Uncertainty and Delay in Bargaining

Abstract

This paper investigates the relationship between uncertainty and delay of agreement in the one-sided offer bargaining model with two-sided uncertainty where the seller makes offers. We construct a weak stationary equilibrium in which different types of the seller charge different prices in every period. We completely characterize the separating equilibrium by three regularity conditions, and show that the time interval between offers converges to zero, the seller's initial price offer in a separating equilibrium converges with probability 1 to the lowest valuation of the buyer if and only if the gain from trading is common knowledge. We have witnessed a proliferation of sequential bargaining models that, with a few remarkable exceptions (Fudenberg and Tirole (1983), Cramton (1984) and Chatterjee and Samuelson (1987)), are based on models with one-sided uncertainty. The importance of studying bargaining models with one-sided uncertainty is that despite their highly special informational structure, we can derive valuable insights into the role of uncertainty in the delay in reaching an agreement, while keeping the computational exercise tractable. However, the assumption of one-sided uncertainty essentially excludes the situation where the buyer's actual reservation value is lower than the seller's actual reservation value, though they are still engaged in the bargaining process. The primary objective of this paper is to investigate the role of uncertainty about the gain from trading in a sequential bargaining model with two-sided incomplete information. In the models with one-sided incomplete information, important studies (Fudenberg, Levine and Tirole (1985), and Gul, Sonnenschein and Wilson (1986)) are often based on weak stationary equilibria where the equilibrium strategy of each player only depends upon the posterior conjecture about the other player and the previous period's offer. Although the importance of two-sided uncertainty and the weak stationary equilibrium is well recognized, previous research on sequential bargaining models has not combined the two features within the same framework. Studies on the models under two-sided uncertainty have not focused on weak stationary equilibria. For example, Cramton (1984) computes a specific sequential equilibrium with interesting properties, in which the equilibrium strategies of both players depend upon histories in a rather complex fashion. On the other hand, it is only in models under one-sided uncertainty that the weak stationary equilibrium has been extensively studied. This paper computes and analyzes the weak stationary equilibria in a model under two-sided uncertainty in order to investigate the effect of uncertainty on the delay in agreement. We will extend the results from previous studies of one-sided uncertainty models. The computation of the equilibrium in models with two-sided uncertainty is generally very complicated. Furthermore, we must identify a sensible outcome if more than one equilibrium exists in the model. To simplify the analysis, we choose a standard bargaining