Further reflections on non-transitive folk probability
Continuing my thoughts about zero-one probability from here, I come to the recent decision of Standard & Poor’s to lower their rating of US treasury debt. There are plenty of reasons to doubt their judgement, both because they’ve been absurdly wrong in the past (subprime mortgage backed securities were AAA, but treasury bills are risky?), because they can’t read budget estimates or can’t do basic arithmetic, because they are trying to project political trends, which they surely know even less about than about arithmetic, or because the people who work there are generally known to be pretty dim. But from a probabilist’s point of view what’s strange is the timing. Whatever you may think of the recent deal to avoid the US defaulting on its debt, it did avoid defaulting on its debt. Surely the likelihood of a default went down after the deal was passed. So why is the credit rating lower this week than it was last week? Now, this is all perfectly consistent with the view that S&P is not actually making a prediction of future default probability, but simply seeking the best opportunity to promote its wares. Certainly, the way they operate is not the like someone trying to give what will be perceived as neutral advice; they act more like central bankers, timing their announcements to try to move markets and (above all) seem relevant. They’re reminiscent of the folktale of the rooster who threatens to withhold his crowing, which inevitably will forestall the sun rise. The other animals plead with him to relent, but it’s a threat that only works as long as the rooster is modest enough to recognise that he can’t hold out forever. In the case of the US treasury bonds, S&P held out, and still the sun rose.
But there is something about their approach that seems to make sense to intelligent people, and not purely idiosyncratic. I’m reminded of Tversky’s famous conjunction fallacy, with studies seeming to show that people’s everyday probability intuitions don’t necessary satisfy the apparently inevitable law of conjunction: The probability of A or B must be bigger than the probability of A and the probability of B. Here we see intuitions of probability that don’t seem to satisfy the law of total expectation: If are possible future states of the world, and is the probability of event A conditional on happening, then the probability of event A now must be some kind of average of these conditional probabilities.