Reconciling Consistency and Continuity: A Bounded-Population Characterization of the Nash Bargaining Solution
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Reconciling Consistency and Continuity: A Bounded‑Population Characterization of the Nash Bargaining Solution William Thomson1 Received: 5 August 2019 / Accepted: 18 September 2020 © Springer Nature Switzerland AG 2020
Abstract This paper explores the robustness of a characterization of the Nash bargaining solution in a variable-population framework in which the population of players is bounded (Lensberg in J Econ Theory 45:330–341, 1988). A central axiom in this characterization is “consistency”, which says that the solution outcome of any problem should always be “confirmed” by the solution in the “reduced problem” that results after some players have left with their assigned payoffs. Another axiom in this characterization is the standard requirement of Hausdorff continuity. We advocate a weaker version of continuity, a version that is a better conceptual fit with consistency that Hausdorff-continuity, and show that the Nash solution still emerges as the only one to satisfy all of the axioms. Keywords Bargaining theory · Consistency · Continuity · Nash solution JEL Classification C78 · D71 · D74
1 Introduction The goal of this paper is to investigate the robustness of an important characterization of the Nash solution in which the principal axiom is “consistency” (Lensberg 1988). This characterization also involves a standard continuity axiom based on Hausdorff distance. Although this axiom is standard in the literature, we argue that a weaker notion of continuity would be a better conceptual fit with the requirement of consistency, and we ask whether the Nash solution still emerges as the The very useful comments of Youngsub Chun, Terje Lensberg, and two anomymous referees are gratefully acknowledged. (filename: notnas.tex). * William Thomson [email protected] 1
Department of Economics, University of Rochester, Rochester, USA
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only acceptable one when the weaker axiom is imposed instead. The answer is affirmative. The traditional theory of bargaining, as formulated by Nash (1950), was originally written under the assumption that the number of players is fixed.1 The possibility that this number varies was later considered, and characterizations of the solutions that had been central in the classical theory were obtained involving relational axioms pertaining to possible variations in populations (Thomson 2010, is a survey). One of the central axioms in several of these developments is “consistency”. This axiom is the expression for the model of a principle whose implications have now been investigated in a wide range of other contexts. Put in general terms, consistency says that what a solution recommends for each problem in its domain should “agree” with what it recommends for each of the “reduced” problems that results when some of the agents involved initially leave with their “components” of the recommendation and the opportunities open to the remaining agents are reassessed. Depending upon the model, “recommendations” can be vectors of monetary amount
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