For example, U(I) = sqrt(I/$15k), i < $15k, U(I) = 1, $15k < I < $100k, and then U(I) = 2, $100k < I. (I is income. The discontinuity at $100k is not necessary, but the flatness on [$15k, $BIGNUM] is.)
I.e., consider a poor person making $15k/year. If they spend $100/year on lotto (assume 1 in 1 million chance of winning), with virtually no chance of success, that suggests that 1e6 x Utility(big 'thing') > Utility($100 worth of goods/services). In particular, this suggests that the poor person assigns a very low value to an extra $100 worth of goods and services. If this is the case, then the lottery is actually a very efficient tax! It only deprives people of something they barely care about at all.
If correct, this theory would also explain why poor in the US work so little - they don't value the things that the extra money could buy.
 Like PaulJoslin, I am implicitly assuming that lottery ticket buyers are rational and inferring their utility function from their choices. It's also possible that lotto buyers simply don't understand probability, in which case all this speculation is irrelevant.
Now say i=20,010 before playing. Expected utility without playing the game is trivially 20,010. Expected utility with playing is [(20k)*(10^6-1)+(100k+2^20k)]/10^6. This is a stupidly high number, in the 4x10^6020 neighborhood.
It should be possible to produce a more natural form that does this, without going piecewise.
- If I never win, there is no practical loss (total spent sum too low)
- If I win the jackpot, it's life changing
- If I never play, life never changes (or at least there is not such a chance).
- Most of the spent money goes to charity anyway
I never buy lottery tickets or scratch cards (though I love sports betting and casinos), however my father does buy a lottery ticket every week.
He's retired, and has enough money to live fine - he owns the flat he lives in, he has enough money to pay for his £100/month TV/internet package, to smoke a pack of cigarettes a day, and to buy whatever food he fancies without thinking about what's cheaper to eat.
He has a choice of living like that and accepting that's how the rest of his life will remain, or spending a tiny amount each week and having a dream of becomming a millionaire. He's not an idiot, he knows the odds are against him, but his buying lottery tickets doesn't have any impact on his lifestyle, and it lets him imagine that one day he might win big.
He wouldn't notice any difference in his financial situation if he stopped buying the tickets, so really the only downside is that, if he keeps buying them over a twenty year period, the inheritance that comes to myself and my two siblings when he dies will be a bit less. But if you think that way, his quitting the lottery would make a far smaller difference than if he didn't subscribe to extremely expensive sports channels, if he didnt spend £40/week on cigarettes, and so on. I'd far rather see him enjoy his money than save it for my sake - I'll get by fine either way.
Sure, for some people the amount they spend on lottery tickets does take away from money they could spend on other stuff, but there are people who are well off enough to be able to afford the cost, without being so well off that winning the lottery isn't a dream they enjoy having.
I consider that the Pascal's Wager :) Everything to gain, nothing to lose.
Nothing to lose, man.
If a reputable organisation held a one-off lottery, in which you stand a very slim chance of winning £5M, for a £1 stake, then (morals and poverty notwithstanding) then the default position is to enter. £1 doesn't really buy you anything much.
If, on the other hand, the stake were £52, then you'd think twice. £52 is a fairly significant sum it could buy a decent meal out for two, or the latest game with enough change for a couple of beers on the way home.
Playing every week for a year is the same as that second scenario.