Do you have any examples of real economic circumstances under which a sane person (someone who isn't solely concerned with maximizing the number of Porsches they own, e.g.) would have a convex utility/money curve?

(If there is a way to phrase this question so that it seems more curious and less confrontational, please assume that I said that instead.)

From Spetzler and Stael von Holstein (1975), there is a variation of Bet On It that doesn't require risk neutrality.

Say we are going to flip a thumbtack, and it can land heads (so you can see the head of the tack), or tails (so that the point sticks up like a tail). If we want to assess your probability of heads, we can construct two deals.

Deal 1: You win $10,000 if we flip a thumbtack and it comes up heads ($0 otherwise, you won't lose anything). Deal 2: You win $10,000 if we spin a roulette-like wheel labeled with numbers 1,2,3, ..., 100, and the wheel comes up between 1 and 50. ($0 otherwise, you won't lose anything).

Which deal would you prefer? If you prefer deal 1, then you are assessing a probability of heads greater than 50%; otherwise, you are assessing a probability of heads less than 50%.

Then, ask the question many times, using a different number than 50 for deal 2. For example, if you first say you would prefer deal 2, then change it to winning on 1-25 instead, and see if you still prefer deal 2. Keep adjusting until you are indifferent between deal 1 and 2. If you are indifferent between the two deals when deal 2 wins from 1-37, then you have assessed a probability of 37%.

The above describes one procedure used by professional decision analysts; they usually use a physical wheel with a "winning area" that is adjustable continuously rather than using numbers like the above.