Nash Bargaining Theory with Non-Convexity and Unique Solution (original) (raw)

No . 1504 Nash bargaining for log-convex problems

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 all 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.

Single-Valued Nash Bargaining Solutions with Non-Convexity ∗

2018

We consider two-player bargaining problems with compact star-shaped choice sets arising from a class of economic environments. We characterize single-valued solutions satisfying the Nash axioms on this class of bargaining problems. Our results show that there are exactly two Nash solutions with each being a dictatorial (in favor of one player) selection of Nash product maximizers. We also provide an extensive form for implementing these two Nash solutions.

A characterization of convex games by means of bargaining sets

International Journal of Game Theory, 2008

The aim of the paper is to characterize the classical convexity notion for cooperative TU games by means of the Mas-Colell and the Davis-Maschler bargaining sets. A new set of payoff vectors is introduced and analyzed: the max-Weber set. This set is defined as the convex hull of the max-marginal worth vectors. The characterizations of convexity are reached by comparing the classical Weber set, the max-Weber set and a selected bargaining set.

How to Add Apples and Pears: Non-Symmetric Nash Bargaining and the Generalized Joint Surplus

2010

We generalize the equivalence of the non-symmetric Nash bargaining solution and the linear division of the joint surplus when bargainers use different utility scales. This equivalence in the general case requires the surplus each agent receives to be expressed in compatible, or comparable, units. This result is valid in the case of bargaining over multiple-issues. Our conclusions have important implications for comparative static exercises and calibrated work.

Rationality and the Nash Solution to Non-convex Bargaining Problems

Conditions α and β are two well-known rationality conditions in the theory of rational choice. This paper examines the implication of weaker versions of these two rationality conditions in the context of solutions to non-convex bargaining problems. It is shown that, together with the standard axioms of efficiency, anonymity and scale invariance, they characterize the Nash solution. This result makes a further connection between solutions to non-convex bargaining problems and rationalizability of choice functions in the theory of rational choice.

Multilateral non-cooperative bargaining in a general utility space

International Journal of Game Theory, 2007

We consider an n-player bargaining problem where the utility possibility set is compact, convex, and stricly comprehensive. We show that a stationary subgame perfect Nash equilibrium exists, and that, if the Pareto surface is differentiable, all such equilibria converge to the Nash bargaining solution as the length of a time period between offers goes to zero. Without the differentiability assumption, convergence need not hold.

WPO, COV and IIA bargaining solutions for non-convex bargaining problems

International Journal of Game Theory, 2012

We characterize all n-person multi-valued bargaining solutions, defined on the domain of all finite bargaining problems, and satisfying Weak Pareto Optimality (WPO), Covariance (COV), and Independence of Irrelevant Alternatives (IIA). We show that these solutions are obtained by iteratively maximizing nonsymmetric Nash products and determining the final set of points by so-called LDR decompositions. If, next, we assume the (set-theoretic) Axiom of Determinacy, then this class coincides with the class of iterated Nash bargaining solutions; but if we assume the Axiom of Choice then we are able to construct an additional large set of discontinuous and even nonmeasurable solutions. We show however that none of these nonmeasurable solutions can be defined in terms of set theoretic formulae. We next show that a number of existing results in the literature as well as some new results are implied by our approach. These include a characterization of all WPO, COV and IIA solutions -including single-valued ones -on the domain of all compact bargaining problems, and an extension of a theorem of Birkhoff characterizing translation invariant and homogeneous orderings.

A Non­cooperative Solution to the Bargaining Problem

show that the Nash (1950) solution emerges as a limit point of a two player alternating o¤ers bargaining game when the time di¤erence between o¤ers goes to zero. establish the same result in the n¡player cake sharing set up. argue that noncooperative bargaining behavior á la Krishna-Serrano can be compactly described by means of von Neumann-Morgenstern stable set. This paper analyses the general problem. We show that a stable set exists and converges to the Nash solution in any smooth, compact and convex problem. A connection to the generalized Krishna-Serrano game is also established.

Nash bargaining with a nondeterministic threat

Computing Research Repository - CORR, 2008

We consider bargaining problems which involve two participants, with a nonempty closed, bounded convex bargaining set of points in the real plane representing all realizable bargains. We also assume that there is no definite threat or disagreement point which will provide the default bargain if the players cannot agree on some point in the bargaining set. However, there is a nondeterministic threat: if the players fail to agree on a bargain, one of them will be chosen at random with equal probability, and that chosen player will select any realizable bargain as the solution, subject to a reasonable restriction.