Microeconomics question (Game theory)

8.7 An Alternating-Offer Bargaining Game Stahl (1972) and Rubinstein (1982) considered a somewhat different model of bargaining, in which the players alternate making offers until one is accepted. Rubinstein considered a general model in which there is no nite bound on the number of offers that may be made but in which each player has some cost of time, so his payoff depends on both the offer accepted and the number of the round in which this acceptance took place. Rubinstein elegantly characterized the subgameperfect equilibria of these alternating-offer bargaining games for a wide class of cost-of-time formulas and showed that the subgame-perfect equilib- rium is unique when each player's cost of time is given by some discount factor 8. Binmore (see Binmore and Dasgupta, 1987, chap. 4) showed that, with the 8-discounting cost-of-time formula, if the per-round dis- c0unt factor 8 is close to 1, then the outcome of the unique subgame- perfect equilibrium is close to the Nash bargaining solution. In this section, we present a special case of these results. However, instead of assuming discounting or making some other assumptions about players' trade-offs between prizes received at different points in time, we assume here that the cost of delay in bargaining is derived from an exogenous positive probability, after each round, that the bar- gaining process may permanently terminate in disagreement if no offer has yet been accepted. Let (Ev) be any regular two-person bargaining problem, and let p, and 132 be numbers such that 0 v1 + '02. As a function of these parameters (vlg, 11., '02, pl, 132), what is the general formula for the offer that player 1 would always make and the offer that player 2 would always make in the subgame-perfect equilibrium