8, 7 of them broken

Assuming that the dishes in the two sets are of equal quality, which is worth more? This question is easy. You can see that Set A contains all the dishes of Set B, and seven additional intact dishes, and it must be valued more. Indeed, the participants in Hsee's joint evaluation experiment were willing to pay a little more for Set A than for Set B: $32 versus $30.

The results reversed in single evaluation, where Set B was priced much higher than Set A: $33 versus $23. We know why this happened. Sets (including dinnerware sets!) are represented by norms and prototypes. You can sense immediately that the average value of the dishes is much lower for Set A than for Set B, because no one wants to pay for broken dishes. If the average dominates the evaluation, it is not surprising that Set B is valued more. Hsee called the resulting pattern less is more. By removing 16 items from Set A (7 of them intact), its value is improved.

Hsee's finding was replicated by the experimental economist John List in a real market for baseball cards. He auctioned sets of ten high-value cards, and identical sets to which three cards of modest value were added. As in the dinnerware experiment, the larger sets were valued more than the smaller ones in joint evaluation, but less in single evaluation. From the perspective of economic theory, this result is troubling: the economic value of a dinnerware set or of a collection of baseball cards is a sum-like variable. Adding a positively valued item to the set can only increase its value.

The Linda problem and the dinnerware problem have exactly the same structure. Probability, like economic value, is a sum-like variable, as illustrated by this example: probability (Linda is a teller) = probability (Linda is feminist teller) + probability (Linda is non-feminist teller)

This is also why, as in Hsee's dinnerware study, single evaluations of the Linda problem produce a less-is-more pattern. System 1 averages instead of adding, so when the non-feminist bank tellers are removed from the set, subjective probability increases. However, the sum-like nature of the variable is less obvious for probability than for money. As a result, joint evaluation eliminates the error only in Hsee's experiment, not in the Linda experiment.

Linda was not the only conjunction error that survived joint evaluation. We found similar violations of logic in many other judgments. Participants in one of these studies were asked to rank four possible outcomes of the next Wimbledon tournament from most to least probable. Bjorn Borg was the dominant tennis player of the day when the study was conducted. These were the outcomes: A. Borg will win the match.

B. Borg will lose the first set.

C. Borg will lose the first set but win the match.

D. Borg will win the first set but lose the match.

The critical items are B and C. B is the more inclusive event and its probability must be higher than that of an event it includes. Contrary to logic, but not to representativeness or plausibility, 72% assigned B a lower probability than C-another instance of less is more in a direct comparison. Here si again, the scenario that was judged more probable was unquestionably more plausible, a more coherent fit with all that was known about the best tennis player in the world.

To head off the possible objection that the conjunction fallacy is due to a misinterpretation of probability, we constructed a problem that required probability judgments, but in which the events were not described in words, and the term probability did not appear at all. We told participants about a regular six-sided die with four green faces and two red faces, which would be rolled 20 times. They were shown three sequences of greens (G) and reds (R), and were asked to choose one. They would (hypothetically) win $25 if their chosen sequence showed up. The sequences were:RGRRR

GRGRRR

GRRRRR

Because the die has twice as many green as red faces, the first sequence is quite unrepresentative-like Linda being a bank teller. The second sequence, which contains six tosses, is a better fit to what we would expect from this die, because it includes two G's. However, this sequence was constructed by adding a G to the beginning of the first sequence, so it can only be less likely than the first. This is the nonverbal equivalent to Linda being a feminist bank teller. As in the Linda study, representativeness dominated. Almost two-thirds of respondents preferred to bet on sequence 2 rather than on sequence 1. When presented with arguments for the two choices, however, a large majority found the correct argument (favoring sequence 1) more convincing.

The next problem was a breakthrough, because we finally found a condition in which the incidence of the conjunction fallacy was much reduced. Two groups of subjects saw slightly different variants of the same problem:

The incidence of errors was 65% in the group that saw the problem on the left, and only 25% in the group that saw the problem on the right.

Why is the question "How many of the 100 participants..." so much easier than "What percentage..."? A likely explanation is that the reference to 100 individuals brings a spatial representation to mind. Imagine that a large number of people are instructed to sort themselves into groups in a room: "Those whose names begin with the letters A to L are told to gather in the front left corner." They are then instructed to sort themselves further. The relation of inclusion is now obvious, and you can see that individuals whose name begins with C will be a subset of the crowd in the front left corner. In the medical survey question, heart attack victims end up in a corner of the room, and some of them are less than 55 years old. Not everyone will share this particular vivid imagery, but many subsequent experiments have shown that the frequency representation, as it is known, makes it easy to appreciate that one group is wholly included in the other. The solution to the puzzle appears to be that a question phrased as "how many?" makes you think of individuals, but the same question phrased as "what percentage?" does not.

What have we learned from these studies about the workings of System 2? One conclusion, which is not new, is that System 2 is not impressively alert. The undergraduates and graduate students who participated in our thastudies of the conjunction fallacy certainly "knew" the logic of Venn diagrams, but they did not apply it reliably even when all the relevant information was laid out in front of them. The absurdity of the less-is-more pattern was obvious in Hsee's dinnerware study and was easily recognized in the "how many?" representation, but it was not apparent to the thousands of people who have committed the conjunction fallacy in the original Linda problem and in others like it. In all these cases, the conjunction appeared plausible, and that sufficed for an endorsement of System 2.

The laziness of System 2 is part of the story. If their next vacation had depended on it, and if they had been given indefinite time and told to follow logic and not to answer until they were sure of their answer, I believe that most of our subjects would have avoided the conjunction fallacy. However, their vacation did not depend on a correct answer; they spent very little time on it, and were content to answer as if they had only been "asked for their opinion." The laziness of System 2 is an important fact of life, and the observation that representativeness can block the application of an obvious logical rule is also of some interest.

The remarkable aspect of the Linda story is the contrast to the broken-dishes study. The two problems have the same structure, but yield different results. People who see the dinnerware set that includes broken dishes put a very low price on it; their behavior reflects a rule of intuition. Others who see both sets at once apply the logical rule that more dishes can only add value. Intuition governs judgments in the between-subjects condition; logic rules in joint evaluation. In the Linda problem, in contrast, intuition often overcame logic even in joint evaluation, although we identified some conditions in which logic prevails.

Amos and I believed that the blatant violations of the logic of probability that we had observed in transparent problems were interesting and worth reporting to our colleagues. We also believed that the results strengthened our argument about the power of judgment heuristics, and that they would persuade doubters. And in this we were quite wrong. Instead, the Linda problem became a case study in the norms of controversy.

The Linda problem attracted a great deal of attention, but it also became a magnet for critics of our approach to judgment. As we had already done, researchers found combinations of instructions and hints that reduced the incidence of the fallacy; some argued that, in the context of the Linda problem, it is reasonable for subjects to understand the word "probability" as if it means "plausibility." These arguments were sometimes extended to suggest that our entire enterprise was misguided: if one salient cognitive illusion could be weakened or explained away, others could be as well. This reasoning neglects the unique feature of the conjunction fallacy as a case of conflict between intuition and logic. The evidence that we had built up for heuristics from between-subjects experiment (including studies of Linda) was not challenged-it was simply not addressed, and its salience was diminished by the exclusive focus on the conjunction fallacy. The net effect of the Linda problem was an increase in the visibility of our work to the general public, and a small dent in the credibility of our approach among scholars in the field. This was not at all what we had expected.

If you visit a courtroom you will observe that lawyers apply two styles of criticism: to demolish a case they raise doubts about the strongest arguments that favor it; to discredit a witness, they focus on the weakest part of the testimony. The focus on weaknesses is also normal in politicaverl debates. I do not believe it is appropriate in scientific controversies, but I have come to accept as a fact of life that the norms of debate in the social sciences do not prohibit the political style of argument, especially when large issues are at stake-and the prevalence of bias in human judgment is a large issue.

Some years ago I had a friendly conversation with Ralph Hertwig, a persistent critic of the Linda problem, with whom I had collaborated in a vain attempt to settle our differences. I asked him why he and others had chosen to focus exclusively on the conjunction fallacy, rather than on other findings that provided stronger support for our position. He smiled as he answered, "It was more interesting," adding that the Li

nda problem had attracted so much attention that we had no reason to complain.Speaking of Less is More

"They constructed a very complicated scenario and insisted on calling it highly probable. It is not-it is only a plausible story."

"They added a cheap gift to the expensive product, and made the whole deal less attractive. Less is more in this case."

"In most situations, a direct comparison makes people more careful and more logical. But not always. Sometimes intuition beats logic even when the correct answer stares you in the face."

Causes Trump Statistics Consider the following scenario and note your intuitive answer to the question.

A cab was involved in a hit-and-run accident at night.Two cab companies, the Green and the Blue, operate in the city.You are given the following data:

85% of the cabs in the city are Green and 15% are Blue.

A witness identified the cab as Blue. The court tested the reliability of the witness under the circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green?

This is a standard problem of Bayesian inference. There are two items of information: a base rate and the imperfectly reliable testimony of a witness. In the absence of a witness, the probability of the guilty cab being Blue is 15%, which is the base rate of that outcome. If the two cab companies had been equally large, the base rate would be uninformative and you would consider only the reliability of the witness,%"> our w

Causal StereotypesNow consider a variation of the same story, in which only the presentation of the base rate has been altered.

You are given the following data:

The two companies operate the same number of cabs, but Green cabs are involved in 85% of accidents.

The information about the witness is as in the previous version.

The two versions of the problem are mathematically indistinguishable, but they are psychologically quite different. People who read the first version do not know how to use the base rate and often ignore it. In contrast, people who see the second version give considerable weight to the base rate, and their average judgment is not too far from the Bayesian solution. Why?

In the first version, the base rate of Blue cabs is a statistical fact about the cabs in the city. A mind that is hungry for causal stories finds nothing to chew on: How does the number of Green and Blue cabs in the city cause this cab driver to hit and run?

In the second version, in contrast, the drivers of Green cabs cause more than 5 times as many accidents as the Blue cabs do. The conclusion is immediate: the Green drivers must be a collection of reckless madmen! You have now formed a stereotype of Green recklessness, which you apply to unknown individual drivers in the company. The stereotype is easily fitted into a causal story, because recklessness is a causally relevant fact about individual cabdrivers. In this version, there are two causal stories that need to be combined or reconciled. The first is the hit and run, which naturally evokes the idea that a reckless Green driver was responsible. The second is the witness's testimony, which strongly suggests the cab was Blue. The inferences from the two stories about the color of the car are contradictory and approximately cancel each other. The chances for the two colors are about equal (the Bayesian estimate is 41%, reflecting the fact that the base rate of Green cabs is a little more extreme than the reliability of the witness who reported a Blue cab).

The cab example illustrates two types of base rates. Statistical base rates are facts about a population to which a case belongs, but they are not relevant to the individual case. Causal base rates change your view of how the individual case came to be. The two types of base-rate information are treated differently:Statistical base rates are generally underweighted, and sometimes neglected altogether, when specific information about the case at hand is available.

Causal base rates are treated as information about the individual case and are easily combined with other case-specific information.

The causal version of the cab problem had the form of a stereotype: Green drivers are dangerous. Stereotypes are statements about the group that are (at least tentatively) accepted as facts about every member. Hely re are two examples: Most of the graduates of this inner-city school go to college.Interest in cycling is widespread in France.

These statements are readily interpreted as setting up a propensity in individual members of the group, and they fit in a causal story. Many graduates of this particular inner-city school are eager and able to go to college, presumably because of some beneficial features of life in that school. There are forces in French culture and social life that cause many Frenchmen to take an interest in cycling. You will be reminded of these facts when you think about the likelihood that a particular graduate of the school will attend college, or when you wonder whether to bring up the Tour de France in a conversation with a Frenchman you just met.

Stereotyping is a bad word in our culture, but in my usage it is neutral. One of the basic characteristics of System 1 is that it represents categories as norms and prototypical exemplars. This is how we think of horses, refrigerators, and New York police officers; we hold in memory a representation of one or more "normal" members of each of these categories. When the categories are social, these representations are called stereotypes. Some stereotypes are perniciously wrong, and hostile stereotyping can have dreadful consequences, but the psychological facts cannot be avoided: stereotypes, both correct and false, are how we think of categories.

You may note the irony. In the context of the cab problem, the neglect of base-rate information is a cognitive flaw, a failure of Bayesian reasoning, and the reliance on causal base rates is desirable. Stereotyping the Green drivers improves the accuracy of judgment. In other contexts, however, such as hiring or profiling, there is a strong social norm against stereotyping, which is also embedded in the law. This is as it should be. In sensitive social contexts, we do not want to draw possibly erroneous conclusions about the individual from the statistics of the group. We consider it morally desirable for base rates to be treated as statistical facts about the group rather than as presumptive facts about individuals. In other words, we reject causal base rates.

The social norm against stereotyping, including the opposition to profiling, has been highly beneficial in creating a more civilized and more equal society. It is useful to remember, however, that neglecting valid stereotypes inevitably results in suboptimal judgments. Resistance to stereotyping is a laudable moral position, but the simplistic idea that the resistance is costless is wrong. The costs are worth paying to achieve a better society, but denying that the costs exist, while satisfying to the soul and politically correct, is not scientifically defensible. Reliance on the affect heuristic is common in politically charged arguments. The positions we favor have no cost and those we oppose have no benefits. We should be able to do better.Causal SituationsAmos and I constructed the variants of the cab problem, but we did not invent the powerful notion of causal base rates; we borrowed it from the psychologist Icek Ajzen. In his experiment, Ajzen showed his participants brief vignettes describing some students who had taken an exam at Yale and asked the participants to judge the probability that each student had passed the test. The manipulation of causal bs oase rates was straightforward: Ajzen told one group that the students they saw had been drawn from a class in which 75% passed the exam, and told another group that the same students had been in a class in which only 25% passed. This is a powerful manipulation, because the base rate of passing suggests the immediate inference that the test that only 25% passed must have been brutally difficult. The difficulty of a test is, of course, one of the causal factors that determine every student's outcome. As expected, Ajzen's subjects were highly sensitive to the causal base rates, and every student was judged more likely to pass in the high-success condition than in the high-failure rate.

Ajzen used an ingenious method to suggest a noncausal base rate. He told his subjects that the students they saw had been drawn from a sample, which itself was constructed by selecting students who had passed or failed the exam. For example, the information for the high-failure group read as follows: The investigator was mainly interested in the causes of failure and constructed a sample in which 75% had failed the examination.

Note the difference. This base rate is a purely statistical fact about the ensemble from which cases have been drawn. It has no bearing on the question asked, which is whether the individual student passed or failed the test. As expected, the explicitly stated base rates had some effects on judgment, but they had much less impact than the statistically equivalent causal base rates. System 1 can deal with stories in which the elements are causally linked, but it is weak in statistical reasoning. For a Bayesian thinker, of course, the versions are equivalent. It is tempting to conclude that we have reached a satisfactory conclusion: causal base rates are used; merely statistical facts are (more or less) neglected. The next study, one of my all-time favorites, shows that the situation is rather more complex.Can Psychology be Taught?The reckless cabdrivers and the impossibly difficult exam illustrate two inferences that people can draw from causal base rates: a stereotypical trait that is attributed to an individual, and a significant feature of the situation that affects an individual's outcome. The participants in the experiments made the correct inferences and their judgments improved. Unfortunately, things do not always work out so well. The classic experiment I describe next shows that people will not draw from base-rate information an inference that conflicts with other beliefs. It also supports the uncomfortable conclusion that teaching psychology is mostly a waste of time.