Bargaining in Bilateral Oligopoly: An Alternating Offers Representation of the "Nash-in-Nash" Solution

The concept of a Nash equilibrium in Nash bargains, proposed in Horn and Wolinsky (1988), has become the workhorse bargaining model for predicting and estimating the division of surplus in applied analysis of bilateral oligopoly. This paper proposes a non-cooperative foundation for this concept—in which agreements between each pair of firms maximizes their bilateral Nash product conditional on all other negotiated agreements—by extending the Rubinstein (1982) alternating offers model to a setting with multiple upstream and downstream firms. In our model, downstream firms make simultaneous offers to upstream firms in odd periods, and upstream firms make simultaneous offers to downstream firms in even periods. Given restrictions on underlying payoffs, we prove that there exists a perfect Bayesian equilibrium with passive beliefs that generates the "Nash-in-Nash" solution, and that this equilibrium outcome is unique.