Please explain the answer, especially the utility functions, thanks!

Question 12 (a) Describe the set-up of the two-player version of the ultimatum game [ 3 points ] (b) If Alice and Bob are fully rational players with purely self-regarding preferences (i.e. if they don't care about each other's payoffs), what will be the result if they play the game you just described? [ 2 points ] (c) Is the result from part (c) realistic? Why or why not? How would Alice's or Bob's utility functions need to be different to give a different kind of result? (Give a mathematical example.) [5 points ] Source: This question comes from the week 9 tutorial sheet. Explanation: (a) This is given in the lecture slides: a proposer is given a sum of money EM and they must make a one-time-only offer to split this amount with another player called the responder. The responder can either accept the split, or refuse it so that both players get f0. (b) in this case, Alice will offer Bob the absolute minimum, and Bob will accept it. (c) No. In the real world, responders tend to reject offers which are much below 30% of the total pot, and proposers rarely offer so little. Common mistakes: In part (a), many students confused the prisoners' dilemma with the ultimatum game. In part (b), many students seemed confused about what 'rationality' means in economics. Some mistook it for common sense, or "what would be sensible to do in this situation". This sometimes lead to strange reasoning (e.g. it is rational to offer 0, but then people reject it, so we offer a little more). In part (c), there was substantial confusion about what a utility function actually is, and what it represents. Students would often express the utility of one person in terms of the utility of the other, rather than in terms of the respective monetary payoffs (e.g. U1=x-U2 rather than U1=M1-M2). Quite a few students were confused by the importance of altruism vs. fear of rejection in generating large offers. In practice, either motivation can explain large offers, but the two motives will result in very different utility functions. If the proposer is selfish, then they'll have a utility function like Up = Mp or Up = \\Mp or something like that. If the proposer is altruistic, then they'll have a utility function which considers the responder - something like Up = Mp + MR Or Up = Mp + MR or similar. For an altruistic proposer, the nature of the utility function of the responder is not so important (as long as the responder values money). But if the proposer is only giving high offers out of fear of rejection, then it's because the proposer thinks that the responder has a utility function which is something like UR = MR - Mp. If the proposer thinks that the responder has this kind of utility function, then they will make a high offer whether they are altruistic or not. It is important to be clear about these different kinds of motives, whereas many students were not