For Online Publication : Supplementary Appendix for Pirates of the Mediterranean

Since p = β T −1 b in the final bargaining period, we can compute the upper bound on the remaining types such that selling to any possible type (by setting p = β T −2 b) is optimal according to the optimization problem: max p p · F (X) − F (p/β T −2 − βδb 1 − βδ) F (X) + F (p/β T −2 − βδb 1 − βδ) F (X) (β T −2 v + β T −1 δb). Therefore, optimal p in the final period satisfies: (1) v + βδb − p/β T −2 (1 − βδ) f (p/β T −2 − βδb 1 − βδ) − F (p/β T −2 − βδb 1 − βδ) + F (X) ≤ 0. Setting p = β T −2 b (and using F (b) = 0) allows us to determine the upper bound on remaining types: X = F −1 [(b − v 1−βδ)f (b)]. From the way we solved for X, this upper bound on remaining types in the final period can also be interpreted as the minimum type such that if types were hypothetically distributed over [X, ¯ b], the game would end one period earlier than