In experimental tests of the ultimatum bargaining game, subjects who propose the split rarely offer a tiny share of the surplus to the other party. Furthermore, sometimes subjects reject positive offers. These findings seem to contradict our standard analysis of the ultimatum game. Many scholars conclude that the payoffs specified in the basic model do not represent the actual preferences of the people who participate in the experiments. In reality, people care about more than their own monetary rewards. For example, people also act on feelings of spite and the ideal of fairness. Suppose that in the ultimatum game, the responder's payoff is given by y + a(y - z), where y is the responder's monetary reward, z is the offerer's monetary take, and a is a positive constant. That is, the responder cares about how much money he gets and he cares about relative monetary amounts (the difference between the money he gets and the money the other player gets). Assume that the offerer's payoff is as in the basic model.
(a) Represent this game in the extensive form, writing the payoffs in terms of m, the monetary offer of the proposer, and the parameter a.
(b) Find and report the subgame perfect equilibrium. Note how equilibrium behavior depends on a.
(c) What is the equilibrium monetary split as a becomes large? Explain why this is the case