Equilibrium payoffs and proposal ratios in bargaining models

Abstract

We analyze a bargaining model which is a generalization of the model of Rubinstein (Econometrica 50(1):97–109, 1982) from the viewpoint of the process of how a proposer is decided in each period. In our model, a player’s probability to be a proposer depends on the history of proposers and players divide a pie of size 1. We derive a subgame perfect equilibrium (SPE) and analyze how its SPE payoffs are related to the process. In the bilateral model, there is a unique SPE. In the n-player model, although SPE may not be unique, a Markov perfect equilibrium (MPE) similar to the SPE in the bilateral model exists. In the case where the discount factor is sufficiently large, if the ratio of opportunities to be a proposer converges to some value, players divide the pie according to the ratio of this convergent value under these equilibria. This result implies that although our process has less regularity than a Markov process, the same result as in the model that uses a Markov process holds. In addition to these results, we show that the limit of the SPE (or the MPE) payoffs coincides with the asymmetric Nash bargaining solution weighted by the convergent values of the ratio of the opportunities to be a proposer.