(This post is co-written with Matt Mandelkern, based on our joint paper on the topic. 2500 words; 12 minute read.)
It’s February 2024. Three Republicans are vying for the Presidential nomination, and FiveThirtyEight puts their chances at:
Some natural answers: "Pence"; "Pence or Carlson"; "Pence, Carlson, or Haley". In a Twitter poll earlier this week, the first two took up a majority (53.4%) of responses:
But wait! If you answered "Pence", or "Pence or Carlson", did you commit the conjunction fallacy? This is the tendency to say that narrower hypotheses are more likely than broader ones––such as saying that P&Q is more likely than Q—contrary to the laws of probability. Since every way in which "Pence" or "Pence or Carlson" could be true is also a way in which “Pence, Carlson, or Haley” would be true, the third option is guaranteed to be more likely than each of the first two.
Does this mean answering our question with “Pence” or “Pence or Carlson” was a mistake?
We don’t think so. We think what you were doing was guessing. Rather than simply ranking answers for probability, you were making a tradeoff between being accurate (saying something probable) and being informative (saying something specific). In light of this tradeoff, it’s perfectly permissible to guess an answer (“Pence”) that’s less probable––but more informative––than an alternative (“Pence, Carlson, or Haley”).
Here we'll argue that this explains––and partially rationalizes––the conjunction fallacy.