13* Ra'.ionj] Choice :n »n Uncertain World Hawkins, S. A., Jk I tank, R.; 1990'. Hindsight: B-.ised judgments of past events after the miWWnft are known. Psychological Bulletin. 107, 311-32". Heath. C, 3c Heath. D. (2007), Made (0 tffcfa My iowe surviwand others die. New York: Random Home. Jardcl Co. v. Hughes. Del. Supr.. 523 A.2d 51S - 195",. 11 m mi . r. m., & Rotenbaum, M 19*7 . Recall oi paternal behavior by acute depressive*, remitted depressives, and nnndcpicss.'kcv Journal of Personality and Social Psychology. 52, 611-620. Much, J, C. (1972). Model bias in social action. Rexieu- of Education Research, 42. 413-429. Neisser, U. < 1981). John Dean's memory: A case study. Cognition. 9, 1-22. Pennington, N., &: Hastie, R. 1193SI. Explanation bumi decision making: Effects of memory structure or. judgment. Journal of Experimental Psychology: Learning, Memory, and Cognition. 14, 521-533. Pennington, N'., 3c Hastie. R. 119911. A cognitive theory of juror decision making: Trie story model. Cardozo Liu Res ten; H, 519-55". Schank, R. C, & Abelson, R. P. 11995t. Knowledge and memory: The real story. Ir. K. VCycr, Jr. iFd.i, Advances in social cognition iVol. 8. pp. l-86i. Hi.lsdalc. NJ: Lawrence Eiibaum. Silkwood v. Kcrr-McGcc Corp., 464 U.S. 238 11984.1. Spcivce, f). f, 119S2:. Narramv truth and historical truth; Meaning and interpretation in psychoanalysis. New York: Notion. Spcncc, G. 11994, November 29:. winning attorneys. Sew York Times, p. 11. Tversky. A., & Kahneman, D. 119831. Extensional versus Intuitive tcasoning: The conj unction fallacy in ptobabiliry judgment. Psychological Bulletin, 90, 293-315. Van den Broek, P.. & Thurlow. R. 119911. The role and structure of personal narratives. Journal of Cognitive Psychotherapy, S, 257-2"-. WttMfUMO, D., Umpert. R. O., &c Hastic. R. (1991). Hindsight and causality. Personality and Social Psychology Bulletin, 17, 30-35. 7 Chance and Cause Say you're thinking about a plate of shrimp. Suddenly someone says plate, or shrimp, or plate of shrimp. Out of the blue. So use looking for one either. It's part of the lattice of coincidence that lays on tap of everything. —From the film Repo Man, written and directed by Alex Cox, 1984 7.1 Misconceptions About Chance On January 26, 1972, Vesna Vulovic, a 22-year-old Yugoslavian flight ndant, was serving drinks to passengers on JAT Flight 367 when the plane was demolished by a bomb planted by a Croatian nationalist group. Most people would think she was extremely unlucky—first, to be on a rare flight destroyed by a terrorist bomb and second, because of a name COflfu' ". she had been assigned to work the wrong flight. Definitely the wrong flight. But there is a positive side to this story. Ms. Vulovic lived and now holds the world record for surviving the highest fall without a parachute— •*3,000 feet (10,000 meters). Just a little more than a year alter the fall, she *clarcd herself ready to return to work, a self-described "optimist"r with a "pvfound belief in God. So, many people would describe her as exceptionally lucky. Ms. Vulovic goes with our first assessment, "I'm not lucky. Everybody thinks I am lucky, but they are mistaken. If I were luck* I would ■•ever have had this accident" (Bilefsky, 2008). 140 Rational Choice In an ItacerUifl World Chir.LC jr.J Cause HI li is not surprising th.u people often think an J talk about unexpictcd event* in different. sometimes contradictory ways. After all, tlu-%.- eunt> unpredictable and by definition mysterious and poorly understood. But. even beyond that, mi.' -rands do nut seem to he designed to reason sysrematK.il.'-, about chance and uncertainty. Perhaps for evolutionary reasons, we arc inclined to over-explain uncertain events and. even when we recognize they arc inherently unpredictable, we have some queer notions about how they behave, including many superstitious beliefs iSagan, 1997). Because we hive natural mifconccptions about uncertainty and randomness, this is one case in which learning about the rudiments of a technical framework—proba'?; jn theory—can make a big difference in how we sec the world. But without special training, no one thinki about the world in terms of probabilities. Raiher, the world seems to be a bunch of events and objects glued together by causal relationships, and most of us think about causation deterministically and in terms of degrees of causal force, but no: in terms <>i probabilities. We have been careful not to refer to the world as probabilistic or random. Probability theory is a language we can use to describe the world or. more precisely, to describe the relationships among our beliefs about the world. It is an unfamiliar language to most people, with a special symbolic vocabulary and rules of grammar .see the Appendix for an introduction to probability theory I. As we noted earlier, probability theory was not invented until recently in the history of Western civilization, and words like probability don't seem to have • ii ed the English lexicon until the l"th century. | Lexicography is b, I... ii was derived from the expression "approvable," e.g., a proktNe husband was originally an acceptable or morally "approvab'.c" husband.' Sometimes we do talk about chance, luck, probability, or randomness in everyday events—we say, "she was lucky," -it happened by chance." "that was a random event." But the most sensible interpretation of these expressions is that they indicate the state of knowledge within the mind of the person speaking. Harking back to a very wise essay on the nature of chance by the philosopher Potncarc (1914/1952), the events that we refer to in everyday life are all brought about by deterministic, phy>k.i processes. What singles our the events that we refer to as random, chance, or probabilistic is that the causal context is hidden, complex, nt unknown to the person who describes the event as such. We can't specify the physical events that occurred to preserve Vesna Vulovic's life, but we believe that she survived because of physical conditions that could be specified, if we had enough ] information. If we'd been able to observe her fall, including the minute details concerning her contact with the ground and her internal body state immediately before contact, we should be able to account for her remarkable escape from death in terms of physical causality. For another example, wc refer to the toss of a fair coin as a random rocess and assign the i ideal} probability value of .50 to the event of beads, ugh wc believe that the hidden biological and physical events that cause l outcome of rhc toss arc all deterministic. In fact, skilled slcight-oHi ind _ igkians, like the mathematician Persi Diaconis, have developed their manual skills to the point where they can execute apparently uncontrolled coin tosses and ic'.iably produce the desired result, heads or tails iBayer Diaconis, 1992; Diaconis. Holmes. & Montgomery, 20071. Of course, there arc levels of physical analysis, for example, at the quantum level, where id enlists do nor believe causaliry maps directly onto the me.ban-, d pi incipli s of causality wc experience. But we do not experience the world at that level, and it is a rare conversation that refers to those events. Of course, there are parts of our environment thai ipp OXhnaie the idealized behavior of theoretical random processes; events in casinos and lotteries are "caused" by deterministic physical processes, but the causal mechan.sui is so complex and the determinants of the events are so subtle that the best way to think about these situations is in terms of probability theory. An important message of this book is that we should use probability theory to organize our ihnking about all judgments under uncertainty, even where wc know much more lor less) about the relevant causes than we do in a casino. But we tend to deny the random components even in trivial events that wc know to be the result of chance. There is a wonderful story about the winner of a national lottery in Spain. When interviewed about how he won, the winner said that he had deliberately selected a ticket that ended with the numbers 4 and S. He explained, "I dreamed of the number 7 for seven straight nights. And 7 times " is 48" i.Mcislcr, 1977}. 7.2 Illusions of Control In a clever scries of experiments, Ellen Langcr 11975) of Harvard University demonstrated that—automatically, without any conscious awareness—wc often treat chance events as if they involve skill and are hence controllable. For example, gamblers tend to throw dice with greater force when they arc attempting to roll high numbers than when they are attempting to roll lower numbers. Langcr conducted a lottery in which each participant was given a card containing the name and picture of a National Football League player; *n identical card was put into a bag; and the person holding the card match-i'-S the one drawn from the bag won the lottery. In fact, Langcr conducted 'wo lotteries. In one, the participants chose which player would constitute eir ticket; in the other, players were assigned to the participants by the M2 Rational China in an '. '::•. fit..::: \W.I experimenter. Of course, whether or not the entrants were able lo choose their own players had no effect or. the probability or' th;-.: u inning the lottery, because the winning cards were drawn it random from the hag. Nevertheless, when an experimenter appci u, he d the p n lk p "its offering to buy their card, (host who had chosen their own player on the avenge demanded »;< 4 times as much money for their card as did those with randomly assigned cards. Upon questioning, no one claimed that being allowed to choosi player influenced h:s or her probability of winning. The participants just behaved as if it had. In another striking experiment, I .anger and Susan Roth 11975} were abk to convince Ya!c undergraduates that they were better or worse than the average person at predicting the outcome of coin tosses. The subjects were given rigged feedback that indicated they did not perform any better than .. chance level—that they were correct on 15 of 30 trials. What the experimenters did was manipulate whether the subjects tended to be correct tow 11 the beginning of the 30-trial sequence or toward the end. Consistent with a primacy effect [01 iiieli:ir:ng-and-|ituufficiejK|-adjustmenti, those subject who tended to be correct toward the beginning were apt to think of themselves as "better than average" at predicting, while those who did not do wt ' at the beginning judged themselves to be worse. I Of course, due to rant fluctuations, the probability of success in predicting the outcome of coin tosses cannot be \p . ted to be invariant across a sequence as short as 30 trials.) In addition, "over 25% of the subjects reported that performance would be hampered by distraction and 40% of all the subjects felt that performa-ce would improve with practice." Thu», noi or.;. d.. jn<••.*!.• Ivlme i> r'thev can control random events; they also express the conscious belief that doing so IS a skill, which, like other skills, i- hampered by distractions and improve*, with practice. It is important ro remember that these subjects were from one of the most elite universities in the world, yet they treated the prediction of coin tosses as if it involved some type of ability, not just dumb luck. Moreover, as with most everyday applications of psychology, practitioners like the managers of casinos and lotteries already have an intuitive understanding of these principles. Commercial games of chance often contain decepth (-I. II elements, deliberately designed to confuse the players about the skill and opportunity tor c< ■iinul inv.i'.vcJ in games of chattel In most st."' lottery players can choose the numbers tin;. bet then money on. and the lotteries often have skill-v.. 1 .....i >:;•!.<•: Hi: a home run and win Major League bucks." "Just by buying a Bowling for Bucks ticket, you're a winner." A more serious consequence of the illusion of control is revealed in i ur preference for driving over (lying. At least part of this irrational—from a survival point of view—habit is due to the fact that we "feel in control" when Chance and Cause M3 driving* but not when flying. The probability of dying in a cross-country flight is approximately equal to the probability of dying in a 12-mile drive— in many cases, the most dangerous part of the trip is over when you reach the airport iSivak &: Flamtagan, 2iNlsi. (ierd Gigcrcnzcr 120061 estimates that the post-9/11 shift from Hying to driving in the United States N..... an additional 1,500 deaths, beyond the original 3,000 immediate victims of the terrorist attacks. One of file must compelling studies of the illusion of control demonstrated that it was related to consequential, pour performance in a real-world investment situation. I British finance experts asked traders from four investment banks to play a computer game in which they attempted to influence the price of a fictional investment index (Fenton-O'Creevy, Nicholson, Sloane, & Willman, 2003). The movements of the index were complete!*. :ukpendent of the actions by the trader-players—it was a random walk with a slight positive trend. 1 he r.iders played the game for four rounds and rated their personal success in raising the index—because the index movements were independent of the actions of the traders, this is a measure of individual illusions of control. On average, the traders fell prey to the illusion that they had influenced the movement of the pi ice index. More interesting, (he level of individual illusions of control negatively predicted (he traders' earnings and their managers' ratings of ili. - nil irs and performance. Traders with a greater illusion of control earned substantially less than their more iv.i!i>' , |v.-s >l n'."i: M; they contributed less to their bank's profi(S; and their managers rated them lower on risk management, analytical ability, and people skills. 7.3 Seeing Causal Structure Where It Isn't A pernicious result of representative and scenario-based thinking is that they make us see struct m (itotuandotnness) where none exists. This occurs because our naive conceptions of randomness involve too much variation— often to the point where we conclude that a generating process in MOt random, even when it represents an ideal random trial. Consider one of the 1 nplest, most familiar processes we would describe as random, a coin toss, w ' n asked to "behave like a coin" and to generate a sequence of heads and tails th.it would be typical of the behavior of a fairly tossed coin, most people produce too much alternation—nonrandomly too many heads-tails and tails-heads transitions. . ["hey exhibit the same bias when shown sequences and asked to pick the "real coin" (Lopes. 1982).I Representativeness enters in because when we arc faced with the task of distinguishing between mh dorn and nnnrandom "generators" of events, we rely on our stereotype of a M4 Rational Choice in an Uncertain World random process :aualng:i.,% m ...n stereotype of a feminist or a bank tcllCf or an art history majon and use ibniijrity to judge or produce a sequence Thus, when we encounter a truly random sequence, we are likely to decide it is nonrandom became it does not look haphazard enough—bee.hi » jt shows lc» alternation :h.in (unpublished rescarchi discovered that the fit to the theoretical random prediction based on a constant p was almost perfect. Yet crashes seem to occur Chance and Cause 145 I bunches." Why? Because (1 -pY> \l-p\lp when / < k. Hence, truly ran Join sequences actually contain "bunches" of events. The problem is that representative thinking leads us to conclude that such random patterns are >:•.< r.ndom. Instead, we hypothesize positive feedback mechanises lUch as iKntum" to account for them. (Those of us hypothesizing "Jungian syn-Jimnicity" arc in a minority! W hie, for example, the maxim that "nothing mccceds like success or fails like failure" may be true, phony evidence for it can be (bund In the bunching of nxecsses m patterns of people or organizations with high probabilities ol success, and or failure) in those with high probabilities of failure—even when the pattern ii of independent i venti A well-defined situation in which people clearly see pattern* that are not in the data is the ban J phenomenon in basketball. I U: In: !■ o >■ d.'i-1, : merely refer to the fact that some players arc more accurate shooters than others, but to the I hypothetical I positive feedback performance pn>,vss that makes players more likely to score after scoring and to miss after missing. (Note that the same term—a hot hand—is used to describe successful u.ip shooters, despite general acknowledgment that in well-run games, they n, ot control the outcome of a roll.) Tom Gilovich, Robert Vallonc, ar.d Amos Tversky 11985) demonstrated empirically that the hot hand does not exist; that a success following a success is just as likely for an individual player a> a success following a failure. At least, neither the floor shots of the Philadelphia '76crs, the free throws of the Boston Celtics, or the experimentally controlled floor shots of men and women on the Cornell varsity basketball teams showed evidence of a hot hand. But the players' predictions of their success showed a hot-hand effect, even though their performance did not. More than 90% of a sample of basketball players and sports reporters answered •yes" to the following question: Does a player have a better chance of making a shot after having just made his last two or three shots than he docs after having just missed his last two or three shots? Jay Koehler and Caryn Conley (20031 followed up the original studies vrith an analysis seeking nonrandom patterns in the NBA Long Distance Shootout Contest from 4 years o: the competition. In this event, the best field goal shooters in the NBA attempt to score as often as possible within a 60-sccond time limit from the 3-point shot arc Ithe area of the court from which shots will count for 3 points instead of 2i. Again, there v. is no evidence 'I nonrandomncss. Even when the researchers conditioned their analysis on the announcers' assertions of "hotness," there were no patterns. It is notable that nonrandom streaks have been verified in other sports such as bowling, archery, billiards, and golf, suggesting that the statistics arc sensitive enough to pick up patterns if they arc there in the data. 1 It looks like there might be a bigger picture here: In nonreactive, uniform-playing-field sports, subtle 146 Rational Choke in an Uncertain World sequciui.il dependencies manifest themselves in pci1oiiiur.ee; ir. chaotic, in your-face, player-on-player reactive sports, there are no such patterns, i These studies do not prove the una trsal nonexistence of the hot hand in basketball (which would be difficult to do, if you think about it}, but their results imply thai il i: exists, il is small, unreliable, or very rare. The claim thai any particular si i or data is random is tenuous; il is more defensible to claim that a process that generates the data is random, in ihc sense that the observers of the data could not know the information neces in n: predict the events m the data with any degree of specificiry—that to these observers. :1k- beat description is a probabilistic or random process. The example of i I hot hand in hiskcthal! is cspeciallv surprising because it is so easy to imagine i causal process thar might generate the expected {but not obsen MJ • terns. For example, one reply to Gilovich ct al.'s 119851 and Tversky and Gilovich's 119891 original claim was that they had missed the true hot-hand pattern thar was hidden in their da:a because they had ignored the timing of baskets. Patrick l.arkey. Richard Smith, and Jay Kadane : I9S9' published a reanalysis consisting only of runs of shuts occurring in close temporal proximity. They found one player, Vinr.ir "Microwave" Johnson of the Detroit Pistons, who departed from the random model. Microwave earned his nickname because ol his reputation for streak (hooting. I lnvvevcr. Gilovich et al. (1985), in rebuttal, noted that the reanalysis found only one "hot" player, and thar h:s statistically distinctive srreakiness was due entirely to I lingli run ol seven baskets. Then they pointed out that a review of the original .;;-ot.ipcs d-io-.v, J t>-..-.: the v.-.t-ii :-.ivL-; n . • .ohm.. In Microwave had a run of four baskets, missed a shot but scored on his own rebound, and then made one more score. After correcting for this data collection error, even Microwave did not depart from the random model. I'i i good weeks 111 3 n:v. dkYVI I t.ipciltlC S'UCn " o ' 11 ,i pill.-.'.:: I 1 ' 3 bad weeks indicate failure lor, more sanguinely, "coming to face problems"!? Does losing three games in a row mean the coach should be fired? Or do three down quart)1 n ( in a CEO should be fined? No, no more than three heads in a row within a sequence of coin tosses that the coin is biased. Yet, knowing the person's base rate for success—and expecting more alternation than in fact occurs if these weeks or quarters are totally unrelated—makes the temptation to impute causal factors to such strings almost overpowering, especially causal factors related to the actor's own behavior. {Another speculation is this: Could it be that the perceptual salience :il "streaks" nl hits and misses is the key temptation to sec "hot" • • "cold" patterns in performance? In professional basketball where fans talk avidly about "hot hands," the success rate for shots is well over sDr., Mid so, runs of "hits" would be common and violate our expectation for too I Chance and Cause 147 many reversals (hit-miss and miss-hit transitions!. But consider baseball batting where the fans arc hkely to talk about "slumps" and where batting averages arc ail well below 50% so thai runs ol "misses" would be most salient.) Why do we e\|-. .: i< >u much alternation: Tversky and kahneman \ 1974> ascribe this expectation to the belief that even very small sequences must be representative of a population, that is, the proportion of events in a small frame must match—be representative of—the proportion in the population. When, for example, we are tossing a lair coin, we know that the entire population of possible sequences contains 50% heads: therefore, we expect 50% heads in a sample of four tosses. That requires mure alternation than is found when each toss .s ind< pei .k-nt. iAt the extreme, 50% heads in a sequence of two losses requires that each head is followed by a tail and vice versa.) Here, representative thinking takes us from schema to chaiactciistic, rather than the reverse. Again, however, the basic belief is due to similarity matching—that is, to association. Moreover, the effect is compounded by our relatively brief span of attention—we want the short sequences we can remember or imagine to be representative tKarccv, 1992). Consider the following question from a study In Tversky and Kahneman <1974): All fannies of six children in a city were surveyed. In 72 families, ihe exact order of births of boys and g:rl* was G B G B B G. What is your estimate of the number of families in which the exact order of births was B G B B B B? What about the number of families with the exact order B B B G G G? Almost everyone iiiO% or more of respondents! judges the latter birth sequences to be less likely than the first. However, all exact sequences arc equally likely (the probability of any exact sequence is simply .5 x .5 x .5 x ,5 x .5 x .5 or 0.015625, implying approximately 16 families out of a sample of 1,000 six-child families). Why do people have the strong intuition that G BG B B (j is much more frequent? Brc.uiic this s-:ni: sequence captures all ol nur intuitions about what the result ol a random process will look like: The sequence exhibits the correct proportion I hall boys, half girlsi, it looks haphazard, and it has lots of alternation—in short, it looks "really random " ilt is also the kind of sequence of hits and misses we would expect an ordinary basketball player to generate—too many short alternating runs, so that when we see a performance with longer runs, we arc prone to soy. "That can't be random. This player must really be 'hot.'"i In contrast, the second sequence looks less likely because it violates the law of small numbers by having the wrong ratio of births |ioo many boysj, while the third sequence is okay for proportion, but looks too orderly ithrcc in a row-, then tluei in a rowi. N8 Ration*! Choice In an Uncertain World Occasionally, this belief in alternation in random sequences ;thc gambler's fallacy that "red is due" because the last 6 outcomes on the roulette wheel were black 1 reaches ludicrous extremes. Consider, for example, the beginning of a "Dear Abby" letter: 11 \[{ v !•..••) hi. i i ' I just had our eighth child. Another girl, and I am really one disappointed woman. I suppose I should thank God that she wa» healthy, bi:t, Ahby, this one was sapposed tu have been .1 \\>:. I •.<.- ihr .'<«:• 1 m!c 1 n l.iw i.- .i\c-i.-;ix v-.u in u.ir tavu.' 00 tu one. A "graphic" example of the tendency to sec patterns (and infer caiisesi where there surely weren't .my occurred during the World War II bombing of London by German V-l and V-2 missiles. London newspapers publil maps of the missile impact sites isee Figure 7.1 *, and citizens immediately mu clusters of strikes and interpreted them with :«fi pence to the intentions of the hostile forces. What kind of stories did they tell to explain these patterns? The British citizens reasoned that the patterns they saw were the result of deliberate efforts to miss the areas of the city in which German spies lived. However, a classic probability modeling analysis demonstrated thar the clusters were completely consistent with a random Poisson process-generating device, that there was no reason to infer a systematic motive or cause behind the pattern! (see William Feller's classic texthook. An Introduction to Probability Theor) and Its Applications, Vol. 1, pp. 160f, for a mathematical analysis). A t.mcly example of this tendency to infer causes for geographic patterns is part of the psychology of "unai duster" hysterias. During the past two decades, report! it 1.. immunities in which there seem to be an unusual 11. tuber of cancer incidents li r,, .. J isee Gawande. 1999|. A community that notices an unusual number of cancers quite naturally looks for a cause in the environment—something in the water or the ground or the air. But investigating isolated neighborhood cancer clusters is almost always an exercise in futility. Public health .1.4.ncies deploy thouvin,!. : l " 1 p; :f *:u.li-> every year in response to reports of raised local cancer rates. But Raymond Richard Neutra, C alifmnia's chief environmental health investigator |in 1999), notes that among the h.n.l . .1- it p.:MM-id reports of such investigations, not (WW has convincingly identified an environmental cause icited in Gawande, 1999i. And only one investigation resulted in the discovery of an unrecognized «. arcir.ogcn. Neutra points out that in a typical Public Health Service registry of 80 different cancers, probability theory predicts you would expect to observe 2,75(1 of < llifom »'l $,000 census tracts to have statistically significant I - • it random elevations of some form of cancer. So, if you check to see it \mu neighborhood has a statistically signifkrant elevation in the rate of at least I of the 80 cancers, the chances arc better than .50 it Chance and Cause 149 KEY V-1 Fy-g Bomb Incidents V-2 Focm-bolkally important events and discovering many Mise correlations between clusters and their contexts. The strategy of analyzing individual clusters and looking for correlations with some [an) | environmental cause is called the Texas sharps!: -.Vr f.i!!ar. ly epidemiologists, after the story about a rifleman who shoots a cluster of bullet holes in the side of a barn and then draws a bull's-eye around the holes. This is a ease where we should go with the advice of statistically sophisticated experts and only respond when there are ISO Rational Choice in an Uncertain World Chance and Cause 151 good a priori reasons to hypothesize an environmental cause, or then u truly cxtr.iordin.uy itarjstical patterns. The much-publici/vd case of the cancer cluster in Woburn, Massachusetts, described in the book .ind movie A Civil Action, was never resolved by the identification of a Scientifically credible causal pathway relating the po'.iut.tnts fro:- the Riley Tannery to the incidences of cancel in the neighborhood surrounding the factory. 7.4 Regression Toward the Mean A lui.il pr.••>!;— v.ii -t|'n.Mi mm i'--ink:tig about events with a random (unknown causes) component is that it leads to non-rcgre^>i>« pn did i iu To understand why. it is necessary first to undei>t.md regressive prediction. Consider VCTJ tall fathers. On the average, their sons are tall, but about in inch shorter than their fathers. Also, the fathers of very tall sons arc on the average shorter than their sons. Examine the vert ( ll solid lir.c representing Tall Fathers in Figure 7.2. The average son's height for tall fathers is ir.dic.iti .1 by tracing the horizontal broken line labeled "average for tall fathers'* to the ordinate—the y-axis—in the graph. iThe horizontal line is slightly hi/ than tin- midpn.ni of the vertical line between the top and bottom edges •>: the ellipse representing "the da:.i" \\\ .uise the disirihurion of sons' heights on that vi.nu.il J •••.-si:i:i is probably not exactly symmetric, but is likely tn have a longer tail downward toward shorter sons' heights.!1 Tracing this path for a typical "Tall Father" simply works through the logic of ider.tifyir.i; mean height for sons of such fathers and shows that the mean "regresses"— that is, it is less extreme than the extreme father's height. 1 he difference between D and d' is an index of the degree of regression for this data set. Fxactly the vim.-. .-.i . u ...\t:.rn in reverse is revealed if we work from Tall Sons, following the horizontal solid line for a typical Tall Son and tracing the vertical broken line path downward to the abscissa—the x-axis—for the average father's height for Tail Sons. The British scientist Sir Francis Gabon •1886! was the first to not-.a this tela tionship. which he lain:led "filial regression towards mediocrity" tp. 246!'. At litst, he thought the relationship was the result of sorr.e genetic process thai made organisms shift toward average attributes, but after considering the reverse relation- i.< b . kward in t imes he concluded it was a statistical property of all i relational relationships. Hie relationship is illustrated in Figure ".2. What you see is a simple averaging effect. Because the heights of fathers and sons are not perfectly correlated (for whatever reasonsi, there is regression. Sion-regresshv prediction refers to people's tendency to miss the subtle regression relationship and to predict that extreme values will be associated with too-extreme values— as we will see :n a moment. ~ean of Fa,"0,s fathers' heights figure ?.2 lla.-n .• <>-,, i ..il regression Consider another example (based on the work of Qumn McNemar (1940), a psychologist who was one of the first to point out this statistical result and ik implications for research on human behavior); Suppose that an intelligence test is administered to all the children in an orphanage on two occasions, a year apart. Assume, plausibly, that the group mem and standard deviation are the same on both tests; but that the correlation between scores on the two tests i- not perfect ithc actual correlation would be about -t-.SOl. Now consider only the children with the highest scores on the first test: Their scores on the WOOttd test will he on average lower. iSmcc the correlation is below +1.00, we expect Some change; since the two distributions of stores are the s.vne. the first test high scorers must get lower scores on average. The same was true for the children with the lowest scores: The average of the lowest-scoring children on the 152 Rational Choice m .in '. . 1u-.-.-. WW.A Chance and Cause 153 hot u>: will be higher on the second. What it we reverse time and look backward from the Second to the first test? I lit- same relationships will apply: Extreme scores will be less extreme. Regression toward the mean .s Inevitable for Kaled variables that are not perfectly correlated. Perhaps u is easiest to understand regression hy considering the extreme case in which we obtain perfect regression. Toss a fair coin 8 times; now toss it another 8 times. No matter how many heads are obtained in the ftrsi sequence of losses, the expected [average) number of heads in the second sequence is 4. Because the coin is fair, the number or heads in the first scqccr.ee is totally uncorrected with the number in the second—hence, average, of 4. Tha: is total regression to the mean. As variables become more predictable from each other, there is less regression; fur example, on average, the son. of very tall fathers are taller than the average person, but not as ta'.i as their fathers. It is only when on; variable is perfectly predictable from the other that there is no regression. In fact, the (squared valne of the) standard correlation coefficient can be defined quite simply as the degree to which a l:n-eai prediction of one variable from another is no: regressive. 'Ihe technic.: definition of regression toward the mean is the difference between a perfeci relationship (+M.00) and the linear correlation: regression = perfect relationship - correlation There are many examples of failure to appreciate regression toward the mean in everyday judgments. We arc constantly surprised when an exceptional performance on Wall Street, a hit movie, a #1 pop song, or a sports achievement is followed by something mnu- mediocre. The Sports tibutrattd t. ft f/M.v is one of the classic examples. Readers noticed that when an athlete or a team was featured on the cover of 5porrs Illustrated, always tor some exceptional achievement, the individual or ream was likely to experience a slump in performance or some other misfortune afterward. Statistical analy-s s stned to reinfi Id thl im «rsv, m, m. I in- umu.I nv.-y ;>'m> blc explanations for the phenomenon—the athlete became overconfident because of the publicit), the athlete was distracted by the media attention, and so forth. Of course, we knnw that mošt if not all of "the effect" v. ,i% J í to selecting extreme cases and observing regression toward the mean. No special explanation beyond noting "selection for exceptionality" i- needed A classic academic example is provided by Horace Secrist's 1933 book. Tht Triumph of Mediocrity in Business. Secrist's thesis was that successful and onsucci >sful businesses "tend towards medtocrity," [*he tht tis b supported by hundreds of graphics shuwin^ ill n v. Inn |> ..j-,nm . are selected in Year 1 for exceptional performance, on average the most successful become less successful and the least successful become more successful. Howard Hotelling, a prominent statistician, cu::-1n. ii ..-. I u -.-ui, n,: s.. nee is a statistical fallacy, resulting from the method of grouping. These diagrams really prove nothing more than that the ratios in question have a tendency to wander about." He points out that the true test of convergence toward mediocrity would be i consistent decrease in the variance among the groups over tunc— which was not observed This same mistake was manifested in Tom Pctcrs's ii d Robert Waterman's 1984 best-seller In Search ofExceQena. These management consultants selected 43 exception.:!', successful .••••ip. • and reviewed the cUiinui'..- Uatun thai they believed made them "excellent." But. 5 years later, BusinessWeek's cover story, "Oops! Who's Excellent Now?" pointed out that over one-third of the original, sampled-becausc-thcy-werc-extrcme companies were in financial difficulty or bankrupt. In many cases, wc are interested in the effects of some treatment on performance—an educational enrichment treatment for low-performing schoolchildren, bonuses for high-performing employees, a dietary Supplement for the least healthy. Again, there is a problem of separating the true effects of a treatment, applied only to extreme cases, from simple regression. Some of t In subsequent errors can be quite subtle. For example, when Daniel Kalmeman iTversky 8c Kahncman. 1974;. was explaining to Israeli Defense (■ice flight instructors in the mid-1960s that reward is a better motivator than punishment, he was told by one instructor that he was wrong. With all due respect, Sir, what you are saying is literally fur the birds. I've often praised people warmly for beautifully executed maneuvers, and the next tunc they almost always do worse. Ar.f I've s.ic.uicd at pup.U tor badly executed rnaneviver», and hy and luxe, the nevt tar.r rhey -.ir.pn .<. i't :-l tr.e th.il reward works and punishment doesn't. My experience contradict* it. This flight instructor had witnessed a regression effect. People tend to do worse after a "beautifully executed maneuver" because performance u one time is not perfectly correlated with performance the next lagain, for whatever reason:. Performances also tend to improve each time artcr "badly executed maneuvers"—once more, simply becaUM performance is not perfectly correlated from one occasion to the next. (The easiest way to obtain an award for "academic improvement" is to be right near the bottom of the class the semester prior to the one for which such awards are given, and the way to be labeled an "underachicver" is to score brilliantly on an aptitude test.I Unfortunately, as the flight instructor anecdote illustrates, teachers who do not understand regression effects may be s\»tun-atically reinforced I by regression to better performance i for punishing students and disappointed tby regression to worse performance) for rewarding them. I Regression alone may be a sufficient explanation for 154 Rational Choice In an Uncertain World Chance and Cause 155 some people's preference, like the flight instructor's, !m punishment n\: reward as a means of behavior control.I 11' i unhappy byproduct of our ignorance of the inevitability of regression effects is our overconfidence in the success of interventions like firing coaches and CEOs. Consider the prototypical situation: A team performs poorly during the first half of the season. The owner reacts by firing then I, and the team performs better during the second half of the sea sun. Should we attribute the improvement to the firing and replacement of the coach or to simple regression effects? After all, mid-season firings are usual!) condhiotied i an extreme, poor performance. Absent an experiment in which coaches stt i mi i m!y fired, we cannot be sure (and such an experiment is unlikely to be performcdl. But careful statistical analyses consistently show that most of the improvement is due to regression iKomng. 2003), and the same is true tor the firing of busii i »•> exe< utrves, ; H» tea ity in sports is that, if a team performs extremely poorly during the first half of the season, it is likely to have been pitted against stronger teams, and the second half will involve weaker oppur.er.ts. exaggerating the apparent success of the replacement coach even furthei. The rat.on.il way o: dealing with regression effects is to "regress" when making predictions. Then, if there is some need or desire to evaluate discrepancy leg., to give awards tor "overachieverncnt" or therapy for "und?.'-achievement"), compare the actual value to the predicted value—not with the actual value of the variable used to make the pred ctton. For examp i. to determine patient "improvement" by comparing Minnesota Mmtiphas:, Personality Inventory [MMPI] profiles at time 1 and time 2, firs: correl in i profiles to determine a {regressed) predicted score for each patient a* rime 2; then compare the actual profile with this predicted score, not with the score a: time 1. Otherwise, patients who have high i.paihnlomcali profiles at time 1 may be mistakenly labeled "improved" !"thcy had nowhere to go but down" i, while those with normal MMPI profiles may be mistaken!) regarded as unresponsive to treatment. Representative thinking, in contrast, leads to comparing discrepancies without regress ng r'.ist, and rhc results are pre dictable. For example. "Of particular significance was the fact that those scoring highest on symptom reductions .. . were those whose symptom-. 11 initially more severe, and who were the less promising candidates for conventional types of therapy" (Dawes. 1986t. i While Dawes was a clinical ps\ chologist trainee, he asked the psychologists and psychiatrists at the hospital to dichotomize patients whose improvement was above • >> I age at discharge and those whose improvement was below average. Those they categorized as above average in improvement had higher scores on most of the MMPI scales on admission—significantly higher on the major clinical ones i Regression toward ihc mean is particularly insidious when we are crying to assess the success of some kind of intervention designed to improve the state of affairs—like the flight instructor's efforts to improve student pertor-mance by intervening to punish poor performances. The worst case scenarios for understanding the effects of interventions occur when the intervention is introduced because "we've got a problem." For instance, it is almost impossible li> aa ii sscss the causal effects of the introduction of a strict traffic enforce mcnt program after a flurry of tragic traffic accidents, or the hiring of a new ( HO after several poor corporate performances, or the hiring of a new coach after a losing streak. The chances arc, the interventions are going to show improvements, and it is almost certain that some or most of the effect will be due to regression toward the mean. 7.5 Reflections on Our Inability to Accept Randomness Some of the errors in judgment we have just described are pro\il-b noi so Surprising. Why would we be smarter than casino operators who have spent hundreds of years perfecting diabolical probability games to trap unwary customers? Or why wouldn'r sports fans confuse conditions under which Streaks do occur (in some sports events) with similar situations in which they do not? But the pervasive tendency to sec much more structure than is i. tu-ally present and to imagine we have much more control over events than we do in hundreds of important naturally occurring situations is still a puzzle. In the next chapter, we'll introduce the best remedy we know for these hard-to-cradicate bad habits—thinking like a probability theorist. References Bayer. D.. \- l>i;...i <., I' |sv2 •. Ii: '• a the J- eetai slu.ttk- m its L:r. ,'t >.••.•..'.- Applied Probability. 2. 29-1-313. Bilefsky. D. i200S, April 26'.. Serbia's most fantous survivor feats th.ii recent I; •: i . | I repeat itself. Sen- Yurie Timer. Retrieved Jane 20. 2009. from hnpi'.'seww.nytimes ^o«V20CiS.t)4/2f.'worlii'curope/26suIos-k.html Dawes, R. M. 119861. Representative thinking in clinical judgment. Clinical Psycholoyy Review, 6, 425-141. Diaconis, P., Holmes. S.. ic Montgomery, R. 1200*1. Dynamical bias in the com t; Anthr.&■:.>! i'/i-fi'.Vi.v <".»,.:■• Britain and Inland, )\ 246-26 1. Cuwar.de. A. 11999. February 8l. The canccr-clu-tter myth. Sew Vorter, pp. 34»37. Gigerenzer, C. i2006l Out i»f the fry jig pan into the fire: behavioral reactions to tcr- rorist attacks. Risk Analysis, 26, 347-351. Gibvich. T.. Vallor.e. R., fic Tversky, A. :19S5.». The hot hand in ba*kctha!l: On the misperccption of random sequences. Cognitive Psychology, 17. 295-314, Hotelling. H. i!933l. Review of TXv Triumph . * M.-dn..vi.';. in Business. Journal,„•" the AimHCMM ftrtfttfftl1 AHOdttiOH, 28. 463-465. Karcev. Y. 119921. Nut that Kid after ali: Generation of random sequences./onz«.-.i.' Experiments! PsycMtgy: Petceplt-.m and Per,' -ruunce. IS. 1189-1194. Koehler, J. J., & Cor.ley, C. A. (20031. The "hot hand" myth in professional basketball, journal of Sport & Exercise Psychology. IS, 253-259. Kor.ing. R. i20O3;. An econometric evaluation of the effect of firing a coach on team performance Applied Economies, SS, 555-564, Langer, H. J. Il9"".si The d'nsion of COftttoL Journal of Personality and Social Psychology, 32. 311-328. Langer, L J.. Sc Roth. J. • li'H: Head* I wir., tail* chance: The illusion of control is a function of the sequence of outcomes in a pmely chance task. Journal •■' Perso'ulityrmtd So.;...' I'sxciotogy, 32, 951-955-Larkcy, P. D„ Smith, R. A.. 8c Kadanc. J. B.. 1989 . \:\ okay to believe i- tl :• I .t hand." Chance. 2(4), 22-30. Lope*. II. 1^321. Doing the iwyonibfe: A note on indution and the experience ol nth dom.ness. journal of Experimental Psychology: learning. Memory, and Cogn:t>-«;. 8,626-636. M.N'amar, Q. 11940. A critical examination of the U:i •.«..-% ty of Iowa studies of environmental influences on II}. Psychological Bulletin. IS. 63-92. Meisler, S. (1977, Decembct 30;. Spain lottery—Not even war stop« it. Los Angeles ! wm*, p. 1)1 Oops: Vt'ho's cxieilent now? ;1984. November 5.'. BusinessWeek. 76-88. Peters. T, X Waterman, R„ Jr. I9Ü4 ,'»> '..:»nt-\l tn.rld: Science ->s a candle in the dark. Htm York: Ballar.tine. Secrist. H. 11933:. The triumph of med;■-<-. nly tn business. Chicago: Bareau of Business Research. Northwestern Ur.iveisitv. Sivak, M., Sc Flannagan. M. J. (2003). Flying and driving after the September 11 attacks. AmerUan "iewmtst. <• s. Tversky, A., 5c Gilovkh. T. i 1989;. The "hot hand": Statistical reality or cognitive iI1l»:imi. Chance, 2 4:, 31-34. Tversky, A. Äc kahneman, |>. i19"4i. Judgment under uivertvnty: llcaristics and biases. Science. 115, 1124-1131. 8 Thinking Rationally About Uncertainty The actual science of hgic i> out ersant at present only with things either certain, or impossible, or entirely dwalnful, of' which ifortunatelyl we hare to reason on. Therefore the true logic for this world h the Calculus of Probabilities. U bh >> takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind. —James Clerk Maxwell 8.1 What to Do About the Biases Ulysses wisely had himself chained to his ship's mast before corning within earshot of the Sirens. He did so not because he feared the Sirens per sc. bur because he feared his own reaction to their singing. In effect, he took a precaution against himself, because he knew what he would be likely to do if he heard the Sirens. Vm . ., ilu- ..iigr.it ive biases of automatic thinking can lead us astray, in a predictable direction. We must take precautions to avoid the pitfalls dI such unexamined judgment. One of the goals of this book is to teach analytical thinking about judgment processes. The best way we know to think systematkally about judgment is to learn tb.;- ftr .1 in en. a Is nl probability theory and statistics and to apply those l.i:-..i-p:- win n in iktttg import or jin ..nn-n:-. 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