Extensional Versus Intuitive Reasoning

Let's revisit a famous psychology experiment from 1974. In fact, you should play along yourself by reading the following description of a person and then ranking the 8 statements that follow according to their probability, with 1 being the most probable and 8 being the least probable.

"Linda is 31 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 anti-nuclear demonstrations.

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

OK, all done?

Notice anything interesting about this list?

What the experimenters are really interested in is seeing how you rank the items "active in the feminist movement" (F), "bank teller" (B), and "bank teller and active in the feminist movement" (B+F).

Those of us who took probability and statistics will remember something called the conjunction rule: P(A+B)<=P(B). This means simply that the probability of two things together being true must be less than or equal to the probability of one of the two things being true by itself. For example, the probability that "my pet is a rabbit named Leo" (R+L) cannot be greater than the probability that "my pet is a rabbit" (R). This is because the pet has to be a rabbit (R) in order for it to be a rabbit named Leo (R+L) and it is possible for R to be true while L is not true: I could have a rabbit, but his name is Blackberry or Hazel or Bigwig or ....

Although looking at the formula can be confusing, I think the central idea is pretty easy to grasp. The probability that "Joe has a computer that is a Mac" can't be higher than the probability that "Joe has a computer." The probability that "Kevin had a burger and fries at lunch" cannot be higher than the probability that "Kevin had a burger at lunch." And so we see, the probability that "Linda is a bank teller and is active in the feminist movement" cannot be higher than the probability that "Linda is a bank teller."

So how did you do? Did you rank "bank teller + feminist" higher than "bank teller" alone? If so, you have plenty of company. In the experiments, 88% of students thought it was more likely that Linda was a feminist bank teller than a bank teller, thus violating the conjunction rule. When they tested students with higher levels of "statistical sophistication" (i.e. had taken more statistics classes), 85% of them violated the conjunction rule.

They did another experiment in which students looked at the same list, but instead of ranking the probability of the events, they ranked how well Linda resembles the typical member of the class. In this case, 85% ordered the descriptions Feminist > Feminist Bank Teller > Bank Teller, which matched up very well with the rankings in the probability experiment.

So it appeared that when people were asked to think about the probabilities of various things being true, they thought about how well the description matched up with what they knew about Linda. Since Linda had many characteristics associated with the stereotypical feminist, and few traits associated with the stereotypical bank teller, it seemed natural to assume that she was more likely a feminist than a bank teller, even to the point of finding "feminist bank teller" more plausible, and hence probable, than just "bank teller," despite the mathematical incoherence of such a thing.

The experimenters were a bit dismayed by these results and hence proceeded to create "increasingly desperate manipulations designed to induce subjects to obey the conjunction rule." One of these desperate manipulations was to see if people could at least recognize that they should be using the conjunction rule in determining these probabilities. Subjects were shown two arguments and asked to indicate which they found more convincing:

"Argument 1: Linda is more likely to be a bank teller than she is to be a feminist bank teller, because every feminist bank teller is a bank teller, but some women bank tellers are not feminists, and Linda could be one of them.

Argument 2: Linda is more likely to be a feminist bank teller than she is to be a bank teller, because she resembles an active feminist more than she resembles a bank teller."

65% of subjects found Argument 2 more convincing.

Then the experimenters changed the experiment, giving the same description of Linda's personality and background, but then saying: "If you couuld win $10 by betting on an event, which of the following would you bet on? (Check one.)" and listing the bank teller and feminist bank teller descriptions. 56% of subjects selected feminist bank teller.

I thought this was particularly interesting:

"Why do intelligent and reasonably well-educated people fail to recognize the applicability of the conjunction rule in transparent problems? Postexperimental interviews and class discussions with many subjects shed some light on this question. Naive as well as sophisticated subjects generally noticed the nesting of the target events [i.e. that "bank teller" includes "feminist bank teller"] in the direct-transparent test, but the naive, unlike the sophisticated, did not appreciate its significance for probability assessment. However, most naive subjects did not attempt to defend their responses. As one subject said after acknowledging the validity of the conjunction rule, 'I thought you only asked for my opinion.' "

The experimenters then liken the naive subjects to children in the preconservative stage of cognitive development who recognize the validity of concepts such as conservation of volume but do not see that the conservation argument is "decisive" and should over-rule their impression that when you pour a given quantity of liquid from a short, wide glass into a tall, skinny glass, there is a greater amount of liquid because it looks like more.

I find it rather endlessly fascinating and depressing that even when the error is pointed out to them, so many adults could believe, "Well, yes, technically she is more likely to be a bank teller, but it's my opinion that she's probably a feminist bank teller." While this may seem like a trivial example, the implications are disturbing. Even in situations in which the person's belief is not even possible, they stick to it as a matter of "opinion." So imagine the situations in which the belief is possible - like, Barack Obama has the middle name Hussein so he is probably a Muslim. How much evidence would be required to make the person change their "opinion" about that? And while in experiments with other types of things (like a gambling scenario with a sequence of die rolls given) subjects were able to be more logical with their reasoning, person perception scenarios seem to be particularly prone to triggering this intuitive (and faulty) reasoning based on stereotypes. What's more, the thinking process doesn't have to be motivated by bad intentions to come up with these impossible beliefs.

The researchers conclude their article by observing, "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 maximation of expected utility. The presence of bias and incoherence does not diminish the normative force of these principles [i.e. that people should follow them], but it reduces their usefulness as descriptions of behavior and hinders their prescriptive applications." Economists, they are looking at you.

The researchers, psychologists Amos Tverksy and Daniel Kahneman, pioneered the hugely influential area of "heuristics and biases" in judgment (of which these experiments give a taste of the impact of the "representativeness heuristic") and also collaborated on "prospect theory" (another theory that showed deficiencies in the expected utility model in decision-making under uncertainty), for which Kahneman was awarded the Nobel Prize in Economics in 2002. (Tversky died in 1996 and was not eligible for the award.) Apparently, Kahneman has never taken a course in economics.

Source: Tversky, A. & Kahneman, D. Extensional versus intuitive reasoning: the conjunction fallacy in probability judgment (originally published 1984), in Heuristics and Biases, eds. Thomas Gilovich, Dale Griffin, Daniel Kahneman, 2002.