“Nash-in-Nash” Bargaining: A Microfoundation for Applied Work
A “Nash equilibrium in Nash bargains” has become a workhorse bargaining model in applied analyses of bilateral oligopoly. This paper proposes a non-cooperative foundation for “Nash-in-Nash” bargaining that extends the Rubinstein (1982) alternating offers model to multiple upstream and downstream firms. We provide conditions on firms’ marginal contributions under which there exists, for sufficiently short time between offers, an equilibrium with agreement among all firms at prices arbitrarily close to “Nash-in-Nash prices”—i.e., each pair's Nash bargaining solution given agreement by all other pairs. Conditioning on equilibria without delayed agreement, limiting prices are unique. Unconditionally, they are unique under stronger assumptions.
This paper was previously circulated under the title “Bargaining in Bilateral Oligopoly: An Alternating Offers Representation of the ʽNash-in-Nashʼ Solution.” We would like to thank Elliot Lipnowski, Sebastián Fleitas, and Eli Liebman for excellent research assistance; Attila Ambrus, John Asker, Catherine de Fontenay, Joshua Gans, Patrick Greenlee, Heski Bar-Isaac, Rachel Kranton, Volcker Nocke, Janine Miklos-Thal, Dan O'Brien, Alexander Raskovich, Stan Reynolds, Michael Riordan, Christopher Snyder, Mike Whinston, Tom Wiseman, Ali Yurukoglu, and numerous seminar audiences for useful discussion; and the editor and three anonymous referees for helpful comments. Gowrisankaran acknowledges funding from the National Science Foundation (Grant SES-1425063). The usual disclaimer applies. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.
Allan Collard-Wexler & Gautam Gowrisankaran & Robin S. Lee, 2019. "“Nash-in-Nash” Bargaining: A Microfoundation for Applied Work," Journal of Political Economy, vol 127(1), pages 163-195.