Consider a three-player version of the bargaining game discussed in class:

(1) In round 1, player 1 offers a division (x1, x2, x3) of the dollar (i.e., x1 + x2 + x3 = 1). Players 2 and 3 simultaneously accept or reject the offer; if both 2 and 3 accept, the game ends and each player keeps their share of the offer. If any one of them rejects the offer, the game goes to the next round.

(2) In round 2, player 2 offers a division (y1, y2, y3) of the dollar (i.e., y1 + y2 + y3 = 1). Players 1 and 3 simultaneously accept or reject the offer; if both 1 and 3 accept, the game ends and each player keeps their share of the offer. If any one of them rejects the offer, the game goes to the next round.

(3) In round 3, player 3 offers a division (z1, z2, z3) of the dollar (i.e., z1 + z2 + z3 = 1). Players 1 and 2 simultaneously accept or reject the offer; if both 1 and 2 accept, the game ends and each player keeps their share of the offer. If any one of them rejects the offer, the game goes to the next round. (4) In round 4, the players receive payoffs (e1, e2, e3) at time t = 4. (Assume e1+e2+e3 = 1 as well.)

Suppose the players use a common discount factor 0 < < 1.

(a) Find a subgame perfect Nash equilibrium strategy for each player. Note that your strategy should fully specifiy what each player does at each round in which they are asked to play. That is, in round 2, player 2's strategy should describe the offer she makes, and player 1's strategy should specify which offers he'll accept and which ones he'll reject, etc.

(b) Using a reasoning similar to what was employed in class, find a subgame perfect equilibrium strategy for all players in the infinite horizon bargaining gamethe one in which player 1 makes an offer in rounds 1, 4, 7, 10, etc.; player 2 makes an offer in rounds 2, 5, 8, etc.; and player 3 makes an offer in rounds 3, 6, 9, etc.