Question: (a) Consider the two-person Rubinstein-Stahl model of section 4.4. The two players bargain to divide a pie of size 1 and take turns making offers. The discount factor is 6. Introduce "outside options" in the following way: At each period, the player whose turn it is to make the offer makes the offer; the other player then has the choice among

(1) accepting the offer,

(2) exercising his outside option instead, and

(3) continuing bargaining (making an offer the next period). Let x0 denote the value of the outside option. Show that, if x0 ≤δ/(1 + δ), the outside option has no effect on the equilibrium path. Comment. What happens if x0 > δ/(1 + δ)?

(b) Consider an alternative way of formalizing outside options in bargaining. Suppose that there is an "exogenous risk of breakdown" of re-negotiation (Binmore et al. 1986). At each period t, assuming that bargaining has gone on up to date t, there is probability (1 - x) that bargaining breaks down at the end of period t if the period-t offer is turned down. The players then get xo each. Show that the "outside opportunity" x0 matters even if it is small, and compute the subgame-perfect equilibrium.

(c) In their study of supply assurance, Bolton and Whinston (1989) consider a situation in which the outside option is endogenous. Suppose that there are three players: two buyers (i = 1, 2) and a seller (i = 3). The seller has one indivisible unit of a good for sale. Each buyer has a unit demand. The seller's cost of departing from the unit is 0 (the unit is already produced). The buyers have valuations v1 and v2, respectively. Without loss of generality, assume that v1 ≥ v2. Bolton and Whinston consider a gen-eralization of the Rubinstein-Stahl process. At dates 0, 2, ..., 2k, ..., the seller makes offers; at dates 1, 3, ..., 2k + 1, the buyers make offers. Buyers' offers are prices at which they are willing to buy and among which the seller may choose. The seller can make an offer to only a single buyer, as she has only one unit for sale (alternatively one could consider a situation in which the seller organizes an auction in each even period). Consider a stationary equilibrium and show that, if parties have the same discount factor and as the time between offers tends to 0, the parties' perfect-equilibrium payoffs converge to v1/2 for both the seller and buyer 1 and to 0 for buyer 2 if v1/2 > v2, and to v2 for the seller, v1 - v2 for buyer 1, and 0 for buyer 2 if v1/2 <>2. (For a uniqueness result see Bolton and Whinston 1989.)