Bargaining with Uncertain Disagreement Points

Abstract

MUCH OF THE UNCERTAINTY concerning the likely outcome of a typical management-labor conflict pertains to the cost of possible conflict to the two sides. In this paper, we consider situations of this kind, where the cost of conflict is not known with certainty. However, we will assume the benefits from cooperation to be known. We place our analysis in the abstract framework formulated by Nash (1950): Nash described a bargaining problem as a pair consisting of a feasible set (the amount to be divided among management and labor) and a disagreement point (giving the payoffs to both sides when they fail to reach agreement on a division, that is, the strike). Nash investigated the existence of solutions to such problems that would satisfy a certain list of appealing properties. In his analysis both feasible set and disagreement point were assumed to be known. Here, we assume only the feasible set to be known. Several studies have appeared of bargaining situations where the feasible set is unknown but the disagreement point is known. While we are of course not denying the relevance of such studies, we believe that an analysis of situations where it is the consequences of conflict that are unknown might be equally, and perhaps even more, relevant to industrial experience. Indeed, consider a management-labor conflict over wages and benefits. In many industries, the future profitability of the enterprise can be predicted with reasonable accuracy on the basis of its performance in the previous years, whereas the impact of a strike might depend on a number of factors that are significantly harder to evaluate. This is because strikes are infrequent and conjectures about these factors are often not put to the test (a strike is a threat that is often not carried out), and because they involve a number of parameters that are difficult to quantify, such as the psychological readiness of the strikers, the support they might receive from the population and the media, and the likely response of competing and related industries. We impose on solutions a new condition of disagreement point concavity guaranteeing that agents will agree on a compromise before the uncertainty concerning the disagreement point is resolved. To illustrate this requirement somewhat more concretely, suppose that bargaining takes place today, without the precise location of the disagreement point being known, this uncertainty being resolved tomorrow. The bargainers have two options: the first option is simply to wait until tomorrow and solve then whatever problem has come up. Unfortunately, the resulting pair of contingent compromises, evaluated today, is in general strictly Pareto-dominated. The other possibility is to solve today the problem obtained by replacing the uncertain disagreement point by its expected value (this represents the cost of conflict evaluated today) and to solve the resulting problem today. This second option has the advantage of yielding Paretoundominated compromises (provided, of course, that agents would, in the case of no uncertainty, select such compromises), but unfortunately, it may make one of the agents worse off than under the first option. In order to ensure that all agents agree to reaching a compromise today, we require that the second option always Pareto-dominate the first one. We show that disagreement point concavity, when used in conjunction with three standard properties that are satisfied by virtually all of the solutions commonly discussed