Relative Disagreement-Point Monotonicity of Bargaining Solutions (original) (raw)

Monotonicity and equal-opportunity equivalence in bargaining

Mathematical Social Sciences, 2005

In this paper we study two-person bargaining problems represented by a space of alternatives, a status quo point, and the agents' preference relations on the alternatives. The notion of a family of increasing sets is introduced, which reflects a particular way of gradually expanding the set of alternatives. For any given family of increasing sets, we present a solution which is Pareto optimal and monotonic with respect to this family, that is, makes each agent weakly better off if the set of alternatives is expanded within this family. The solution may be viewed as an expression of equal-opportunity equivalence as defined in Thomson [19]. It is shown to be the unique solution that, in addition to Pareto optimality and the monotonicity property mentioned above, satisfies a uniqueness axiom and unchanged contour independence. A non-cooperative bargaining procedure is provided for which the unique backward induction outcome coincides with the solution. the participants at the XV Bielefeld FoG Meeting and two anonymous referees for their valuable suggestions and comments.

Monotonicity in bargaining networks

2010

We study bargaining networks, discussed in a recent paper of Kleinberg and Tardos , from the perspective of cooperative game theory. In particular we examine three solution concepts, the nucleolus, the core center and the core median. All solution concepts define unique solutions, so they provide testable predictions. We define a new monotonicity property that is a natural axiom of any bargaining game solution, and we prove that all three of them satisfy this monotonicity property. This is actually in contrast to the conventional wisdom for general cooperative games that monotonicity and the core condition (which is a basic property that all three of them satisfy) are incompatible with each other. Our proofs are based on a primal-dual argument (for the nucleolus) and on the FKG inequality (for the core center and the core median). We further observe some qualitative differences between the solution concepts. In particular, there are cases where a strict version of our monotonicity property is a natural axiom, but only the core center and the core median satisfy it. On the other hand, the nucleolus is easy to compute, whereas computing the core center or the core median is #P-hard (yet it can be approximated in polynomial time).

Bargaining with endogenous disagreement: The extended Kalai–Smorodinsky solution

Games and Economic Behavior, 2012

Following Vartiainen we consider bargaining problems in which no exogenous disagreement outcome is given. A bargaining solution assigns a pair of outcomes to such a problem, namely a compromise outcome and a disagreement outcome: the disagreement outcome may serve as a reference point for the compromise outcome, but other interpretations are given as well. For this framework we propose and study an extension of the classical Kalai-Smorodinsky bargaining solution. We identify the (large) domain on which this solution is single-valued, and present two axiomatic characterizations on subsets of this domain.

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.

On the Sensitivity Matrix of the Nash Bargaining Solution

2006

In this note we provide a characterization of a subclass of bargaining problems for which the Nash solution has the property of disagreement point monotonicity. While the original d-monotonicity axiom and its stronger notion, strong d-monotonicity, were introduced and discussed by Thomson [15], this paper introduces local strong d-monotonicity and derives a necessary and sufficient condition for the Nash solution to be locally strong d-monotonic. This characterization is given by using the sensitivity matrix of the Nash bargaining solution w.r.t. the disagreement point d. Moverover, we present a sufficient condition for the Nash solution to be strong d-monotonic.

Efficient solutions to bargaining problems with uncertain disagreement points

Social Choice and Welfare, 2002

We consider a cooperative model of bargaining where the location of the disagreement point may be uncertain. Based on the maximin criterion, we formulate an ex ante efficiency condition and characterize the class of bargaining solutions satisfying this axiom. These solutions are generalizations of the monotone path solutions. Adding individual rationality yields a subclass of these solutions. By employing maximin efficiency and an invariance property that implies individual rationality, a new axiomatization of the monotone path solutions is obtained. Furthermore, we show that an efficiency axiom employing the maximax criterion leads to an impossibility result.

Nash Bargaining Theory with Non-Convexity and Unique Solution

2009

We characterize a class of bargaining problems allowing for non-convexity on which all of Nash axioms except for that of symmetry uniquely characterize the asymmetric Nash bargaining solution. We show that under some basic conditions, the uniqueness of the solution is equivalent to the choice sets of the bargaining problems being “log-convex”. The well-recognized non-convex bargaining problems arising from duopolies with asymmetric constant marginal costs turn out to belong to this class. We compare the Nash bargaining solution with some of its extensions that appeared in the literature.

Nash bargaining theory, nonconvex problems and social welfare orderings

Theory and decision, 2000

Nash's (1950) bargaining solution was axiomatized on a domain of convex problems, where the rationalizability of a bargaining solution raises some technical difficulties. Recently, however, the convexity assumption has been questioned, and the Nash Bargain-ing ...

person non-convex bargaining: Efficient proportional solutions

Operations Research Letters, 2010

For n-person bargaining problems the family of proportional solutions (introduced and characterized by Kalai) is generalized to bargaining problems with non-convex payoff sets. The so-called ''efficient proportional solutions'' are characterized axiomatically using natural extensions of the original axioms provided by Kalai. (M. Tvede).

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.