A theory of unstructured bargaining using distribution-valued solution concepts

In experiments it is typically found that many joint utility outcomes arise in any given unstructured bargaining game. This suggests using a positive unstructured bargaining concept that maps a bargaining game to a probability distribution over outcomes rather than to a single outcome. We show how to "translate" Nash's bargaining axioms to apply to such distributional bargaining concepts. We then prove that a subset of those axioms forces the distribution over outcomes to be a power-law. Unlike Nash's original result, our result holds even if the feasible set is finite. When the feasible set is convex and comprehensive, the mode of the power law distribution is the Harsanyi bargaining solution, and if we require symmetry it is the Nash bargaining solution. However in general these modes of the joint utility distribution are not Bayes-optimal predictions for the joint uitlity, nor are the bargains corresponding to those outcomes the most likely bargains. We then show how an external regulator can use distributional solution concepts to optimally design an unstructured bargaining scenario. Throughout we demonstrate our analysis in computational experiments involving flight rerouting negotiations in the National Airspace System.

[1]  J. Nash.  THE BARGAINING PROBLEM , 1950, Classics in Game Theory.

[2]  T. Schelling,et al.  The Strategy of Conflict. , 1961 .

[3]  R. Selten,et al.  A Generalized Nash Solution for Two-Person Bargaining Games with Incomplete Information , 1972 .

[4]  G. Owen,et al.  Two-person bargaining: An experimental test of the Nash axioms , 1974 .

[5]  E. Kalai.  Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons , 1977 .

[6]  A. Roth,et al.  Game-Theoretic Models and the Role of Information in Bargaining , 1979 .

[7]  M. Kaneko.  An extension of the Nash bargaining problem and the Nash social welfare function , 1980 .

[8]  S. Tijs,et al.  Probabilistic bargaining solutions , 1984 .

[9]  V. Smith,et al.  Bubbles, Crashes, and Endogenous Expectations in Experimental Spot Asset Markets , 1988 .

[10]  M. Herrero.  The nash program: Non-convex bargaining problems , 1989 .

[11]  A. Rubinstein,et al.  On the Interpretation of the Nash Bargaining Solution and Its Extension to Non-expected Utility Preferences , 1992 .

[12]  Ken Binmore,et al.  Focal points and bargaining , 1993 .

[13]  John P. Conley,et al.  An Extension of the Nash Bargaining Solution to Nonconvex Problems , 1996 .

[14]  Lin Zhou.  THE NASH BARGAINING THEORY WITH NON-CONVEX PROBLEMS , 1997 .

[15]  M. Mariotti.  Extending Nash's Axioms to Nonconvex Problems , 1998 .

[16]  M. Mariotti.  Nash bargaining theory when the number of alternatives can be finite , 1998 .

[17]  Colin Camerer.  Behavioral Game Theory: Experiments in Strategic Interaction , 2003 .

[18]  Naoki Yoshihara,et al.  Alternative characterizations of three bargaining solutions for nonconvex problems , 2006, Games Econ. Behav..

[19]  H. Peters,et al.  WPO, COV and IIA bargaining solutions , 2006 .

[20]  Murat R. Sertel,et al.  Bargaining over a finite set of alternatives , 2007, Soc. Choice Welf..

[21]  Kevin Leyton-Brown,et al.  Beyond equilibrium: predicting human behaviour in normal form games , 2010, AAAI.

[22]  Hans Peters,et al.  WPO, COV and IIA bargaining solutions for non-convex bargaining problems , 2012, Int. J. Game Theory.