDSON, D., & MARSHAK, J. Experimental testsof stochastic decision theories. In C. W. Church-man & P. Ratoosh (Eds.), Measurement: Defini-tions and theories. New York: Wiley, 1959.

DAVIDSON, D., MCKINSEY, J. C. C., & SUPPES, P.Outlines of a formal theory of value. Philosophyof Science, 1955, 22, 140-160.

DAVIS, J. M. The transitivity of preferences.Behavioral Science, 1958, 3, 26-33.

EDWARDS, W. Probability-preferences in gambling.American Journal of Psychology, 1953,59, 290-294.

EDWARDS, W. The theory of decision making.Psychological Bulletin, 1954, 51, 380-416.

EDWARDS, W. Behavioral decision theory.Annual Review of Psychology, 1961, 12, 473-498.

GRISWOLD, B. J., & LUCE, R. D. Choice among un-certain outcomes: A test of a decomposition andtwo assumptions of transitivity. American Jour-nal of Psychology, 1962, 75, 35-44.

KRANTZ, D. H. Conjoint measurement: The Luce-Tukey axiomatization and some extensions.Journal of Mathematical Psychology, 1964, 1, 248-277.

LUCE, R. D. Semiorders and a theory of utilitydiscrimination. Econometrica, 1956, 24, 178-191.

LUCE, R. D. Individual choice behavior. NewYork: Wiley, 1959.

LUCE, R. D. Two extensions of conjoint measure-ment. Journal of Mathematical Psychology, 1966,3, 348-370.

LUCE, R. D., & GALANTER, E. Psychophysicalscaling. In R. D. Luce, R. R. Bush, & E. Gal-anter (Eds.), Handbook of mathematical psychology,Vol. I. New York: Wiley, 1963.

LUCE, R. D., & SUPPES, P. Preference, utility, andsubjective probability. In R. D. Luce, R. R.Bush, & E. Galanter (Eds.), Handbook of mathe-matical psychology, Vol. III. New York: Wiley,1965.

LUCE, R. D., & TUKEY, J. W. Simultaneous con-joint measurement: A new type of fundamentalmeasurement. Journal of Mathematical Psy-chology, 1964, 1, 1-27.

MACCRIMMON, K. R. An experimental study of thedecision making behavior of business executives.Unpublished doctoral dissertation, University ofCalifornia, Los Angeles, 1965.

MAY, K. O. Intransitivity, utility, and the aggrega-tion of preference patterns. Econometrica, 1954,22, 1-13.

MOOD, A. Introduction to the theory of statistics.New York: McGraw-Hill, 1950.


MORRISON, H. W. Intransitivity of paired com-parison choices. Unpublished doctoral disserta-tion, University of Michigan, 1962.

MORRISON, H. W. Testable conditions for triads ofpaired comparison choices. Psychometrika, 1963,28, 369-390.

PAPANDREOU, A. G., SAUERLANDER, 0. H., BROWN-LEE, O, H., HURWICZ, L., & FRANKLIN, W. Atest of a stochastic theory of choice. Universityof California Publications in Economics, 1957, 16,1-18.

QUANDT, R. E. Probabilistic theory of consumerbehavior. Quarterly Journal of Economics, 1956,70, 507-536.

SAMUELSON. P. A. Foundations of economic analy-sis. Cambridge: Harvard University Press, 1953.

SAVAGE, L. J, The theory of statistical decision.Journal of the America! Statistical Association,1951, 46, 55-57.

SCOTT, D. Measurement models and linear in-equalities. Journal of Mathematical Psychology,1964, 1, 233-248.

SHEPARD, R. N. On subjectively optimum selectionamong multiattribute alternatives. In M. W.Shelly & G. L. Bryan (Eds.), Human judgmentsand optimality. New York: Wiley, 1964.

SUPPES, P., & ZINNES, J. L. Basic measurementtheory. In R. D. Luce, R. R. Bush, & E. Gal-anter (Eds.), Handbook of mathematical psychology,Vol. I. New York: Wiley, 1963.

THURSTONE, L. L. A law of comparative judgment.Psychological Review, 1927, 34, 273-286.

TVERSKY, A. Additivity, utility and subjectiveprobability. Journal of Mathematical Psychology,1967, 4, 175-202. (a)

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APPENDIX

Theorem imply

If the additive difference model (Equation 6)is satisfied, then the following assertions holdwhenever the difference functions are denned.

1. For n > 3, transitivity holds if and onlyif all difference functions are linear. That is,0<(5) = tiO for some real d and for all i.

2. For n = 2, transitivity holds if and onlyif 0i(8) = <t>?(t8) for some real t.

3. For w = 1, transitivity is always satisfied.

Proof

By WST, P(x,y) = f and P ( y , z ) = J- implyP(x,z) = J. Hence, according to the additivedifference model there exist functions u\, • • • ,un and increasing continuous functions 0i, • • • ,<t>n defined on some real intervals of the form(- MO such that

= 0.

and

imply

0

= 0

0,

First, suppose n = 1, hence (*) reduces to:

0(a) = 0, 0(|3) = 0 imply 0(a + /3) = 0.

But since 0 is increasing, and 0(0) = — 0(0)= 0, a = ;8 = 0 = a+ /3 , and hence the aboveequation is always satisfied.

Next, suppose n = 2, hence, by (*), 0i(«i)+ 0aM = 0 and ^(fr) + 02032) = 0 imply0i(«! + (30 + 02(«2 + /32) = 0. Since 0,-(- 8)= — 0i(5), the above relation can be rewrittenas 0i(«i) = 0 2(— as) and 0i(/3i) = 02( — /32)imply 0i (ai + $1) = 0 2(— a2 — /Sa). Conse-quently, by letting «i = /3i and a2 = /3a, andrepeating the argument n times, we obtain0i(a) = 0a(/3) implies 0i(wa) = 02(w/3) for anypositive integer n for which both 0i(wa) and<j>i(np) are defined.

Since all difference functions are continu-ously increasing and since they all vanish atzero, one can select positive a, b such that0i(a), 02(&) are defined and such that 0i(a)= 0s(&). Hence, for any positive integers

m, n for which 0i( —a ] and 02l —b] are de-\ « / \ » /

fined

where &(— 5) = — 0i(6). Letting

£»< = w<(x;i)-M<(y<) and Pi = ui(yi')-Ui(Zi')

yields

(*) Efc(a<) = 0 and £0^/3,0 = 0

, 0i( - I and 02( - I are also defined. Fur-\»/ \n/

thermore, 0if - J = 02f - J, for otherwise a

strict inequality must hold. Suppose 0i( - )

< 02( - ), hence there exists c such that 0if - )

48 AMOS TVERSKY

(»)•= 02(c) < 02( - )• Consequently, c < -, or\n I n

nc < b, and 02(we) is defined. Hence, 02(we)= 0i (a) = 02(6) and nc = 6, a contradiction.By the symmetry of the situation, a similar

contradiction is obtained if 02( - ) > 02( - ).^\«/ w

Therefore,0i(a)=02(&)implies0i( - )=0s( - )\n/ \n/

for all n.

Next, let J = - and suppose that both 0i(c)a

and 02(to) are defined. Thus, for any 5 > 0,

there exist m, n such that c—S<—ffi^c, and

, b, .. b m ., 6 _ ,hence -(c — o) < a < -c. Consequently,

o a w ~ a

_[ / s\ ̂ j. (m \ ^ j. i \ j ±\ b, .. "I0i\c ~~ o] ^ G>I i —ct j ^ (pi\c) and<p2| *~(,c — o) i\ n / I a I\ / L. j

( \ / ?i \— 6 J < 0 2 ( - c ) . As 5 approaches 0,M, I \ Q, I

/ \ /

however, 0i(c — 5) = 0i(c) and 02 -(c — 6)

= 02( -c ), and since 0i[ —a] = 0J —& I, by\o / V « / . V " /

hypothesis, 0i(c) = 02(to) as required. Con-versely, if 0i (c) = 02 (to) it follows readily thatEquation (*) is satisfied which completes theproof of this case.

Finally, suppose n > 3. Since we can let allbut three differences be zero, we consider the

case where » = 3. Hence,

*I(QI) + 02(aa) +

and= °

0

imply 0i (ai + /?i) + 02(oi2 + (82) + 0s(a3 + /J3)= 0. By the earlier result, however, 0<(5)= 0;(£,-5) for i,j = 1, 2, 3. Hence, the aboveimplication is expressible as

<f> (a) +<f> $)=<!>(&) and <^(

imply

, , . /\ , . /a i «« _ JL/S i s/\cp(a + a ) + q>(p + p ) = <f>(0 + 5 ).

Define tf> such that 0(a) + <!>(£) = ̂ (a, /3)]for ali a> ^3, Hence,

Hence, ̂ («,«+,A(a',^) - ^(a + «', j8 + /S')and $ is jjnear in a, /3. Therefore, 0(a) + 008)= ^(#a + fflS) for some real p, q. If we letft = 0, we get 0(a) = 0(#a) hence /> = 1.Similarly, if we let a = 0, we get 0(0) = 0(08)hence g = 1. Consequently, 0 (a) + 0(/3) = 0(a+ ̂ and $ if linear as required. The conversefor any M > 3 ;s immediate which completesthe proof of this theorem.

(Received November 20, 1967)

psychological review vol. 76, no. 1, intransitivity of

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