Out of curiosity, what text is this from? Are there any similar examples, or definitions of utility functions, that might help us understand the problem? It seems we are as confused as you are for the time being.
This is from a book of Ignou MA in economics course. Here is the link. The question is in page no. 11.
There is no similar example. This chapter tells us about the expected utility theory:- an individual choose that lottery which gives him maximum expected utility. This chapter is about-
1. How to calculate the 'expected value' and then, the 'expected utility' of a gamble/lottery. With the help of a person's utility function, we can calculate a lottery's expected utility for that person.
2. What is fair gamble and how most people are risk-averse and thus, avoid fair gambles as they don't like risk.
3. What is risk-averse, risk-taker and risk-neutral
4. What is certainty equivalent and risk premium. etc.
Now, about the problem, what I think is-
The lottery is not a fair one as it's expected value is greater than zero. So there is no way, one would deny to play it.(so why would I offer anything to the players to play the lottery???) If it had been a fair one, I would have to offer the risk-averse player certain amount to encourage him to play this lottery(risk-taker doesn't need an offer to play it as he loves risks and accepts fair lotteries gladly). But, it's not a fair one, but a favorably unfair one.
From the utility functions, it's clear that the 1st player is risk averse and the 2nd player is risk-taker. (the 2nd derivative of a utility function tells us the nature of the person).