It was a surprise to me when I learned that sometimes lottery tickets actually have positive expected values. This happens only rarely, and generally only for progressive jackpots. But I was alarmed that it happens at all. So often in the literature people are judged as irrational if they don't act in accordance with expected value. Should I buy a lottery ticket with a positive expected value?
For example, say someone offers you the following bet: we flip a coin, and if it comes up heads you get $10. If it comes up tails you lose $1. This game has expected value of +$10/2 + -$1/2 = +$4.50. Should you play this game?
If I can play this game arbitrarily many times, and I have a handful of dollars to start out with, I will play this game as a full time job, and I will be rich. After playing just six times the odds of losing money overall are under 1%, and that number goes even further to zero the more I play. I am quickly in a world where it is more likely that I am better off than I was before.
However, invite me to play this game just once, and I am ambivalent. Maybe I'll make some money, but it is equally likely that I will lose money. I don't think it is irrational to decline the offer to play just once. (Some might say that it is irrational because it should become one of many positive expected value opportunities that I take, ensuring a positive outcome overall, but (a) positive expected value gambling opportunities are notoriously rare, and (b) the typical slimness of the odds matters, as follows.)
Even the best positive expectation lottery ticket has odds much worse than one in a million, so let's take that as the odds for a $1 ticket. Say the prize for this bet is $2 million. The a ticket has an expected value of nearly +$1. (For simplicity in the following, I treat each lottery ticket as independent, which is not quite true, but is not such an awful simplification since in reality you can't buy enough lottery tickets to benefit significantly by owning a large portion.)
In the coin flip bet above, if you play twice you have over a 50% chance of having more money than you started with. How many times do you have to play this excellent one-in-a-million lottery game to be more likely to have won money than lost? log(0.5)/log(1 - 1/1,000,000) = nearly 700,000 times.
Both the coin flip and lottery here have positive expected value, and if you could play them arbitrarily many times, both would be cash cows. In the case of the coin flip, the generous single-game odds make it fairly easy to ensure that your odds of winning overall are excellent. But as the odds get worse, as in the case of the one-in-a-million bet, (still not bad, by lottery standards) the nature of the physical universe makes it nearly impossible to play enough games to achieve decent odds of a positive outcome.
I think it is rational to consider the probability of the overall outcome of gambling rather than the expected value of the gamble. It is not just about psychological "risk aversion" and it is not just about ignorance of expected value.
I have never seen an army of hedge fund quants scrambling for the nearest 7-11 to invest heavily in high-jackpot Powerball tickets. In any event, I won't be buying any - regardless of expected value.