Most of this exercise does not concern players' preferences about risk, but it is a good example of a game with moves of nature that can be analyzed by using subgame perfection. Consider a T-period bargaining game like the alternating-offer game discussed in Chapter 19, except suppose that in each period, nature chooses which player gets to make an offer.
At the start of a period, nature selects player 1 with probability q1 and player 2 with probability q2 , where q1 + q2 = 1. The selected player proposes a split of some surplus (of size 1). The other player then responds by accepting or rejecting the offer. Acceptance ends the game and yields the payoff associated with the proposal, discounted according to the period in which agreement is made. Rejection leads to the next period, except in period T, in which case the game ends with zero payoffs. Assume the players are risk neutral and have the same discount factor d.
(a) Describe the extensive form of this game.
(b) Find and report the subgame perfect equilibrium.
(c) What are the equilibrium payoffs? Offer an interpretation in terms of the standard bargaining solution (bargaining weights and a disagreement point).
(d) Suppose T = 2 and that the players are not necessarily risk neutral. How would you expect the players' risk preferences to influence the equilibrium proposals that are made in the first period?