Consider the following variations of the ultimatum game: there are $10 to split, with minimum denomination of $1 dollar. The sender only cares about the money he keeps. The receiver cares about the relative money. So if she accepts an offer of 3 dollars for herself, her utility is 3 - 7 = -4. Assume for brevity that when indifferent between accepting and rejecting the receiver always accepts. (a) What is the rollback equilibrium of this game? (b) How does it compare with the rollback equilibrium obtained under the assumption that players care only about the money they earn? Suppose now that the population of receivers is made, in equal share, of people who care about relative money (type B) and people who care about the money they earn for themselves (type A). The sender does not know whether he is facing a type A or B receiver. The sender only knows that they are equally represented in the population of receivers. (a) Find the rollback equilibrium of this game following these steps: • Type A and type B receivers accept the offer when indifferent between accepting and rejecting. • Write the rollback equilibrium strategy for type A receivers. • Write rollback equilibrium strategy for type B receivers. • The sender’s payoff from any offer is the average between the payoff obtained if the receiver is A type and B type. (b) How does your answer change if in the population of receivers type A are represented in a proportion p belonging to the interval (0, 1)? In this case the average payoff for the sender is p times payoff from meeting an A type + (1 - p) times payoff from meeting a B type. [Note you have just done the case p = 1/2]