On the Sensitivity Matrix of the Nash Bargaining Solution (original) (raw)

Relative Disagreement-Point Monotonicity of Bargaining Solutions

2003

Prominent bargaining solutions are disagreement-point monotonic. These solutions’ disagreement-point monotonicity ranking, on the other hand, is impossible to establish. In a large class of bargaining problems, however, a ranking of the relative disagreement-point monotonicity of these prominent bargaining solutions can be obtained. Using the ‘Constant Elasticity of Substitution’ class of bargaining problems, and regardless of the concavity of the Pareto frontier and of the increase in the disagreement point, we find that the Egalitarian solution is most monotonic with respect to changes in disagreement payoffs, followed by the Nash solution. The Equal Sacrifice solution turns out to be the least monotonic, followed by the Kalai/Smorodinsky solution. JEL classification number : C72.

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.

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.

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.

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.

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).

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.

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.

Monotone bargaining is Nash-solvable

Given two finite ordered sets A and B, let O = A × B denote the set of outcomes of the following game: Two players, Alice and Bob, have the sets of strategies X and Y that consist of all monotone non-decreasing mappings x : A → B and y : B → A, respectively. It is easily seen that each pair (x, y) ∈ X × Y produces at least one deal, that is, an outcome (a, b) ∈ O such that x(a) = b and y(b) = a. Denote by G(x, y) ⊆ O the set of all such deals related to (x, y). The obtained mapping G : X × Y → 2 O is a game correspondence. Choose an arbitrary deal g(x, y) ∈ G(x, y) to obtain a mapping g : X × Y → O, which is a game form. We show that each such game form is tight and, hence, Nash-solvable, that is, for any pair u = (u A , u B) of utility functions of Alice and Bob, the obtained monotone bargaining game (g, u) has at least one Nash equilibrium (x, y) in pure strategies. Moreover, |G(x, y)| = 1 and, hence, (x, y) is a Nash equilibrium in game (g, u) for all g ∈ G. We also obtain an efficient algorithm that determines such an equilibrium in time linear in the number of outcomes |O|, although the numbers of strategies are exponential in |O|. Our results show that, somewhat surprisingly, the players have no need to hide or randomize their bargaining strategies, even in the zero-sum case.