Consider the ultimatum game in which player 1 (the proposer ) is given $10. He proposes a split of the $10 between him and player 2 (the receiver ). Player 2 then chooses whether to accept or reject the offer. If he accepts, the game ends and each player gets his proposed share. If he rejects, they both get nothing. Instead of maximizing his own monetary payoff, player 1 maximizes the utility function: u1 = x1 − γx2, where x1 is player 1’s monetary payoff, x2 is player 2’s monetary payoff and γ is a constant between 0 and 1. Similarly, player 2 maximizes u2 = x2 − γx1.
a. What is the subgame perfect equilibrium of this game? Prove that this is the case.
b. Calculate each player’s equilibrium payoff.
c. What happens to player 2’s share of the $10 as γ goes to zero? What happens as γ goes to one?