Question

2. Two players are bargaining, just as in the Rubinstein's alternating offers model studied in class, over the division of a cake of size 1. There are two differences from the standard model: first, there is no discounting. Second, while an acceptance guarantees implementation of the going proposal, following every rejection there is an exogenous probability p > 0 that the game will completely break down. If that happens, each player gets gets 0 <b < 1/2. If not, the rejector gets a chance to propose, and the game goes for one more phase. No agent can deliberately choose to simply take the breakdown payoffs. Assume that the one-shot deviation principle holds here (it does, but you dont have to prove it). a. Prove that there is a unique subgame-perfect equilibrium for this game, and describe the payoff to each proposer. b. Describe what happens as p-0, interpret the outcome.

Answer 1

Players Risk Neutral, delta = 1, perfectly patient, No discount factor, If offer is rejected at any pd, then Game may not start with when an exogenous prob. Rejen in pd 1 {Game goes to pd_2 with prob (1 - p), Game ends at 1 with prob p. i proposes, j responds \r\n as b_i + b_j < 1, so incentive to reach a Negotiation at date 1 itself. Assume unique SPNE - date k rightarrow i(p), j(R) i rightarrow v_i (= pi_i) eqm offer (R - 1): j(p), i(R) No discount, so v_R is valued v_k only, i Rejects he expects to get v_k, prob (1 - p), & b_i with p E_n_i = (1 - p) v_i + pb_i at (R - 1) Risk Neutral players pi_j = v_R - 1 = v_j player j must offer v_j = [1 - ((1 - p)v_i + b_i p)] at date (R - 2) rightarrow Roles reversed v_i = [1 - ((1 - p) v_j + pb_j)]