An Extension of the Nash Bargaining Solution to Nonconvex Problems

Conley, John P.; Wilkie, Simon

doi:10.1006/game.1996.0023

Cited by 75 publications

(50 citation statements)

References 17 publications

(18 reference statements)

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“…We now compare our extension of Nash bargaining with the two other extensions by Conley and Wilkie (1996) and Herrero (1989), respectively. To simplify our comparison, we focus on log-convex problems (S, d • ) satisfying condition (C) in Sect.…”

Section: Comparisonmentioning

confidence: 99%

“…Under Conley and Wilkie's (1996) extension, the solution is determined by the intersection of the Pareto frontier of the choice set with the segment connecting the threat point and the Nash solution for the convexified problem. Although they only consider extensions of the symmetric Nash solution, their method can be also applied to extend the asymmetric Nash solution.…”

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confidence: 99%

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Nash bargaining for log-convex problems

2015

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We introduce log-convexity for bargaining problems. With the requirement of some basic regularity conditions, log-convexity is shown to be necessary and sufficient for Nash's axioms to determine a unique single-valued bargaining solution up to choices of bargaining powers. Specifically, we show that the single-valued (asymmetric) Nash solution is the unique solution under Nash's axioms without that of symmetry on the class of regular and log-convex bargaining problems, but this is not true on any larger class. We apply our results to bargaining problems arising from duopoly and the theory of the firm. These problems turn out to be log-convex but not convex under familiar conditions. We compare the Nash solution for log-convex bargaining problems with some of its extensions in the literature.Professor Shuzhong Shi passed away before the completion of this paper. He was a pleasure and an inspiration to work with and will be deeply missed.We gratefully acknowledge helpful comments from

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Section: Comparisonmentioning

confidence: 99%

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confidence: 99%

Nash bargaining for log-convex problems

2015

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“…Other authors have examined bargaining solutions on non-standard domains. For example, many papers extend the Nash solution to non-convex S (Kaneko, 1980;Herrero, 1989;Conley & Wilkie, 1996;Zhou, Theorem 1. Let µ be an N -dimensional NDB concept such that for any closed, simple, standard problem (T, 0), µ T,0 (x) is differentiable throughout the interior of T .…”

Section: Nash Distributional Bargaining Conceptsmentioning

confidence: 99%

Predicting Behavior in Unstructured Bargaining with a Probability Distribution

Wolpert¹,

Bono²

2013

jair

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In experimental tests of human behavior in unstructured bargaining games, typically many joint utility outcomes are found to occur, not just one. This suggests we predict the outcome of such a game as a probability distribution. This is in contrast to what is conventionally done (e.g, in the Nash bargaining solution), which is predict a single outcome. We show how to translate Nash's bargaining axioms to provide a distribution over outcomes rather than a single outcome. We then prove that a subset of those axioms forces the distribution over utility outcomes to be a power-law distribution. 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 the experimentalist's Bayesoptimal predictions for the joint utility. Nor are the bargains corresponding to the modes of those joint utility distributions the modes of the distribution over bargains in general, since more than one bargain may result in the same joint utility. After introducing distributional bargaining solution concepts, we show how an external regulator can use them to optimally design an unstructured bargaining scenario. Throughout we demonstrate our analysis in computational experiments involving flight rerouting negotiations in the National Airspace System. We emphasize that while our results are formulated for unstructured bargaining, they can also be used to make predictions for noncooperative games where the modeler knows the utility functions of the players over possible outcomes of the game, but does not know the move spaces the players use to determine those outcomes.

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“…There is for instance an extensive literature on the extension of the Nash bargaining solution to non-convex environments (Kaneko (1980), Conley and Wilkie (1996), Mariotti (1997), Zhou (1997), and Xu and Yoshihara (2006)). On the contrary, the literature on strategic bargaining has not paid much attention to non-convexities and non-convexities are only treated for the two-player case.…”

Section: Introductionmentioning

confidence: 99%

Bargaining with non-convexities

Herings

Predtetchinski

2015

Games and Economic Behavior

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We show that in the canonical non-cooperative multilateral bargaining game, a subgame perfect equilibrium exists in pure stationary strategies, even when the space of feasible payoffs is not convex. At such an equilibrium there is no delay. We also have the converse result that randomization will not be used in this environment in the sense that all stationary subgame perfect equilibria do not involve randomization on the equilibrium path. Nevertheless, mixed strategy profiles can lead to Pareto superior payoffs in non-convex cases.

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An Extension of the Nash Bargaining Solution to Nonconvex Problems

Cited by 75 publications

References 17 publications

Nash bargaining for log-convex problems

Nash bargaining for log-convex problems

Predicting Behavior in Unstructured Bargaining with a Probability Distribution

Bargaining with non-convexities

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