Tversky, Amos; Eldar Shafir (ed);
Preference, Belief, and Similarity: Selected Writings
MIT Press (Bradford books), 2004, 1023 pages
ISBN 026270093X, 9780262700931
topics: | psychology | cognitive |
When I play squash, there are days when I find that every shot becomes a winner, I am invincible. Many others have had this feeling. We justify it by saying that the mind wields a lot of power on our body. But is it really true? (Of course this "feeling" can't build up against much stronger players). Tversky and Gilovich show - quite convincingly, that at least for basketball - that this may be a cognitive illusion. The second excerpt below presents this debate : does an athlete have periods where he is "hot" - where anything he does, just works? Particularly in basketball, there is a belief in "hot hands" - among audiences, coaches, and players. But Tversky finds that this is a cognitive illusion - possibly caused by the fact that moments of success are more salient - rather than any underlying reality. What is interesting is that this initial work was challenged by an alternate approach who produced differing statistics based on limiting the time duration over which a "hot hand" even is detected. But in the end, Tversky and Gilovich show that even for the new data, there is no strong correlation between a past set of scores and a new shot. As with any random process it may happen that a player gets nine shots out of ten in a row. However, they show that this sort of series would arise in any probabilistic process - and that there is no correlation between success in previous shots and success in the next. However, as they clarify in their analysis, while the probability of success does not change much, perhaps the player is more energized into attempting more shots than he would have otherwise. As a result, a player may score more points in one period than in another not because he shoots better, but simply because he shoots more often. The absence of streak shooting does not rule out the possibility that other aspects of a player’s performance, such as defense, rebounding, shots attempted, or points scored, could be subject to hot and cold periods. In later work, Tversky identified the clustering illusion as a tendency to erroneously find "streaks" or "clusters" in small samples from random distributions. This is a result of a human tendency to underpredict the amount of variability likely to appear in a small sample of random or semi-random data.
This is but one of the many psychological aberrations that Tversky has ferreted out over a relatively short career. (d. 1996, at age 59) Tversky is one of the giants of cognitive psychology. His work has debunked many myths about how humans function, and has led to new areas such as behavioural economics and decision theory. His work underlined the role of the subconscious in modifying deliberative models that we think are the main wellsprings of behaviour. His work has identified many crucial "psychological tendencies and processes that intrude upon and shape behavior, independently of any deliberative intent." (p.xi)
full title: Extensional vs. Intuitive Reasoning: The Conjunction Fallacy in Probability Judgment p. 221 Amos Tversky and Daniel Kahneman
Our studies of inductive reasoning have focused on systematic errors because they are diagnostic of the heuristics that generally govern judgment and inference. In the words of Helmholtz (1881/1903), "It is just those cases that are not in accordance with reality which are particularly instructive for discovering the laws of the processes by which normal perception originates." The focus on bias and illusion is a research strategy that exploits human error, although it neither assumes nor entails that people are perceptually or cognitively inept. Helmholtz’s position implies that perception is not usefully analyzed into a normal process that produces accurate percepts and a distorting process that produces errors and illusions. In cognition, as in perception, the same mechanisms produce both valid and invalid judgments. Indeed, the evidence does not seem to support a "truth plus error" model, which assumes a coherent system of beliefs that is perturbed by various sources of distortion and error. Hence, we do not share Dennis Lindley’s optimistic opinion that "inside every incoherent person there is a coherent one trying to get out," (Lindley, reference note 3) and we suspect that incoherence is more than skin deep (Tversky & Kahneman, 1981). It is instructive to compare a structure of beliefs about a domain, (e.g., the political future of Central America) to the perception of a scene (e.g., the view of Yosemite Valley from Glacier Point). We have argued that intuitive judgments of all relevant marginal, conjunctive, and conditional probabilities are not likely to be coherent, that is, to satisfy the constraints of probability theory. Similarly, estimates of distances and angles in the scene are unlikely to satisfy the laws of geometry. For example, there may be pairs of political events for which P(A) is judged greater than P(B) but P(A/B) is judged less than P(B/A)—see Tversky and Kahneman (1980). Analogously, the scene may contain a triangle ABC for which the A angle appears greater than the B angle, although the BC distance appears to be smaller than the AC distance. The violations of the qualitative laws of geometry and probability in judgments of distance and likelihood have significant implications for the interpretation and use of these judgments. Incoherence sharply restricts the inferences that can be drawn from subjective estimates. The judged ordering of the sides of a triangle cannot be inferred from the judged ordering of its angles, and the ordering of marginal probabilities cannot be deduced from the ordering of the respective conditionals. The results of the present study show that it is even unsafe to assume that P(B) is bounded by PðA&BÞ. Furthermore, a system of judgments that does not obey the conjunction rule cannot be expected to obey more complicated principles that presuppose this rule, such as Bayesian updating, external calibration, and the maximization of expected utility. The presence of bias and incoherence does not diminish the normative force of these principles, but it reduces their usefulness as descriptions of behavior and hinders their prescriptive applications. Indeed, the elicitation of unbiased judgments and the reconciliation of incoherent assessments pose serious problems that presently have no satisfactory solution (Lindley, Tversky & Brown, 1979; Shafer & Tversky, reference note 4). The issue of coherence has loomed larger in the study of preference and belief than in the study of perception. Judgments of distance and angle can readily be compared to objective reality and can be replaced by objective measurements when accuracy matters. In contrast, objective measurements of probability are often unavailable, and most significant choices under risk require an intuitive evaluation of probability. In the absence of an objective criterion of validity, the normative theory of judgment under uncertainty has treated the coherence of belief as the touchstone of human rationality. Coherence has also been assumed in many descriptive analyses in psychology, economics, and other social sciences. This assumption is attractive because the strong normative appeal of the laws of probability makes violations appear implausible. Our studies of the conjunction rule show that normatively inspired theories that assume coherence are descriptively inadequate, whereas psychological analyses that ignore the appeal of normative rules are, at best, incomplete. A comprehensive account of human judgment must reflect the tension between compelling logical rules and seductive nonextensional intuitions. 1. Lakoff, G. Categories and cognitive models (Cognitive Science Report No. 2). Berkeley: University of California, 1982. 2. Beyth-Marom, R. The subjective probability of conjunctions (Decision Research Report No. 81–12). Eugene, Oregon: Decision Research, 1981.
Amos Tversky and Thomas Gilovich You’re in a world all your own. It’s hard to describe. But the basket seems to be so wide. No matter what you do, you know the ball is going to go in. — Purvis Short, of the NBA’s Golden State Warriors This statement describes a phenomenon known to everyone who plays or watches the game of basketball, a phenomenon known as the "hot hand." The term refers to the putative tendency for success (and failure) in basketball to be self-promoting or selfsustaining. After making a couple of shots, players are thought to become relaxed, to feel confident, and to "get in a groove" such that subsequent success becomes more likely. The belief in the hot hand, then, is really one version of a wider conviction that "success breeds success" and "failure breeds failure" in many walks of life.
[again, a sample of 100 avid basketball fans from Cornell and Stanford; and also the players from Philadelphia 76ers.] Does a player have a better chance of making a shot after having just made his last two or three shots than he does after having just missed his last two or three shots? Yes 91% No 9% When shooting free throws, does a player have a better chance of making his second shot after making his first shot than after missing his first shot? Yes 68% No 32% Is it important to pass the ball to someone who has just made several (2, 3, or 4) shots in a row? Yes 84% No 16% Consider a hypothetical player who shoots 50% from the field. What is your estimate of his field goal percentage for those shots that he takes after a) having just made a shot? Mean : 61% b) after having just missed a shot? Mean : 42%
One reason for questioning the widespread belief in the hot hand comes from research indicating that people’s intuitive conceptions of randomness do not conform to the laws of chance. People commonly believe that the essential characteristics of a chance process are represented not only globally in a large sample, but also locally in each of its parts. For example, people expect even short sequences of heads and tails to reflect the fairness of a coin and to contain roughly 50% heads and 50% tails. Such a locally representative sequence, however, contains too many alternations and not enough long runs. This misconception produces two systematic errors. First, it leads many people to believe that the probability of heads is greater after a long sequence of tails than after a long sequence of heads; this is the notorious gamblers’ fallacy. Second, it leads people to question the randomness of sequences that contain the expected number of runs because even the occurrence of, say, four heads in a row—which is quite likely in even relatively small samples—makes the sequence appear non-representative. Random sequences just do not look random. Perhaps, then, the belief in the hot hand is merely one manifestation of this fundamental misconception of the laws of chance. Maybe the streaks of consecutive hits that lead players and fans to believe in the hot hand do not exceed, in length or frequency, those expected in any random sequence. To examine this possibility, we first asked a group of 100 knowledgeable basketball fans to classify sequences of 21 hits and misses (supposedly taken from a basketball player’s performance record) as streak shooting, chance shooting, or alternating shooting. [Finds that] people perceive streak shooting where it does not exist. The sequence of p(a) = .5, representing a perfectly random sequence, was classified as streak shooting by 65% of the respondents. Moreover, the perception of chance shooting was strongly biased against long runs: The sequences selected as the best examples of chance shooting were those with probabilities of alternation of .7 and .8 instead of .5. What is even more interesting than this quantitative study are the answers by the fans. (above) If a player is occasionally "hot," his record must include more high-performance series than expected by chance. The numbers of high, moderate, and low series for each of the nine Philadelphia 76ers were compared... The results provided no evidence for non-stationarity or streak shooting as none of the nine chi-squares approached statistical significance.
To summarize what we have found, we think it may be helpful to clarify what we have not found. Most importantly, our research does not indicate that basketball shooting is a purely chance process, like coin tossing. Obviously, it requires a great deal of talent and skill. What we have found is that, contrary to common belief, a player’s chances of hitting are largely independent of the outcome of his or her previous shots. Naturally, every now and then, a player may make, say, nine of ten shots, and one may wish to claim — after the fact — that he was hot. Such use, however, is misleading if the length and frequency of such streaks do not exceed chance expectation. Our research likewise does not imply that the number of points that a player scores in different games or in different periods within a game is roughly the same. The data merely indicate that the probability of making a given shot (i.e., a player’s shooting percentage) is unaffected by the player’s prior performance. However, players’ willingness to shoot may well be affected by the outcomes of previous shots. As a result, a player may score more points in one period than in another not because he shoots better, but simply because he shoots more often. The absence of streak shooting does not rule out the possibility that other aspects of a player’s performance, such as defense, rebounding, shots attempted, or points scored, could be subject to hot and cold periods.
Larkey, Smith, and Kadane (1989)]. summary by Eldar Shafir. This critique proposes "a different conception of how observers’ beliefs in streak shooting are based on NBA player shooting performances." Tversky and Gilovich, they suggest, are taking isolated individual-player shooting sequences, whreas the observers in a real game would be seeing "how that player’s shooting activities interact with the activities of other players." For example, LSK propose that two players both with five consecutive field goal successes will be perceived very differently if one’s consecutive successes are interspersed throughout the game, whereas the other’s occur in a row, without teammates scoring any points in between. For their revised analyses, LSK devise a statistical model of players’ shooting behavior in the context of a game. They find that Vinnie Johnson — a player with the reputation for being "the most lethal streak shooter in basketball" — "is different than other players in the data in terms of noticeable, memorable field goal shooting accomplishments," and reckon that "Johnson’s reputation as a streak shooter is apparently well deserved." "Basketball fans and coaches who once believed in the hot hand and streak shooting and who have been worried about the adequacy of their cognitive apparatus since the publication of Tversky and Gilovich’s original work," conclude LSK, "can relax and once again enjoy watching the game." Reference Larkey, P., Smith, R., and Kadane, J. B. (1989). "It’s Okay to Believe in the Hot Hand," Chance, pp. 22–30. (from Editor’s Introductory Remarks to Chapter 11)
Amos Tversky and Thomas Gilovich Myths die hard. Misconceptions of chance are no exception. Despite the knowledge that coins have no memory, people believe that a sequence of heads is more likely to be followed by a tail than by another head. ... Larkey, Smith, and Kadane (LSK) challenged our conclusion [that the probability of hitting a shot is not higher following a hit than following a miss... and that belief in the "hot hand" or "streak shooting" is a cognitive illusion.] Like many other believers in streak shooting, they felt that we must have missed something... LSK collected a new data set consisting of 39 National Basketball Association (NBA) games from the 1987–1988 season and analyzed the records of 18 outstanding players. LSK's [analysis focuses on] "cognitively manageable chunks of shooting opportunities" on which the belief in the hot hand is based. Their argument confounds the statistical question of whether the hot hand exists with the psychological question of why people believe in the hot hand — whether it exists or not. We shall address the two questions separately, starting with the statistical facts. LSK argue, in effect, that the hot hand is a local (short-lived) phenomenon that operates only when a player takes successive shots within a short time span. By computing, as we did, a player’s serial correlation for all successive shots, regardless of temporal proximity, we may have diluted and masked any sign of the hot hand. The simplest test of this hypothesis is to compute the serial correlation for successive shots that are in close temporal proximity. LSK did not perform this test but they were kind enough to share their data. Using their records, we computed for each player the serial correlation r1 for all pairs of successive shots that are separated by at most one shot by another player on the same team. This condition restricts the analysis to cases in which the time span between shots is generally less than a minute and a half. The results, presented in the first column of table 11.1, do not support the locality hypothesis. The serial correlations are negative for 11 players, positive for 6 players, and the overall mean is -.02. None of the correlations are statistically significant. The comparison of the local serial correlation r1, with the regular serial correlation r, presented in the second column of table 11.1, shows that the hot-hand hypothesis does not fare [sic fair] better in the local analysis described above than in the original global analysis. (Restricting the local analysis to shots that are separated by at most 3, 2, or 0 shots by another teammate yielded similar results.)
Introduction and Biography ix Sources xv
1 Features of Similarity 7 Amos Tversky 2 Additive Similarity Trees 47 Shmuel Sattath and Amos Tversky 3 Studies of Similarity 75 Amos Tversky and Itamar Gati 4 Weighting Common and Distinctive Features in Perceptual and Conceptual Judgments Itamar Gati and Amos Tversky 5 Nearest Neighbor Analysis of Psychological Spaces 129 Amos Tversky and J. Wesley Hutchinson 6 On the Relation between Common and Distinctive Feature Models 171 Shmuel Sattath and Amos Tversky
7 Belief in the Law of Small Numbers 193 Amos Tversky and Daniel Kahneman 8 Judgment under Uncertainty: Heuristics and Biases 203 Amos Tversky and Daniel Kahneman 9 Extensional vs. Intuitive Reasoning: The Conjunction Fallacy in Probability Judgment 221 Amos Tversky and Daniel Kahneman 10 The Cold Facts about the "Hot Hand" in Basketball 257 Amos Tversky and Thomas Gilovich 11 The "Hot Hand": Statistical Reality or Cognitive Illusion? 269 Amos Tversky and Thomas Gilovich 12 The Weighing of Evidence and the Determinants of Confidence 275 Dale Gri‰n and Amos Tversky 13 On the Evaluation of Probability Judgments: Calibration, Resolution, and Monotonicity 301 Varda Liberman and Amos Tversky 14 Support Theory: A Nonextensional Representation of Subjective Probability 329 Amos Tversky and Derek J. Koehler 15 On the Belief That Arthritis Pain Is Related to the Weather 377 Donald A. Redelmeier and Amos Tversky 16 Unpacking, Repacking, and Anchoring: Advances in Support Theory 383 Yuval Rottenstreich and Amos Tversky
Probabilistic Models of Choice 411 17 On the Optimal Number of Alternatives at a Choice Point 413 Amos Tversky 18 Substitutability and Similarity in Binary Choices 419 Amos Tversky and J. Edward Russo 19 The Intransitivity of Preferences 433 Amos Tversky 20 Elimination by Aspects: A Theory of Choice 463 Amos Tversky 21 Preference Trees 493 Amos Tversky and Shmuel Sattath Choice under Risk and Uncertainty 547 22 Prospect Theory: An Analysis of Decision under Risk 549 Daniel Kahneman and Amos Tversky 23 On the Elicitation of Preferences for Alternative Therapies 583 Barbara J. McNeil, Stephen G. Pauker, Harold C. Sox, Jr., and Amos Tversky 24 Rational Choice and the Framing of Decisions 593 Amos Tversky and Daniel Kahneman 25 Contrasting Rational and Psychological Analyses of Political Choice 621 George A. Quattrone and Amos Tversky 26 Preference and Belief: Ambiguity and Competence in Choice under Uncertainty 645 Chip Heath and Amos Tversky 27 Advances in Prospect Theory: Cumulative Representation of Uncertainty 673 Amos Tversky and Daniel Kahneman 28 Thinking through Uncertainty: Nonconsequential Reasoning and Choice 703 Eldar Shafir and Amos Tversky 29 Conflict Resolution: A Cognitive Perspective 729 Daniel Kahneman and Amos Tversky 30 Weighing Risk and Uncertainty 747 Amos Tversky and Craig R. Fox 31 Ambiguity Aversion and Comparative Ignorance 777 Craig R. Fox and Amos Tversky 32 A Belief-Based Account of Decision under Uncertainty 795 Craig R. Fox and Amos Tversky Contingent Preferences 823 33 Self-Deception and the Voter’s Illusion 825 George A. Quattrone and Amos Tversky 34 Contingent Weighting in Judgment and Choice 845 Amos Tversky, Shmuel Sattath, and Paul Slovic 35 Anomalies: Preference Reversals 875 Amos Tversky and Richard H. Thaler 36 Discrepancy between Medical Decisions for Individual Patients and for Groups 887 Donald A. Redelmeier and Amos Tversky 37 Loss Aversion in Riskless Choice: A Reference-Dependent Model 895 Amos Tversky and Daniel Kahneman 38 Endowment and Contrast in Judgments of Well-Being 917 Amos Tversky and Dale Gri‰n 39 Reason-Based Choice 937 Eldar Shafir, Itamar Simonson, and Amos Tversky 40 Context-Dependence in Legal Decision Making 963 Mark Kelman, Yuval Rottenstreich, and Amos Tversky Amos Tversky’s Complete Bibliography 995