Negotiation when the size of the pie depends on how it is cut

Abstract

Rubinstein’s (1982) infinitely repeated ultimatum game between two players provides a paradigm to analyze negotiation. We extend his análisis to the case in which the benefits to be distributed depend on the result of the negotiation, that is, on whose proposal is finally accepted and on how the benefits are distributed. Each player draws her proposals from a different set of possibilities, which is not assumed to be convex. We obtain general theorems classifying and calculating the results of this kind of negotiatio game becomes a principal agent game. Most of the paper is devoted to the proof of these results. We also give two brief examples of their application. The first example is one of the problema which motivated our reseach. We analyze the simple case of negotiation (sharing a fixed pie) when the players have a concept of fairness. In this case, their final utility depends not only on their share of the pie but also on how fairly they feel it was cut; unfairness causing a loss of utility which may be different fo each player. We find that the rule of sharing in half holds for a range of payers including morally deficient players and playes with weak characters (that is, not very willing to punish the other for being unfair). The second example is Cournot duopoly. We consider a market supplied by two firms which are deciding how much to produce. Their proposals consist of taking observable actions in preparation for production, which announce their intended output. The other firm accepts the porposal if it decide to maximize its profits taking as given to other firm’s output as implied by its actions. Instead, it may op for a different level of production, and act accordingly. In such a situation, which is modeled by the infinitely repeated ultimatum game, not all Cournot equilibria (at wich each firm has no incentives to deviate from its prodction level given the other firm’s production level) are equilibria inthe negotiaion game. The plan of the paper is as follows. In section II we describe the general negotiation game which we solve, and write down the main results. In section III we apply these results to the simple case of negotiation when the players have a concept of fairness. In section IV we give our examples on duopoly. Section V constructs the proof of the general negotiation game. Then follow the final remarks and references. The Appendix contains the details of the proofs of the proceding sections.