Q1. Consider the ultimatum game in which player 1 (the proposer ) is given $10.

Question

Q1. 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 x2, where x1 is player 1’s monetary payoff, x2 is player 2’s monetary payoff and is a constant between 0 and 1. Similarly, player 2 maximizes u2 = x2 x1.

a. What is the subgame perfect equilibrium of this game? Prove that this is the case. (6 marks)

b. Calculate each player’s equilibrium payoff. (2 marks)

c. What happens to player 2’s share of the $10 as goes to zero? What happens as goes to one? (2 marks)

please show me how to do it. thanks a lot

Explanation / Answer

a. the perfect equilibrium shall occur when x1 = x2 so that both of them can maximise their pay offs. This can be proved by analysing the fact that total utility is highest when marginal utility is equal to zero.
b. Each player;s equilibrium payoff shall be at 5$ each
c. If player 2's goes to zero then player 1 shall not give any money to player 2 and if it goes to 1, then player 1 shall give the entire 10$ to player 2.

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