Terence Tao on Nostr: It has been noted in the behavioral economics literature that actual human decisions ...

It has been noted in the behavioral economics literature that actual human decisions do not, actually, behave like this: external shocks do in fact have psychological impacts that can distort the decision making process. But there is actually a rational basis for having such shocks alter one's decisions, *if* the shocks also themselves carry risk. Suppose for instance that the external shock does not simply add -5 to all outcomes, but rather −5±10; the added impact of the external shock in fact ranges in most scenarios from a very negative addition of -15 to even a slight positive outcome of +5. How does this change the calculus of risk and reward?

Thanks to the law of linearity of expectation in probability, the reward side of the equation is easy to analyze: just add together the two component rewards. For instance, the safe action initially had a reward of 5, and the shock imposes an additional reward of -5, so the combined reward is 0. Similarly, the combined reward of the bold action is now 9-5=4. But what about the risk? Risk behaves differently from reward in that it is not additive; it is possible for one risk to sometimes cancel another. A more realistic rule is given by the law of additivity of variance, that says that if two risks behave in an uncorrelated fashion, then it is the *square* of the combined risk which is the sum of the *squares* of the individual risks. This should remind one of the Pythagorean law for computing the length of the hypotenuse of a right-angled triangle, and indeed in some sense the additivity of variance is an instance of the Pythagorean law (but now in an infinite-dimensional Hilbert space of random variables); but I digress. (3/4)