Plausible detail versus logical consistency

From Thinking, Fast and Slow by Daniel Kahneman. One of his more elegant examples of how hard it can be to thinking logically when plausibility is in the air.

Linda: Less Is More

The best-known and most controversial of our experiments involved a fictitious lady called Linda. Amos and I made up the Linda problem to provide conclusive evidence of the role of heuristics in judgment and of their incompatibility with logic. This is how we described Linda:
Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.
 
The audiences who heard this description in the 1980s always laughed because they immediately knew that Linda had attended the University of California at Berkeley, which was famous at the time for its radical, politically engaged students. In one of our experiments we presented participants with a list of eight possible scenarios for Linda. As in the Tom W problem, some ranked the scenarios by representativeness, others by probability. The Linda problem is similar, but with a twist.

Linda is a teacher in elementary school.
Linda works in a bookstore and takes yoga classes.
Linda is active in the feminist movement.
Linda is a psychiatric social worker.
Linda is a member of the League of Women Voters.
Linda is a bank teller.
Linda is an insurance salesperson.
Linda is a bank teller and is active in the feminist movement.

The problem shows its age in several ways. The League of Women Voters is no longer as prominent as it was, and the idea of a feminist “movement” sounds quaint, a testimonial to the change in the status of women over the last thirty years. Even in the Facebook era, however, it is still easy to guess the almost perfect consensus of judgments: Linda is a very good fit for an active feminist, a fairly good fit for someone who works in a bookstore and takes yoga classes—and a very poor fit for a bank teller or an insurance salesperson.

Now focus on the critical items in the list: Does Linda look more like a bank teller, or more like a bank teller who is active in the feminist movement? Everyone agrees that Linda fits the idea of a “feminist bank teller” better than she fits the stereotype of bank tellers. The stereotypical bank teller is not a feminist activist, and adding that detail to the description makes for a more coherent story.

The twist comes in the judgments of likelihood, because there is a logical relation between the two scenarios. Think in terms of Venn diagrams. The set of feminist bank tellers is wholly included in the set of bank tellers, as every feminist bank teller is necessarily a bank teller. Therefore the probability that Linda is a feminist bank teller must be lower than the probability of her being a bank teller. When you specify a possible event in greater detail you can only lower its probability. The problem therefore sets up a conflict between the intuition of representativeness and the logic of probability.

Our initial experiment was between-subjects. Each participant saw a set of seven outcomes that included only one of the critical items (“bank teller” or “feminist bank teller”). Some ranked the outcomes by resemblance, others by likelihood. As in the case of Tom W, the average rankings by resemblance and by likelihood were identical; “feminist bank teller” ranked higher than “bank teller” in both.

Then we took the experiment further, using a within-subject design. We made up the questionnaire as you saw it, with “bank teller” in the sixth position in the list and “feminist bank teller” as the last item. We were convinced that subjects would notice the relation between the two outcomes, and that their rankings would be consistent with logic. Indeed, we were so certain of this that we did not think it worthwhile to conduct a special experiment. My assistant was running another experiment in the lab, and she asked the subjects to complete the new Linda questionnaire while signing out, just before they got paid.

About ten questionnaires had accumulated in a tray on my assistant’s desk before I casually glanced at them and found that all the subjects had ranked “feminist bank teller” as more probable than “bank teller.” I was so surprised that I still retain a “flashbulb memory” of the gray color of the metal desk and of where everyone was when I made that discovery. I quickly called Amos in great excitement to tell him what we had found: we had pitted logic against representativeness, and representativeness had won!

In the language of this book, we had observed a failure of System 2: our participants had a fair opportunity to detect the relevance of the logical rule, since both outcomes were included in the same ranking. They did not take advantage of that opportunity. When we extended the experiment, we found that 89% of the undergraduates in our sample violated the logic of probability. We were convinced that statistically sophisticated respondents would do better, so we administered the same questionnaire to doctoral students in the decision-science program of the Stanford Graduate School of Business, all of whom had taken several advanced courses in probability, statistics, and decision theory. We were surprised again: 85% of these respondents also ranked “feminist bank teller” as more likely than “bank teller.”

In what we later described as “increasingly desperate” attempts to eliminate the error, we introduced large groups of people to Linda and asked them this simple question:

Which alternative is more probable?

Linda is a bank teller.
Linda is a bank teller and is active in the feminist movement.

This stark version of the problem made Linda famous in some circles, and it earned us years of controversy. About 85% to 90% of undergraduates at several major universities chose the second option, contrary to logic. Remarkably, the sinners seemed to have no shame. When I asked my large undergraduatnite class in some indignation, “Do you realize that you have violated an elementary logical rule?” someone in the back row shouted, “So what?” and a graduate student who made the same error explained herself by saying, “I thought you just asked for my opinion.”

The word fallacy is used, in general, when people fail to apply a logical rule that is obviously relevant. Amos and I introduced the idea of a conjunction fallacy, which people commit when they judge a conjunction of two events (here, bank teller and feminist) to be more probable than one of the events (bank teller) in a direct comparison.
As in the Müller-Lyer illusion, the fallacy remains attractive even when you recognize it for what it is. The naturalist Stephen Jay Gould described his own struggle with the Linda problem. He knew the correct answer, of course, and yet, he wrote, “a little homunculus in my head continues to jump up and down, shouting at me — ‘but she can’t just be a bank teller; read the description.’” The little homunculus is of course Gould’s System 1 speaking to him in insistent tones. (The two-system terminology had not yet been introduced when he wrote.)

The correct answer to the short version of the Linda problem was the majority response in only one of our studies: 64% of a group of graduate students in the social sciences at Stanford and at Berkeley correctly judged “feminist bank teller” to be less probable than “bank teller.” In the original version with eight outcomes (shown above), only 15% of a similar group of graduate students had made that choice. The difference is instructive. The longer version separated the two critical outcomes by an intervening item (insurance salesperson), and the readers judged each outcome independently, without comparing them. The shorter version, in contrast, required an explicit comparison that mobilized System 2 and allowed most of the statistically sophisticated students to avoid the fallacy. Unfortunately, we did not explore the reasoning of the substantial minority (36%) of this knowledgeable group who chose incorrectly.

The judgments of probability that our respondents offered, in both the Tom W and Linda problems, corresponded precisely to judgments of representativeness (similarity to stereotypes). Representativeness belongs to a cluster of closely related basic assessments that are likely to be generated together. The most representative outcomes combine with the personality description to produce the most coherent stories. The most coherent stories are not necessarily the most probable, but they are plausible, and the notions of coherence, plausibility, and probability are easily confused by the unwary.

The uncritical substitution of plausibility for probability has pernicious effects on judgments when scenarios are used as tools of forecasting. Consider these two scenarios, which were presented to different groups, with a request to evaluate their probability:

A massive flood somewhere in North America next year, in which more than 1,000 people drown
 
An earthquake in California sometime next year, causing a flood in which more than 1,000 people drown

The California earthquake scenario is more plausible than the North America scenario, although its probability is certainly smaller. As expected, probability judgments were higher for the richer and more detailed scenario, contrary to logic. This is a trap for forecasters and their clients: adding detail to scenarios makes them more persuasive, but less likely to come true.

To appreciate the role of plausibility, consider the following questions:

Which alternative is more probable?

Mark has hair.
Mark has blond hair.

and

Which alternative is more probable?

Jane is a teacher.
Jane is a teacher and walks to work.

The two questions have the same logical structure as the Linda problem, but they cause no fallacy, because the more detailed outcome is only more detailed—it is not more plausible, or more coherent, or a better story. The evaluation of plausibility and coherence does not suggest and answer to the probability question. In the absence of a competing intuition, logic prevails.