Random-settlement arbitration and the generalized Nash solution: one-shot and infinite-horizon cases

We study bilateral bargaining á la Nash (Econometrica 21:128–140, 1953) but where players face two sources of uncertainty when demands are mutually incompatible. First, there is complete breakdown of negotiations with players receiving zero payoffs, unless with probability p, an arbiter is called upon to resolve the dispute. The arbiter uses the final-offer-arbitration mechanism whereby one of the two incompatible demands is implemented. Second, the arbiter may have a preference bias toward satisfying one of the players that is private information to the arbiter and players commonly believe that the favored party is player 1 with probability q. Following Nash’s idea of ‘smoothing,’ we assume that $$1-p$$ 1 - p is larger for larger incompatibility of demands. We provide a set of conditions on p such that, as p becomes arbitrarily small, all equilibrium outcomes converge to the Nash solution outcome if $$q=1/2$$ q = 1 / 2 , that is when the uncertainty regarding the arbiter’s bias is maximum. Moreover, with $$q\ne 1/2$$ q ≠ 1 / 2 , convergence is obtained on a special point in the bargaining set that, independent of the nature of the set, picks the generalized Nash solution with as-if bargaining weights q and $$1-q$$ 1 - q . We then extend these results to infinite-horizon where instead of complete breakdown, players are allowed to renegotiate.