You have a coin, and each time you don't get heads the payout doubles. Get heads and get the payout. What do you pay to play this?
It turns out that you can't work out an E[X] for this, as it's a series that doesn't converge. But also if you go around and ask people what they'd pay, surprisingly they don't say infinity. Expectation is thus not the last word in decision making.
Turns out utility is a thing that matters, and we can come up with some thoughts on it (von Neumann-Morgenstern etc).
I'm not sure it's that surprising, the expectation is only infinity if you think the person making the payout has the capacity to pay an infinite amount. The second you think this amount is finite (which must always be the case in reality), then the expectation becomes finite and probably quite small.
I agree in the more general point, expectation is not always the best tool for extremely rare / extremely significant scenarios.
In general money has a diminishing utility, your first dollar is worth more to you than your millionth dollar. In order for it to be worth it to play a -EV lottery not only does you utility graph need to be nonlinear, it cannot decrease monotonically. At some point your nth dollar needs to be worth more (to you) than your first dollar. For example, you have one dollar and a life saving drug for your terminal disease costs 1 million dollars. You would see a sudden spike in utility at the 1 million dollar mark. It would be rational to play the lottery in this case.
Another possibility is that the event of winning money or even just the thought of possibly winning that money gives you more value than simply having the money alone, and this tips the scale. This would be like the opposite of insurance. For some people the experience of losing, or worrying about the possibility of losing money is so negative that it still makes it worth it to buy insurance.
Your millionth dollar might have less marginal utility, but you were never going to win 10^6 -1 dollars anyway. So really to consider marginal utility you need 'margins' as wide as the payouts. (Really stretching/breaking the meaning of marginal, but hopefully you see what I mean.)
Yes, and I sometimes think about the inverse - that people like to hate on lottery players (which I'm not) with a clever sounding argument like E[X] < 1, so don't play, you'll lose.
But this fails to acknowledge the harder to model 'but I can pay £1 to play, I won't miss it, who cares, it's insignificant, and on the tiny chance I win, the win is significant'. I.e. 'but Var[X] is massive'.