[arcsin]

I think there's two factors. One is the nonlinear utility of money, and the other is the cost/value of losing/winning.

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.

[OJFord]

Payouts are also non-linear though, with enormous discontinuities.

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.)