Slate reposted an old piece of mine about the lottery, on the occasion of tonight’s big Mega Millions drawing. This prompted an interesting question on Math Overflow:
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.
Quomodocumque 
I highly recommend Derek Parfit’s “Reasons and Persons” to you and anyone else interested in these kind of questions from the standpoint of moral philosophy. He considers the kind of arguments that the Less Wrong people do, and pushes them to very uncomfortable places.
You used the term “expected utility” in the title of this post, but MathOverflow question is about “expected profit”. As you point out in your Slate article, these are different things. Indeed, the evidence is that, at least in developed countries, most people’s utility function for money is very flat once you pass into the solid middle class (e.g. a household income of $60-70k in the US). So in fact the utility payoff of these big lotteries is probably effectively constant even as the pot fluctuates between $1 and $100 million, and at the same time varies greatly from person to person depending on their current income (the utility value of money is very high at the lower end of the scale, e.g. below the poverty line).
Say you can play once a year with a payoff that in the mean breaks even (1/N chance at sole winner of $N). For N very large, since the median payoff is zero, you should really take the net present value of the payoff arriving in year k and average across the entire future to value this bet. I should work out what this converges to, but I’ll bet it’s less than a dollar (actually, since I didn’t buy a ticket, I implicitly *did* bet that it’s less than a dollar.)
The other way that I think about it on nights like this when I want to resist buying a lottery ticket: suppose you could make even as little as one one-hundredth of a percent per year above inflation in a risk-free investment. Waiting just one million years, your dollar would grow to about 2×10^43, while you’d probably still be tens of millions of years away from your first measly hundred-million-dollar lottery win.
I think you need to think in terms of risk profiles too. If you play lotto every week all your life, there’s an excellent chance you will not win big and in fact lose almost all of the money you “invested”. The same goes for putting money into startups vs gilt-edged securities etc.
You can draw a probability distribution for any type of investment. Unless you’ve discovered something amazing, there’s a good chance the distribution will be centred around the 3% mark (or whatever the current prevailing interest rate is) however the shape of the distribution can vary considerably.
If you talk to a financial advisor (a real one anyway) they will always ask about your risk profile because in an “efficient” market all investments should have about the same expected return, just the prob distribution will be different.
For lotto, it’s almost all on the negative side with negligible but non-zero values all the way up to the millions.
I bite this bullet in some contexts but not others.
First, there’s a really good reason we have cultural memes like “a bird in the hand is better than two in the bush”: personal utility (in terms of life outcomes) is extremely sublinear as a function of wealth (or other tangible and easily quantifiable goods), and so in personal decisions it doesn’t make sense to pursue some strategies with positive expected profit but extremely high variance. That’s such a robust heuristic, in fact, that going against it in other contexts feels wrong to us.
(On the other hand, we also have the biases that come from a qualitative intuition about probabilities; we seem to act as if the options for probability are (in order) certain/almost certain/likely/50-50/unlikely/very unlikely/impossible. A raffle with 300 tickets and a lottery with 100 million tickets, in other words, don’t feel very different in probability to us. That’s part, I think, of why people do play the lottery, because the payoff is saliently enormous and the probability isn’t so saliently infinitesimal.)
But when it comes to things like your asteroid example, then yes, one should follow one’s best calculation of expected utility. The real dangers there, it seems to me, stem not from the magnitudes of the payoffs and probabilities so much as human foibles in doing the calculations- for instance, the tendency to think too much about one particular unlikely possibility to the exclusion of a much broader class. (A classic example is Pascal’s Wager; a modern example is what Bruce Schneier calls a waste of resources on stopping particular “Hollywood terrorist scenarios” instead of focusing on the kind of detective work that more effectively guards against a wider range of attacks.) If there were a government or entity not liable to fall into this trap, I’d advise it to calculate expected utilities rather than ignoring small probabilities.
Cosma Shalizi is not only intuitively right but echoes very closely the justifications that I’ve heard from people who do buy lottery tickets on a regular basis, and who give them as gifts (stocking stuffers, little presents, that sort of thing), to other people most of whom don’t think such gifts weird (much less ironic). The lottery is that wonderful thing, a tax on the willing, as Jefferson said; it’s also a form of entertainment, an excuse for ticket buyers (including those who give the tickets as gifts) to imagine what they might do (or what their friends or family might do) with a giant windfall. I think I should start reading Cosma Shalizi’s blog.
YES — you should read Cosma’s blog.
[…] asked a while back how seriously we should take expected utility computations that rely on multiplying very large […]
Somebody needs to go back to school.In Slate when you wrote about buying a lottery ticket,you put up a very long equation,jackpot multiplied by various numbers multiplied by 0.Anything multiplied by zero,will equal zero,grade school math and yet somehow your equation was “proof” that a lottery ticket wasn’t a suckers bet.