I have often seen discussions of what actions to take in the context of rare events in terms of expected value. For example, if a lottery has a 1 in 100 million chance of winning, and delivers a positive expected profit, then one “should” buy that lottery ticket. Or, in a an asteroid has a 1 in 1 billion chance of hitting the Earth and thereby extinguishing all human life, then one “should” take the trouble to destroy that asteroid.
This type of reasoning troubles me.
Typically, the justification for considering expected value is based on the Law of Large Numbers, namely, if one repeatedly experiences events of this type, then with high probability the average profit will be close to the expected profit. Hence expected profit would be a good criterion for decisions about common events. However, for rare events, this type of reasoning is not valid. For example, the number of lottery tickets I will buy in my lifetime is far below the asymptotic regime of the law of large numbers.
Is there any justification for using expected value alone as a criterion in these types of rare events?
This, to me, is a hard question. Should one always, as the rationality gang at Less Wrong likes to say, “shut up and multiply?” Or does multiplying very small probabilities by very large values inevitably yield confused and arbitrary results?
Update: Cosma Shalizi’s take on lotteries and utilities, winningly skeptical as usual.