Ultimatum bargaining with envy under incomplete information

We propose an ultimatum bargaining model in which the parties experience an envy-based externality that is private information. Our results indicate that there is a threshold for the proposer's envy which determines whether there will be either a perfectly equitable, certain agreement or an uncertain, inequitable agreement, and that this threshold rises as the distribution of the responder's envy level improves in a first-order stochastic-dominance sense. In addition, conditionally on the scenario ruling out a perfectly equitable agreement, we show that the proposer's envy level plays a dual role: (i) it increases the probability of a negotiation breakdown, and (ii) it constitutes a source of bargaining power. Numerical simulations also allow us to explore some properties of the role played by the responder's envy and by changes in the envy distributions of the two players. Overall, our theoretical results are consistent with the main evidence from ultimatum experiments conducted in behavioral and neuroscience settings. In addition, we provide testable implications of our model for future experiments.