Reputational Wars of Attrition with Complex Bargaining Postures
Dilip Abreu and David Pearce October 6, 2005
Abstract Consider a two-person intertemporal bargaining problem in which players choose actions and collect payo¤s while bargaining proceeds. Theory is silent regarding how the surplus is likely to be split, because a folk theorem applies. Perturbing such a game with a rich set of behavioral types for each player yields a speci…c asymptotic prediction for how the surplus will be divided, as the perturbation probabilities approach zero. Behavioral types may follow nonstationary strategies and respond to the opponent' play. s How much should a player try to get, and how should she behave while waiting for the resolution of bargaining In both respects she should build her strategy around the advice given by the ' Nash bargaining with threats' theory developed for two-stage games. The results suggest that there are forces at work in some dynamic games that favor certain payo¤s over all others. This is in stark contrast to the classic folk theorems, to the further folk theorems established for repeated games with two-sided reputational perturbations, and to the permissive results obtained in the literature on bargaining with payo¤s-as-you-go.
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Introduction
What kind of reputation should a bargainer try to establish Should she claim that her demand will never change, or that she will become more aggressive over time Should improvements in her opponent' o¤er be punished s as signs of weakness or should she promise to reward them with a softening of her own position Is it useful to announce deadlines after which o¤ers will be withdrawn This paper addresses these questions in an essentially full-information two-person bargaining model in which there is a small possibility that each player might be one of a rich variety of behavioral types. For example, to use the terminology of Myerson (1991), rather than optimizing as a fully rational player would, the player might use an "r-insistent strategy" that always demands the amount r and never accepts anything less. But the player might instead employ a complex history-dependent strategy, a possibility not considered by previous papers in the behavioral bargaining literature.1 Now think about broader bargaining problems in which the players interact in payo¤-signi…cant ways BEFORE an agreement is reached. Such considerations were introduced by Fernandez and Glazer (1991) and Haller and Holden (1990).2 For example, before two countries sign a treaty on trade or pollution abatement, their unilateral policies a¤ect one another' s payo¤s. Here, possibilities for strategic posturing are even more interesting. Does each party maximize its immediate payo¤ before agreement, or is some degree of cooperation possible during negotiations As time passes without agreement, do players treat one another more harshly Is a player' s behavior related to her demand, and to the opponent' demand s Since our framework will generalize the model of Abreu and Gul (2000) in two ways, we pause now to summarize their work. An exogenous protocol speci…es the times at which each of two impatient bargainers can make o¤ers about how a …xed surplus will be divided. When an o¤er is made, the

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