A minor philosophical niggle, this. ‘The Conjunction Problem’ is what Daniel Kahneman, in his Thinking, Fast and Slow (2011), calls a particular problem in probabilistic thinking. It goes like this:
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. Which is more probable?Kahneman says people will choose (b) as more probable, because they are wrongheaded wrongy wrong-wrongs, the berks. Phil Edwards (in a post discussing something else) summarises: ‘The correct answer is—logically has to be—(a); “A and also B” cannot be more probable than “A with or without B”, whatever A and B are. But we’re not hard-wired to be good at probability; we seem to read the question as an invitation to fill in the blank in the way that gives the most satisfying story, in this case option (b).’
a) Linda is a bank teller.
b) Linda is a bank teller and is active in the feminist movement.
Something about this smells fishy to me. I take the force of Kahneman's point, of course; and we can make it clearer by rephrasing the original question.
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. Which is more probable?Put like that, (a) certainly looks more probable. But there seem to me two problems with it. One has to do with the vagueness with which the question is framed, a haziness that devolves upon being asked to compare a simple with a compound probability. The other has to do with the different valences of probability itself. ‘Given this information about a person, what do you consider plausible extrapolations as to the person's life or work?’ is a different kind of question to ‘this tossed coin has come up heads ten times in a row, what are the odds of an eleventh head?’
a) Linda is a bank teller and either is or is not active in the feminist movement
b) Linda is a bank teller and is active in the feminist movement.
It may be, as Edwards suggests, that people plump for (b) because they prefer its implied story, rather than for reasons of pure probabilistic estimation. But then again, that might have nothing to do with it. Conceivably people choose (b) because they simply consider it more probable, irrespective of narrative. Reframe the question another way:
Here's quite a lot of information about Linda, designed to give you a sense of the kind of person she is, so as to feel confident making informed guesses as to the sorts of things she likes and what she does. Which is more probable?The ‘likelihood’ angle is there in the original formulation as, if you like, Kahneman's sleight of hand, to try and nudge people to the ‘wrong’ answer. But it seems to me, when you frame it this way, it's reasonable to select (b). The thinking would be: (a) is, by definition, unlikely, so it doesn't make much sense to choose it. Of course (b) also includes that unlikely thing, but it also includes something much more likely, so if I select it I'll at least get 50%, instead of the all-or-nothing, weighted heavily towards nothing, unlikeliness of (a). The fuzziness is in the question not specifying whether we are selecting for the likelihood of the statement, or the likelihood of that statement plus literally everything else that might or might not be the case. And when you put it like that (a) becomes not so much unlikely as impossible with regard to anything save the Supreme Being. Quite apart from anything else, that's simply not how we assess statements for likelihood. Is it?
a) Linda is doing something you consider unlikely.
b) Linda is doing something you consider unlikely. Linda is also doing something you consider she'd be likely to do.
It puts me in mind of a version of the celebrated twins-at-the-fork-in-the-road logic puzzle, I think by Raymond Smullyan. You know the original puzzle. You have to get to Blogtown, urgently. But you've come to a fork in the road and don't know the way. Happily there are two people right there, identical twins, who do know the way. Unhappily, for obscure reasons (perhaps to do with their religion) one of them always tells the truth and the other always lies; and, moreover, they will only answer one question from you. What do you ask? The standard answer is: you pick one at random and ask ‘if I asked your brother which road to follow, what would he say?’ and then proceed up the other road than the one indicated. Fair enough. But the Smullyan solution, which seems to me to have a bearing on the Kahneman dilemma, is: ‘pick one brother and ask him the way to Blogtown. If you have chosen the truth-telling twin, he will point you in the right direction. If you have chosen the liar, he will point along the wrong fork in the road and also, with his other hand, point down the road along which you have just come.’ That's bonkers in similar ways to the manner of Kahneman's bonkers explanation. Isn't it?
I disagree with the idea of "bonkers in similar ways" mentioned at the end. To me, you have three very different situations in three examples.
ReplyDeleteRegarding the first, while I might not say, "people will choose (b) as more probable, because they are wrongheaded wrongy wrong-wrongs," I would say that choosing (b) is a sign the person is not thinking probabilistically. Your rephrasing of the idea by adding "and either is or is not active in the feminist movement" captures why, because probabilistically it is the same statement as the first version. The only thing that I would add when using such a rewrite to explain why (a) is clearly the more probable choice is that the added words are redundant--they do not change the nature of choice (a) in terms of likelihood. We could add "and either does or does not have red hair" or any other always true statements. (The classroom approach to this would be to draw some truth tables.)
There might be many valid reasons to approach analyzing a situation other than probabilistically, but once narrowing to what is more probable there is no haziness or vagueness in the framing that I can see. When you posit that "Given this information about a person, what do you consider plausible extrapolations as to the person's life or work?" is a different question than calculating coin-flip odds, that's true, but it's also not what the setup asks, because comparing the likelihood of two events is not asking for plausible extrapolations. Probability looks at both plausible and not plausible events, and tries to model how likely these events are to occur. Unlikely events do happen, and the setup compared two specific (perhaps unlikely) events.
It's here that I think your reframing goes wrong, at least if you wanted it to be similar to the original framing. When you say "Linda is doing something you consider unlikely," the "something" adds ambiguity that doesn't exist in the original, because the something from (a) could be a different something from (b). You need to tie the two choices together to get an equivalent situation. ("Linda is doing [the exact same] something [mentioned in (a)] you consider unlikely" or the like.)
Regarding your third example, that of the fork-in-the-road logic puzzle, I think there is an important point to be made regarding why the final answer is bonkers. Something is wrong in the setup, because even the standard answer you give: "you pick one at random and ask ‘if I asked your brother which road to follow, what would he say?’ and then proceed up the other road than the one indicated" is not necessarily correct. To have the certainty that such a logic puzzle usually requires in order to be solvable, the question typically is phrased as a yes/no question, or one where the answer set is binary, so that there is no possibility of a choice among the false answers. Even with the standard answer as phrased, the liar could direct you to just the road you came in on (it is a road, so belongs in the possible answer set of three roads present, and it is a lie, because it is not the correct direction), and you still do not know the way. Typically, there is no obligation that he point to all false answers, but to just give a false answer, which is why the final answer is bonkers (without a different setup). If you "ask him the way to Blogtown," he need only point in one wrong direction. To get the phrasing for the standard answer right, the question should be to pick one of the forks, and ask, "If I asked your brother if this road leads to Blogtown, would he say 'yes'?" Then go on the road other than the one indicated.