Consider a repeated version of the game in exercise 24.5. In this version, we do not give all the proposal power to one person but rather imagine that the players are bargaining by making different proposals to one another until they come to an agreement. In part A of the exercise we analyze a simplified version of such a bargaining game, and in part B we use the insights from part A to think about an infinitely repeated bargaining game.
A: We begin with a 3-period game in which $100 gets split between the two players. It begins with player 1 stating an amount x1 that proposes she should receive x1 and player 2 should receive (100− x1). Player 2 can then accept the offer—in which case the game ends with payoff x1 for player 1 and (100−x1) for player 2; or player 2 can reject the offer, with the game moving on to period 2. In period 2, player 2 now has a chance to make an offer x2 which proposes player 1 gets x2 and player 2 gets (100−x2). Now player 1 gets a chance to accept the offer—and the proposed payoffs—or to reject it. If the offer is rejected, we move on to period 3 where player 1 simply receives x and player 2 receives (100− x). Suppose throughout that both players are somewhat impatient — and they value $1 a period from now at $δ (< 1). Also suppose throughout that each player accepts an offer whenever he/she is indifferent between accepting and rejecting the offer.
(a) Given that player 1 knows she will get x in period 3 if the game continues to period 3, what is the lowest offer she will accept in period 2 (taking into account that she discounts the future as described above)?
(b)What payoff will player 2 get in period 2 if he offers the amount you derived in (a)? What is the present discounted value (in period 2) of what he will get in this game if he offers less than that in period 2?
(c) Based on your answer to (b), what can you conclude player 2 will offer in period 2?
(d) When the game begins, player 2 can look ahead and know everything you have thus far concluded. Can you use this information to derive the lowest possible period 1 offer that will be accepted by player 2 in period 1?
(e) What pay off will player 1 get in period 1 if she offers the amount you derived in (d)? What will she get (in present value terms) if she offers an amount higher for her (and lower for player 2)?
(f) Based on your answer to (e), can you conclude how much player 1 offers in period 1 — and what this implies for how the game unfolds in sub game perfect equilibrium?
(g) True or False: The more players 1 is guaranteed to get in the third period of the game, the less will be offered to player 2 in the first period (with player 2 always accepting what is offered at the beginning of the game).
B: Now consider an infinitely repeated version of this game; i.e. suppose that in odd-numbered periods — beginning with period 1 — player 1 gets to make an offer that player 2 can accept or reject, and in even-numbered periods the reverse is true.
(a) True or False: The game that begins in period 3 (assuming that period is reached) is identical to the game beginning in period 1.
(b) Suppose that, in the game beginning in period 3, it is part of an equilibrium for player 1 to offer x and player 2 to accept it at the beginning of that game. Given your answer to (a), is it also part of an equilibrium for player 1 to begin by offering x and for player 2 to accept it in the game that begins with period 1?
(c) In part A of the exercise, you should have concluded that —when the game was set to artificially end in period 3 with payoffs x and (100−x), player 1 ends up offering x1 = 100−δ(100− δ x) in period 1, with player 2 accepting. How is our infinitely repeated game similar to what we analyzed in part A when we suppose, in the infinitely repeated game beginning in period 3, the equilibrium has player 1 offer x and player 2 accepting the offer?
(d) Given your answers above, why must it be the case that x = 100−δ (100−δx)?
(e) Use this insight to derive how much player 1 offers in period 1 of the infinitely repeated game. Will player 2 accept?
(f) Does the first mover have an advantage in this infinitely repeated bargaining game? If so, why do you think this is the case?