Judgment under Uncertainty: Heuristics and Biases

Amos Tversky; Daniel Kahneman

Science, New Series, Vol. 185, No. 4157. (Sep. 27, 1974), pp. 1124-1131.

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http://www.jstor.org Thu Apr 5 16:07:46 2007 occupation from a list of possibilities (for example, farmer, salesman, airline pilot, librarian, or physician)? How do people order these occupat~oas from most to least likely? In the representa- Judgment under Uncertainty: tivenes> heuristic, the probability that Steve is a I~brarian, for example, IS Heuristics and Biases assessed by the degree to which he is representative of, or similar to, the stereotype of a librarian. Indeed, re- Biases in judgments reveal some heuristics of search with problems of this type has shown that people order the occupa- thinking under uncertainty. tions by probability and by similarity in exactly the same way (I). This ap- Amos Tversky and Daniel Kahneman proach to the ji~dg~~ientof probability leads to serious errors, because sim- ilarity, or representativeness, is not 1t1- fluenced by several factors that should affect judgments of probability. Many decisions are based on beliefs mated when visibility is good because l1zrer7ritivity to prior probability of concerning the likelihood of uncertain the objects are seen sharply. Thus, the outcomer. One of the factors that have events such as the outcome of an elec- reliance on clarity as an indication of no effect on representat~veness but tion, the guilt of a defendant, or the distance leads to common biases. Such should have a major effect on probabil- future value of the dollar. These beliefs biases are also found in the intuitive ~tyis the prior probability, or base-rate are usually expressed in statements such judgment of probability. This article frequency, of the outcomes. In the case as "I think that . . . ," "chances are describes three heuristics that are em- of Steve, for example, the fact that . . . ," "it is unlikely that . . . ," and ployed to assess probabilities and to there are many more farmers than li- so forth. Occasionally, beliefs concern- predict values. Biases to which these brarians in the population should enter ing iuncertain events are expressed in heuristics lead are enumerated, and the into any reasonable estimate of the numerical form as odds or subjective applied and theoretical implications of probabil~ty that Steve is a librarian probabilities. What determines such be- these observation5 are discussed. rather than a farmer. Considerations of liefs? How do people assess the prob- base-rate frequency, however, do not ability of an uncertain event or the affect the similarity of Steve to the value of an uncertain quantity? This Representativeness stereotypes of librarians and farmers. article shows that people rely on a If people evaluate probability by rep- limited number of heuristic principles Many of the probabilistic questions resentativeness, therefore, prior proba- which reduce the complex tasks of as- with which people are concerned belong bilities will be neglected. This hypothesis sessing probabilities and predicting val- to one of the following types: What is was tested in an experiment where prior ues to simpler judgmental operations. the probability that object A belongs to probabilities were manipulated (I). In general, these heuristics are quite class B? What is the probability that Subjects were shown brief personality useful, but sometimes they lead to severe event A originates from process B? descriptions of several individuals, al- and systematic errors. What is the probability that process R legedly sampled at random from a The subjective assessment of proba- w~llgenerate event A? In answering group of 100 professionals-engineers bility resembles the subjective assess- such questions, people typically rely on and lawyers. The subjects were askcd ment of physical quantities such as the representativeness heuristic, in to assess, for each description, the prob- distance or size. These judgments are which probabilities are evaluated by the ability that it belonged to an engineer all based on data of limited validity, degree to which A is representative of rather than to a lawyer. In one experi- which are processed according to heu- B, that is, by the degree to which A mental condition, subjects were told ristic rules. For example, the apparent resembles B. For example, when A ib that the group from which the descrip- distance of an object is determined in highly representative of B, the proba- tions had been drawn consisted of 70 part by its clarity. The more sharply the bility that A originates from B is judged engineers and 30 lawyers. Tn another object is seen, the closer it appears to to be high. On the other hand, if A is condition, subjects were told that the be. This rule has some validity, hecaust: not similar to B, the probability that A group consisted of 30 engineers and 70 in any given scene the more distant originates from B is judged to be low. lawyers. The odds that any particular objects are seen less sharply than nearer For an illustration of judgment by description belongs to an engineer objects. However, the reliance on this representativeness, consider an indi- rather than to a lawyer should be rule leads to systematic errors in the vidual who has been described by a higher in the first condition, where there estimation of distance. Specifically, dis- former neighbor as follows: "Steve is is a majority of engineers, than in the tances are often overestimated when very shy and withdrawn, invariably second condition, where there is a visibility is poor because the contours helpful, but with little interest in peo- majority of lawyers. Specifically, it can of objects are blurred. On the other ple, or in the world of reality. A meek be shown by applying Bayes' rule that hand, distances are often underesti- and tidy soul, he has a need for order the ratio of these odds should be (.7/.312, and structure, and a passion for detail." or 5.44, for each description. In a sharp The authors are tncmbers of the department of How do people assess the probability violation of Bayes' rule, the subjects psychology at the Hebt'cw University, Jerusalem, I~rael. that Steve is engaged in a particular in the two conditions produced essen-

SCIENCE, VOL. 185 tially the same probability judgments. sample size. Indeed, when subiects In this problem, the correct posterior Apparently, subjects evaluated the like- assessed the distributions of average odds are 8 to 1 for the 4 : 1 sample lihood that a particular description be- height for samples of various sizes, and 16 to 1 for the 12 : 8 sample, as- longed to an engineer rather than to a they produced identical distributions. suming equal prior probabilities. How- lawyer ,by the degree to which this For example, the probability of obtain- ever, most people feel that the first description was representative of the ing an average height greater than 6 sample provides much stronger evidence two stereotypes, with little or no regard feet was assigned the same value for for the hypothesis that the urn is pre- for the prior probabilities of the cate- samples of 1000, 100, and 10 men (2). dominantly red, because the proportion gories. Moreover, subjects failed to appreciate of red balls is larger in the first than in The subjects used prior probabilities the role of sample size even when it the second sample. Here again, intuitive correctly when they had no other infor- was emphasized in the formulation of judgments are dominated by the sample mation. In the absence of a personality the problem. Consider the following proportion and are essentially unaffected sketch, they judged the probability that question: by the size of the sample, which plays an unknown individual is an engineer a crucial role in the determination of to be .7 and .3, respectively, in the two A certain town is served by two hos- the actual posterior odds (2). In ad- pitals. In the larger hospital about 45 base-rate conditions. However, prior babies are born each day, and in $he dition, intuitive estimates of posterior probabilities were effectively ignored smaller hospital about 15 babies are born odds are far less extreme than the cor- when a description was introduced, each day. As you know, about 50 percent rect values. The underestimation of the even when this description was totally of all babies are boys. However, the exact impact of evidence has been observed percentage varies from day to day. Some- uninformative. The responses to the times it may be higher than 50 percent, repeatedly in problems of this type (3, 4). following description illustrate this phe- sometimes lower. It has been labeled "conservatism." nomenon: For a period of 1 year, each hospital Misconceptions of chance. People ex- Dick is a 30 year old man. He is mar- recorded the days on which more than 60 pect that a sequence of events generated ried with no children. A man of high percent of the babies born were boys. by a random process will represent the Which hospital do you think recorded essential characteristics of that process ability and high motivation, he promises more such days? to be quite successful in his field. He is even when the sequence is short. In well liked by his colleagues. b The larger hospital (21) b The smaller hospital (21) considering tosses of a coin for heads This description was intended to convey b A'bout the same (that is, within 5 or tails, for example, people regard the no information relevant to the question percent of each other) (53) sequence H-T-H-T-T-H to be more of whether Dick is an engineer or a The values in parentheses are the num- likely than the sequence H-H-H-T-T-T, lawyer. Consequently, the probability ber of undergraduate students who which does not appear random, and that Dick is an engineer should equal chose each answer. also more likely than the sequence H-H- the proportion of engineers in the Most subjects judged the probability H-H-T-H, which does not represent the group, as if no description had been of obtaining more than 60 percent boys fairness of the coin (2). Thus, people given. The subjects, however, judged to be the same in the small and in the expect that the essential characteristics the probability of Dick being an engi- large hospital, presumably because these of the process will be represented, not neer to be .5 regardless of whether the events are described by the same sta- only globally in the entire sequence, stated proportion of engineers in the tistic and are therefore equally repre- but also locally in each of its parts. A group was .7 or .3. Evidently, people sentative of the general population. In locally representative sequence, how- respond differently when given no evi- contrast, sampling theory entails that ever, deviates systematically from chance dence and when given worthless evi- the expected number of days on which expectation: it contains too many al- dence. When no specific evidence is more than 60 percent of the babies are ternations and too few runs. Another given, prior probabilities are properly boys is much greater in the small hos- consequence of the belief in local rep- utilized; when worthless evidence is pital than in the large one, because a resentativeness is the well-known gam- given, prior probabilities are ignored large sample is less likely to stray from bler's fallacy. After observing a long (1). 50 percent. This fundamental notion run of red on the roulette wheel. for Insensitivity to sample size. To eval- of statistics is evidently not part of example, most people erroneously be- uate the probability of obtaining a par- people's repertoire of intuitions. lieve that black is now due, presumably ticular result in a sample drawn from A similar insensitivity to sample size because the occurrence of black will a specified population, people typically has been reported in judgments of pos- result in a more representative sequence apply the representativeness heuristic. terior probability, that is, of the prob- than the occurrence of an additional That is, they assess the likelihood of ability that a sample has been drawn red. Chance is commonly viewed as a a sample result, for example, that the from one population rather than from self-correcting process in which a devi- average height in a random sample of another. Consider the following ex- ation in one direction induces a devia- ten men will be 6 feet (180 centi- ample: tion in the opposite direction to restore meters), by the similarity of this result the equilibrium. In fact, deviations are to the corresponding parameter (that Imagine an urn filled with balls, of not "corrected" as a chance process is, to the average height in the popula- which % are of one color and 1/3 of unfolds, they are merely diluted. tion of men). The similarity of a sam- another. One individual has drawn 5 balls Misconceptions of chance are not 'from the urn, and found that 4 were red ple statistic to a population parameter and 1 was white Another individual has limited to naive subjects. A study of does not depend on the size of the drawn 20 balls and found that 12 were the statistical intuitions of experienced sample. Consequently, if probabilities red and 8 were white. Which of the two research psychologists (5) revealed a are assessed by representativeness, then individuals should feel more confident that lingering belief in what may be called the urn contains 2/3 red balls and 95 white the judged probability of a sample sta- balls, rather than the opposite? What odds the "law of small numbers," according tistic will be essentially independent of should each individual give? to which even small samples are highly 27 SEPTEMBER 1974 representative of the populations from dent teacher during a particular prac- whose first-year record consists entirely which they are drawn. The responses tice lesson. Some subjects were asked of B's than in predicting the grade- of these investigators reflected the ex- to evaluate the quality of the lesson point average of a student whose first- pectation that a valid hypothesis about described in the paragraph in percentile year record includes many A's and C's. a population will be represented by a scores, relative to a specified population. Highly consistent patterns are most statistically significant result in a sam- Other subjects were asked to predict, often observed when the ~nput vari- ple-with little regard for its size. As also in percentile scores, the standing ables are highly redundant or correlated. a consequence, the researchers put too of each student teacher 5 years after Hence, people tend to have great con- much faith in the results of small sam- the practice lesson. The judgments made fidence in predictions based on redun- ples and grossly overestimated the under the two conditions were identical. dant input variables. However, an replicability of such results. In the That is, the prediction of a remote elementary result in the statistics of cor- actual conduct of research, this bias criterion (success of a teacher after 5 relation asserts that, given input vari- leads to the selection of samples of years) was identical to the evaluation ables of stated validity, a prediction inadequate size and to overinterpretation of the information on which the predic- based on several such inputs can of findings. tion was based (the quality of the achieve higher accuracy when they are Insensitivity to predictability. People practice lesson). The students who made independent of each other than when are sometimes called upon to make such these predictions were undoubtedly they are redundant or correlated. Thus, numerical predictions as the future value aware of the limited predictability of redundancy anlong inputs decreases of a stock, the demand for a commod- teaching competence on the basis of a accuracy even as it increases confidence, ity, or the outcome of a football game. single trial lesson 5 years earlier; never- and people are often confident in pre- Such predictions are often made by theless, their predictions were as ex- dictions that are quite likely to be off representativeness. For example, sup- treme as their evaluations. the mark (I). pose one is given a description of a The illu~ionof validity. As we have Misconceptions of regression. Suppose company and is asked to predict its seen, people often predict by selecting a large group of children has been future profit. If the description of the the outcome (for example, an occupa- examined on two equivalent versions of company is very favorable, a very tion) that is most representative of the an aptitude test. If one selects ten chil- high profit will appear most represen- input (for example, the description of dren from among those who did best on tative of that description; if the descrip- a person). The confidence they have one of the two versions, he will usually tion is mediocre, a mediocre perform- in their prediction depends primarily find their performance on the second ance will appear most representative. on the degree of representativeness version to be somewhat disappointing. The degree to which the description is (that is, on the quality of the match Conversely, if one selects ten children favorable is unaffected by the reliability between the selected outcome and the from among those who did worst on of that description or by the degree to input) with little or no regard for the one version, they will be found, on the which it permits accurate prediction. factors that limit predictive accuracy. average, to do somewhat better on the Hence, if people predict solely in terms Thus, people express great confidence other version. More generally, consider of the favorableness of the description, in the prediction that a person is a two variables X and Y which have the their predictions will be insensitive to librarian when given a description of same distribution. If one selects indi- the reliability of the evidence and to his personality which matches the viduals whose average X score deviates the expected accuracy of the prediction. stereotype of librarians, even if the from the mean of X by k units, then This mode of judgment violates the description is scanty, unreliable, or out- the average of their Y scores will usual- normative statistical theory in which dated. The unwarranted confidence ly deviate from the mean of Y by less the extremeness and the range of pre- which is produced by a good fit between than k units. These observations illus- dictions are controlled by considerations the predicted outcome and the input trate a general phenomenon known as of predictability. When predictability information may be called the illusion regression toward ;he mean, which was is nil, the same prediction should be of validity. This illusion persists even first documented by Galton more than made in all cases. For example, if the when the judge is aware of the factors 100 years ago. descriptions of companies provide no that limit the accuracy of his predic- In the normal course of life, one information relevant to profit, then the tions. It is a common observation that encounters many instances of regression same value (such as average profit) psychologists who conduct selection toward the mean, in the comparison should be predicted for all companies. interviews often experience considerable of the height of fathers and sons, of If predictability is perfect, of course, confidence in their predictions, even the intelligence of husbands and wives. the values predicted will match the when they know of the vast literature or of the performance of individuals actual values and the range of predic- that shows selection interviews to be on consecutive examinations. Neverthe- tions will equal the range of outcomes. highly fallible. The continued reliance less, people do not develop correct in- In general, the higher the predictability, on the clinical interview for selection, tuitions about this phenomenon. First, the wider the range of predicted values. despite repeated demonstrations of its they do not expect regression in many Several studies of numerical predic- inadequacy, amply attests to the strength contexts where it is bound to occur. tion have demonstrated that intuitive of this effect. Second, when they recognize the occur- predictions violate this rule, and that The internal consistency of a pattern rence of regression, they often invent subjects show little or no regard for of inputs is a major determinant of spurious causal explanations for it (I). considerations of predictability (I). In one's confidence in predictions based We suggest that the phenomenon of re- one of these studies, subjects were pre- on these inputs. For example, people gression remains elusive because it is in- sented with several paragraphs, each express more confidence in predicting the compatible with the belief that the describing the performance of a stu- final grade-point average of a student predicted outcome should be maximally SCIENCE, VOL. 185 representative of the input, and, hence, which instances or occurrences can be begin w~thr (road) and words that that the value of the outcome variable brought to mind. For example, one may have r in the third position (car) and should be as extreme as the value of assess the risk of heart attack among assess the relative frequency by the the input variable. middle-aged people by recalling such ease with which words of the two types The failure to recognize the import occurrences among one's acquaintances. come to mind. Because it is much easier of regression can have pernicious con- Similarly, one may evaluate the proba- to search for words by their first letter sequences, as illustrated by the follow- bility that a given business venture will than by their third letter, most people ing observation (I). In a discussion fail by imagining various difficulties it judge words that begin with a given of flight training, experienced instruc- could encounter. This judgmental heu- consonant to be more numerous than tors noted that praise for an exception- ristic is called availability. Availability words in which the same consonant ap- ally smooth landing is typically followed is a useful clue for assessing frequency pears in the third position. They do so by a poorer landing on the next try, or probability, because instances of even for consonants, such as r or k, while harsh criticism after a rough large classes are usually recalled better that are more frequent in the third landing is usually followed by an im- and faster than instances of less fre- position than in the first (6). provement on the next try. The instruc- quent classes. However, availability is Different tasks elicit different search tors concluded that verbal rewards are alliected by factors other than frequency sets. For example, suppose you are detrimental to learning, while verbal and probability. Consequently, the re- asked to rate the frequency with which punishments are beneficial, contrary to liance on availability leads to predicta- abstract words (thought, love) and con- accepted psychological doctrine. This ble biases, some of which are illustrated crete words (door, water) appear in conclusion is unwarranted because of below. written English. A natural way to the presence of regression toward the Biases due to the retrievabitity of in- answer this question is to search for mean. As in other cases of repeated stances. When the size of a class is contexts in which the word could ap- examination, an improvement will usu- judged by the availability of its in- pear. It seems easier to think of ally follow a poor performance and stances, a class whose instances are contexts in which an abstract concept a deterioration will usually follow an easily retrieved will appear more nu- is mentioned (love in love stories) than outstanding performance, even if the merous than a class of equal frequency to think of contexts in which a concrete instructor does not respond to the whose instances are less retrievable. In word (such as door) is mentioned. If trainee's achievement on the first at- an elementary demonstration of this ef- the frequency of words is judged by the tempt. Because the instructors had fect, subjects heard a list of well-known availability of the contexts in which praised their trainees after good land- personalities of both sexes and were they appear, abstract words will be ings and admonished them after poor subsequently asked to judge whether the judged as relatively more numerous than ones, they reached the erroneous and list contained more names of men than concrete words. This bias has been ob- potentially harmful conclusion that pun- of women. Different lists were presented served in a recent study (7) which ishment is more effective than reward. to different groups of subjects. In some showed that the judged frequency of Thus, the failure to understand the of the lists the men were relatively more occurrence of abstract words was much effect of regression leads one to over- famous than the women, and in others higher than that of concrete words. estimate the effectiveness of punish- the women were relatively more famous equated in objective frequency. Abstract ment and to underestimate the effec- than the men. In each of the lists, the words were also judged to appear in a tiveness of reward. In social interaction, subjects erroneously judged that the much greater variety of contexts than as well as in training, rewards are typ- class (sex) that had the more famous concrete words. ically administered when performance personalities was the more numerous Binses of irnaginnbility. Sometimes is good, and punishments are typically (6). one has to assess the frequency of a administered when performance is In addition to familiarity, there are class whose instances are not stored in poor. By regression alone, therefore, other factors. such as salience, which memory but can be generated accord- behavior is most likely to improve after affect the retrievability of instances. For ing to a given rule. In such situations, punishment and most likely to deterio- example, the impact of seeing a house one typically generates several instances rate after reward. Consequently, the burning on the subjective probability of and evaluates frequency or probability human condition is such that, by chance such accidents is probably greater than by the ease with which the relevant in- alone, one is most often rewarded for the impact of reading about a fire in punishing others and most often pun- stances can be constructed. However, ished for rewarding them. People are the local paper. Furthermore, recent oc- the ease of constructing instances does generally not aware of this contingency. currences are likely to be relatively not always reflect their actual frequency, In fact, the elusive role of regression more available than earlier occurrences. and this mode of evaluation is prone in determining the apparent conse- It is a common experience that the to biases. To illustrate, consider a group quences of reward and punishment subjective probability of traffic accidents of 10 people who form committees of seems to have escaped the notice of stu- rises temporarily when one sees a car k members, 2 < k < 8. How many dents of this area. overturned by the side of the road. different committees of k members can Biases drte to the eflectiveness of a be formed? The correct answer to this search set. Suppose one samples a word problem is given by the binomial coef- Availability (of three letters or more) at random ficient (If)which reaches a maximum from an English text. Is it more likely of 2.52 for k = 5. Clearly, the number There are situations in which people that the word starts with r or that of committees of k members equals assess the frequency of a class or the r is the third letter? People approach the number of committees of (10 - k) probability of an event by the ease with this problem by recalling words that members, because any committee of k 27 SEPTEMBER 1974 members defines a unique group of natural associates, such as suspicious- That is, different starting points yield (10 - k) nonmembers. ness and peculiar eyes. This effect was different estimates, which are biased One way to answer this question with- labeled illusory correlation. In their er- toward the initial values. We call this out computation is to mentally con- roneous judgments of the data to which phenomenon anchoring. struct committees of k members and they had been exposed, naive subjects InsufJicierlt adju~ttnent.In a demon- to evaluate their number by the ease "rediscovered" much of the common, stration of the anchoring effect, subjects with which they come to mind. Corn- but unfounded, clinical lore concern- were asked to estimate various quanti- mittees of few members, say 2, are ing the interpretation of the draw-a- ties, stated in percentages (for example, more available than committees of many person test. The illusory correlation the percentage of African countries in members, say 8. The simplest scheme ctrect was extremely resistant to con- the United Nations). For each quantity, for the construction of committees is a tradictory data. It persisted even when a number between 0 and 100 was deter- partition of the group into disjoint sets. the correlation between symptotn and mined by spinning a wheel of fortune One readily sees that it is easy to con- diagnosis was actually negative, and it in the subjects' presence. The subjects struct five disjoint committees of 2 prevented the judges from detecting were instructed to indicate first whether members, while it is impossible to gen- relationships that were in fact present. that number was higher or lower than erate even two disjoint committees of Availability provides a natural ac- the value of the quantity, and then to 8 members. Consequently, if fre- count for the illusory-correlation effect. estimate the value of the quantity by quency is assessed by imaginability, or The judgment of how frequently two moving upward or downward from the by availability for construction, the events co-occur could be based on the given number. Different groups were small committees will appear more num- strength of the associative bond between given different numbers for each quan- erous than larger committees, in con- them. When the association is strong, tity, and these arbitrary numbers had a trast to the correct bell-shaped func- one is likely to conclude that the events marked effect on estimates. For example, tion. Indeed, when naive subjects were have been frequently paired. Conse- the median estimates of the percentagc asked to estimate the number of distinct quently. strong associates will be judged of African countries in the United Na- committees of various sizes, their esti- to have occurred together frequently. tions were 25 and 45 for groups that re- mates were a decreasing monotonic According to this view, the illusory ceived I0 and 65, respectively, as start- function of committee size (6). For correlation between suspiciousness and lng points. Payoffs for accuracy did not example, the median estimate of the peculiar drawing of the eyes, for ex- reduce the anchoring effect. number of committees of 2 members ample, is due to the fact that suspi- Anchoring occurs not only when the was 70, while the estimate for com- ciousness is more readily asqociatcd with starting point is given to the subject, mittees of 8 members was 20 (the cor- the eyes than with any other part of but also when the subject bases his rect answer is 45 in both cases). the body. cstimate on the result of some incom- Tmaginability plays an important role Lifelong experience has taught us plete computation. A study of intuitive in the evaluation of probabilities in real- that, in general, instances of large numerical estimation illustrates this ef- life situations. The risk involved in an classes are recalled better and faster fect. Two groups of high school students adventurous expedition, for example, is than instances of less frequent classes; estimated, within 5 seconds, a numerical evaluated by imagining contingencies that likely occurrences are easier to expression that was written on the with which the expedition is not imagine than unlikely ones; and that blackboard. One group estimated the equipped to cope. If many such difficul- the associative connections between product ties are vividly portrayed, the expedi- events are strengthened when the events tion can be made to appear exceedingly frequently co-occur. As a result, man dangerous, although the ease with which has at his disposal a procedure (the while another group estimated the disasters are imagined need not reflect availability heuristic) for estimating the product their actual likelihood. Conversely, the numerosity of a class, the likelihood of risk involved in an undertaking may be an event, or the frequency of co-occur- grossly underestimated if some possible rences, by the ease with which the To rapidly answer such questions, peo- dangers are either difficult to conceive relevant mental operations of retrieval, ple may perform a few steps of compu- of, or simply do not come to mind. construction, or association can be tation and estimate the product by Illlrsory correlation. Chapman and performed. However, as the preceding extrapolation or adjustment. Because ad- Chapman (8)have described an interest- examples have demonstrated, this valu- justments are typically insufficient, this ing bias in the judgment of the fre- able estimation procedure results in procedure should lead to underestima- quency with which two events co-occur. systematic errors. tion. Furthermore, because the result of They presented naive judges with in- the first few steps of multiplication (per- formation concerning several hypothet- formed from left to right) is higher in ical mental patients. The data for each Adjustment and Anchoring the descending sequence than in the patient consisted of a clinical diagnosis ascending sequence, the former expres- and a drawing of a person made by In many situations, people make esti- sion should be judged larger than the the patient. Later the judges estimated mates by starting from an initial value latter. Both predictions were confirmed. the frequency with which each diagnosis that is adjusted to yield the final answer. The median estimate for the ascending (such as paranoia or suspiciousness) The initial value, or starting point, may sequence was 512, while the median had been accompanied by various fea- be suggested by the formulation of the estimate for the descending sequence tures of the drawing (such as peculiar problem, or it may be the result of a was 2,250. The correct answer is 40,320. eyes). The subjects markedly overesti- partial computation. In either case, Bia~esin the evaluation of conjunc- mated the freq~~encyof co-occurrence of adjustments are typically insufficient (4). tive and disjlrnctive events. In a recent SCIENCE, VOL. 185 study by Bar-Hillel (9) subjects were large. The general tendency to overesti- tained probability distributions for many given the opportunity to bet on one of mate the probability of conjunctive quantities from a large number of two events. Three types of events were events leads to unwarranted optimism in judges. These distributions indicated used: (i) simple events, such as drawing the evaluation of the likelihood that a large and systematic departures from a red marble from a bag containing 50 plan will succeed or that a project will proper calibration. In most studies, the percent red marbles and 50 percent be completed on time. Conversely, dis- actual values of the assessed quantities white marbles; (ii) conjunctive events, junctive structures are typically encoun- are either smaller than X,, or greater such as drawing a red marble seven tered in the evaluation of risks. A com- than X,, for about 30 percent of the times in succession, with replacement, plex system, such as a nuclear reactor problems. That is, the subjects state from a bag containing 90 percent red or a human body, will malfunction if overly narrow confidence intervals which marbles and 10 percent white marbles; any of its essential components fails. reflect more certainty than is justified by and (iii) disjunctive events, such as Even when the likelihood of failure in their knowledge about the assessed drawing a red marble at least once in each component is slight, the probability quantities. This bias is common to seven successive tries, with replacement, of an overall failure can be high if naive and to sophisticated subjects, and from a bag containing 10 percent red many components are involved. Be- it is not eliminated by introducing prop- marbles and 90 percent white marbles. cause of anchoring, people will tend to er scoring rules, which provide incentives In this problem, a significant majority underestimate the probabilities of failure for external calibration. This effect is at- of subjects preferred to bet on the con- in complex systems. Thus, the direc- tributable, in part at least, to anchoring. junctive event (the probability of which tion of the anchoring bias can some- To select X,, for the value of the is .48) rather than on the simple event times be inferred from the structure of Dow-Jones average, for example, it is (the probability of which is .50).Sub- the event. The chain-like structure of natural to begin by thinking about one's jects also preferred to bet on the simple conjunctions leads to overestimation, the best estimate of the Dow-Jones and to event rather than on the disjunctive funnel-like structure of disjunctions adjust this value upward. If this adjust- event, which has a probability of .52. leads to underestimation. ment-like most others-is insufficient, Thus, most subjects bet on the less likely Anchoring in the assessment of sub- then X,, will not be sufficiently extreme. event in both comparisons. This pattern jective probability distributions. In deci- A similar anchoring effect will occur in of choices illustrates a general finding. sion analysis, experts are often required the selection of XI,, which is presumably Studies of choice among gambles and to express their beliefs about a quantity, obtained by adjusting one's best esti- of judgments of probability indicate such as the value of the Dow-Jones mate downward. Consequently, the con- that people tend to overestimate the average on a particular day, in the fidence interval between X,, and X,, probability of conjunctive events (10) form of a probability distribution. Such will be too narrow, and the assessed and to underestimate the probability of a distribution is usually constructed by probability distribution will be too tight. disjunctive events. These biases are asking the person to select values of In support of this interpretation it can readily explained as effects of anchor- the quantity that correspond to specified be shown that subjective probabilities ing. The stated probability of the percentiles of his subjective probability are systematically altered by a proce- elementary event (success at any one distribution. For example, the judge dure in which one's best estimate does stage) provides a natural starting point may be asked to select a number, X,,, not serve as an anchor. for the estimation of the probabilities of such that his subjective probability that Subjective probability distributions both conjunctive and disjunctive events. this number will be higher than the for t given quantity (the Dow-Jones Since adjustment from the starting point value of the Dow-Jones average is .90. average) can be obtained in two differ- is typically insufficient, the final esti- That is, he should select the value X,, ent ways: (i) by asking the subject to mates remain too close to the probabili- so that he is just willing to accept 9 to select values of the Dow-Jones that ties of the elementary events in both 1 odds that the Dow-Jones average will correspond to specified percentiles of cases. Note that the overall probability not exceed it. A subjective probability his probability distribution and (ii) by of a conjunctive event is lower than distribution for the value of the Dow- asking the subject to assess the prob- the probability of each elementary Jones average can be constructed from abilities that the true value of the event, whereas the overall probability of several such judgments corresponding to Dow-Jones will exceed some specified a disjunctive event is higher than the different percentiles. values. The two procedures are formally probability of each elementary event. By collecting subjective probability equivalent and should yield identical As a consequence of anchoring, the distributions for many different quanti- distributions. However, they suggest dif- overall probability will be overestimated ties, it is possible to test the judge for ferent modes of adjustment from differ- in conjunctive problems and underesti- proper calibration. A judge is properly cent anchors. In procedure (i), the mated in disjunctive problems. (or externally) calibrated in a set of natural starting point is one's best esti- Biases in the evaluation of compound problems if exactly II percent of the mate of the quantity. In procedure (ii), events are particularly significant in the true values of the assessed quantities on the other hand, the subject may be context of planning. The successful falls below his stated values of Xn. For anchored on the value stated in the completion of an undertaking, such as example, the true values should fall question. Alternatively, he may be an- the development of a new product, typi- below X,, for 1 percent of the quanti- chored on even odds, or 50-50 chances, cally has a conjunctive character: for ties and above X,, for 1 percent of the which is a natural starting point in the the undertaking to succeed, each of a quantities. Thus, the true values should estimation of likelihood. In either case, series of events must occur. Even when fall in the confidence interval between procedure (ii) should yield less extreme each of these events is very likely, the X,,, and X,, on 98 percent of the prob- odds than procedure (i). overall probability of success can be lems. To contrast the two procedures, a quite low if the number of events is Several investigators (11) have ob- set of 24 quantities (such as the air dis- 27 SEPTEMBER 1974 tance from New Delhi to Peking) was It is not surprising that useful heuris- subjective interpretation of probability presented to a group of subjects who tics such as representativeness and that is applicable to unique events and assessed either XI, or X,, for each prob- availability are retained, even though is embedded in a general theory of ra- lem. Another group of subjects re- they occasionally lead to errors in pre- tional decision. ceived the median judgment of the first diction or estimation. What is perhaps It should perhaps be noted that, while group for each of the 24 quantities. surprising is the failure of people to subjective probabilities can sometimes They were asked to assess the odds that infer from lifelong experience such be inferred from preferences among each of the given values exceeded the fundamental statistical rules as regres- bets, they are normally not formed in true value of the relevant quantity. In sion toward the mean, or the effect of this fashion. A person bets on team A the absence of any bias, the second sample size on sa.mpling variability. Al- rather than on team B because he be- group should retrieve the odds specified though everyone is exposed, in the nor- lieves that team A is more likely to to the first group, that is, 9 : 1. How- mal course of life, to numerous ex- win; he does not infer this belief from ever, if even odds or the stated value amples from which these rules could his betting preferences. Thus, in reality, serve as anchors, the odds of the sec- have been induced, very few people subjective probabilities determine pref- ond group should be less extreme, that discover the principles of sampling and erences among bets and are not de- is, closer to 1 : 1. Indeed, the median regression on their own. Statistical prin- rived from them, as in the axiomatic odds stated by this group, across all ciples are not learned from everyday theory of rational decision (12). problems, were 3 : 1. When the judg- experience because the relevant in- The inherently subjective nature of ments of the two groups were tested stances are not coded appropriately. For probability has led many students to the for external calibration, it was found example, people do not discover that belief that coherence, or internal con- that subjects in the first group were too successive lines in a text differ more in sistency, is the only valid criterion by extreme, in accord with earlier studies. average word length than do successive which judged probabilities should be The events that they defined as having pages, because they simply do not at- evaluated. From the standpoint of the a probability of .10 actually obtained in tend to the average word length of in- formal theory of subjective probability, 24 percent of the cases. In contrast, dividual lines or pages. Thus, people any set of internally consistent probabil- subjects in the second group were too do not learn the relation between sample ity judgments is as good as any other. conservative. Events to which they as- size and sampling variability, although This criterion is not entirely satisfactory, signed an average probability of .34 the data for such learning are abundant. because an internally consistent set of actually obtained in 26 percent of the The lack of an appropriate code also subjective probabilities can be incom- cases. These results illustrate the man- explains why people usually do not patible with other beliefs held by the ner in which the degree of calibration detect the biases in their judgments of individual. Consider a person whose depends on the procedure of elicitation. probability. A person could conceivably subjective probabilities for all possible learn whether his judgments are exter- outcomes of a coin-tossing game reflect nally calibrated by keeping a tally of the the gambler's fallacy. That is, his esti- Discussion proportion of events that actually occur mate of the probability of tails on a among those to which he assigns the particular toss increases with the num- This article has been concerned with same probability. However, it is not ber of consecutive heads that preceded cognitive biases that stem from the reli- natural to group events by their judged that toss. The judgments of such a per- ance on judgmental heuristics. These probability. In the absence of such son could be internally consistent and biases are not attributable to motiva- grouping it is impossible far an indivi- therefore acceptable as adequate sub- tional effects such as wishful thinking or dual to discover, for example, that only jective probabilities according to the the distortion of judgments by payoffs 50 percent of the predictions to which criterion of the formal theory. These and penalties. Indeed, several of the he has assigned a probability of .9 or probabilities, however, are incompatible severe errors of judgment reported higher actually came true. with the generally held belief that a earlier occurred despite the fact that The empirical analysis of cognitive coin has no memory and is therefore in- subjects were encouraged to be accurate biases has implications for the theoreti- capable of generating sequential de- and were rewarded for the correct cal and applied role of judged probabili- pendencies. For judged probabilities to answers (2, 6). ties. Modern decision theory (12, 13) be considered adequate, or rational, in- The reliance on heuristics and the regards subjective probability as the ternal consistency is not enough. The prevalence of biases are not restricted quantified opinion of an idealized per- judgments must be compatible with the to laymen. Experienced researchers are son. Specifically, the subjective proba- entire web of beliefs held by the in- also prone to the same biases-when bility of a given event is defined by the dividual. Unfortunately, there can be they think intuitively. For example, the set of bets about this event that such a no simple formal procedure for assess- tendency to predict the outcome that person is willing to accept. An inter- ing the con~patibilityof a set of proba- best represents the data, with insufficient nally consistent, or coherent, subjective bility judgments with the judge's total regard for prior probability, has been probability measure can be derived for system of beliefs. The rational judge observed in the intuitive judgments of an individual if his choices among bets will nevertheless strive for compatibility, individuals who have had extensive satisfy certain principles, that is, the even though internal consistency is training in statistics (1, 5). Although axioms of the theory. The derived prob- more easily achieved and assessed. In the statistically sophisticated avoid ability is subjective in the sense that particular, he will attempt to make his elementary errors, such as the gambler's different individuals are allowed to have probability judgments compatible with fallacy, their intuitive judgments are different probabilities for the same event. his knowledge about the subject mat- liable to similar fallacies in more in- The major contribution of this ap- ter, the laws of probability, and his own tricate and less transparent problems. proach is that it provides a rigorous judgmental heuristics and biases. SCIENCE, VOL. 185 Summary and usually effective, but they lead to 8. L. J. Chapman and J. P. Chapman, I. Abnorm. Psvchol. 73. 193 (1967): ibid... 74.. systematic and predictable errors. A 271 (1969). This article described three heuristics better understanding of these heuristics 9. M. Bar-Hillel, Organ. Behav. Hurn. Per- formance 9, 396 (1973). that are employed in making judgments and of the biases to which they lead 10. J. Cohen, E. I. Chesnick, D. Haran, Br. J. Ps~chol.63. 41 (1972). under uncertainty: (i) representativeness, could improve judgments and decisions 11. M: Alpert and H. ~iiffa,unpublished manu- which is usually employed when peo- in situations of uncertainty. script; C. A. S. von Holstein, Acta Psychol. 35, 478 (1971); R. L. Winkler, J. Am. Slat. ple are asked to judge the probability Assoc. 62, 776 (1967). References and Notes that an object or event A belongs to 12. L. J. Savage, The Foirndatior~s of Statistics class or process B; (ii) availability of in- 1. D. Kahneman and A. Tversky, Psychol. Rev. (Wiley, New York, 1954). SO, 237 (1973). 13. B. De Finetti, in International Encyclopedia of the Social Sciences, D. E. Sills, Ed. (Mac- stances or scenarios, which is often em- 2. Cognitive Psychol. 3, 430 (1972). ---, miUan, New York, 1968), vol. 12, pp. 496- ployed when people are asked to assess 3. W. Edwards, in Formnl Representation of 504. Human Judgment, B. Kleinmuntz, Ed. (Wiley, the frequency of a class or the plausibil- 14. This research was supported by the Advanced New York, 1968), pp. 17-52. Research Projects Agency of the Department ity of a particular development; and 4. P. Slovic and S. Lichtenstein, Organ. Behav. of Defense and was monitored by the Ofice 6, (iii) adjustment from an anchor, which Hum. Performance 649 (1971). of Naval Research under contract N00014- 5. A. Tversky and D. Kahneman, Psychol. Bull. 73-C-0438 to the Oregon Research Institute, is usually employed in numerical predic- 76, 105 (1971). Eugene. Additional support for this research 6. -, Cognitive Psychol. 5, 207 (1973). was provided by the Research and Develop- tion when a relevant value is available. 7. R. C. Galbraith and B. J. Underwood, ment Authority of the Hebrew University, These heuristics are highly economical Mem. Cognition 1, 56 (1973). Jerusalem, Israel.

accidents (2). In 1970 there were 34,- 107 doctors in Mexico (2). The ratio of inhabitants to doctors, which is 1423.7, is not a representative index Rural Health Care in Mexico? of the actual distribution of resources because there is a great scarcity of Present educational and administrative structures must be health professionals in rural areas and a high concentration in urban areas changed in order to improve health care in rural areas. (Fig. 1) (7, 8). In order to improve health at a na- Luis Cafiedo tional level, this situation must be changed. The errors made in previous attempts to improve health care must be avoided, and use must be made of the available manpower and resources The present health care structure in The continental and insular area of of modern science to produce feasible Mexico focuses attention on the urban Mexico, including interior waters, is answers at the community level. Al- population, leaving the rural communi- 2,022,058 square kilometers (1,2). In though the main objective of a special- ties practically unattended. There are 1970 the population of Mexico was ist in community medicine is to control two main factors contributing to this 48,377,363, of which 24,055,305 per- disease, such control cannot be situation. One is the lack of coordina- sons (49.7 percent) were under 15 achieved unless action is taken against tion among the different institutions years of age. The Indian population the underlying causes of disease; it has responsible for the health of the com- made up 7.9 percent of the total (2, 3). already been observed that partial solu- munity and among the educational As indicated in Table 1, 42.3 percent tions are inefficient (9). As a back- institutions. The other is the lack of of the total population live in commu- ground to this new program that has information concerning the nature of nities of less than 2,500 inhabitants, and been designed to provide health care the problems in rural areas. In an at- in such communities public services as in rural communities, I shall first give tempt to provide a solution to these well as means of communication are a summary of the previous attempts problems, a program has been designed very scarce or nonexistent. A large per- that have been made to provide such that takes into consideration the en- centage (39.5 percent) of the econom- care, describing the various medical in- vironmental conditions, malnutrition, ically active population is engaged in stitutions and other organizations that poverty, and negative cultural factors agriculture (4). are responsible for the training of med- that are responsible for the high inci- The country's population growth rate ical personnel and for constructing the dences of certain diseases among rural is high, 3.5 percent annually, and it facilities required for health care. populations. It is based on the develop- seems to depend on income, being ment of a national information system higher among the 50 percent of the The author is an investigator in the department for the collection and dissemination of population earning less than 675 pesos of molecular biology at the Instituto de Investi- information related to general, as well ($50) per family per month (5). The gaciones Biomkdicas, Universidad Nacional Aut6- noma de Mexico, Ciudad Universitaria, Mexico as rural, health care, that will provide majority of this population lives in the 20, D.F. This article is adapted from a paper the basis for a national health care sys- rural areas. The most frequent causes presented at the meeting on Science and Man in the Americas, jointly organized by the Consejo tem, and depends on the establishment of mortality in rural areas are malnu- Nacional de Ciencia y Tecnologia de Mkxico and of a training program for professionals trition, infectious and parasitic diseases the American Association for the Advancement of Science and held in Mexico City, 20 June to in community medicine. (6, 7), pregnancy complications, and 4 July 1973. http://www.jstor.org

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You have printed the following article: Judgment under Uncertainty: Heuristics and Biases Amos Tversky; Daniel Kahneman Science, New Series, Vol. 185, No. 4157. (Sep. 27, 1974), pp. 1124-1131. Stable URL: http://links.jstor.org/sici?sici=0036-8075%2819740927%293%3A185%3A4157%3C1124%3AJUUHAB%3E2.0.CO%3B2-M

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11 The Assessment of Prior Distributions in Bayesian Analysis Robert L. Winkler Journal of the American Statistical Association, Vol. 62, No. 319. (Sep., 1967), pp. 776-800. Stable URL: http://links.jstor.org/sici?sici=0162-1459%28196709%2962%3A319%3C776%3ATAOPDI%3E2.0.CO%3B2-E

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