person non-convex bargaining: Efficient proportional solutions (original) (raw)

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.

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.

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.

The Kalai–Smorodinsky bargaining solution with loss aversion

Mathematical Social Sciences, 2011

We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z.

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.

Twofold optimality of the relative utilitarian bargaining solution

Given a bargaining problem, the relative utilitarian (RU) solution maximizes the sum total of the bargainer's utilities, after having first renormalized each utility function to range from zero to one. We show that RU is 'optimal' in two very different senses. First, RU is the maximal element (over the set of all bargaining solutions) under any partial ordering which satisfies certain axioms of fairness and consistency; this result is closely analogous to the result of . Second, RU offers each person the maximum expected utility amongst all rescaling-invariant solutions, when it is applied to a random sequence of future bargaining problems generated using a certain class of distributions; this is recalls the results of and .

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.

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.

Dynamics and axiomatics of the equal area bargaining solution

International Journal of Game Theory, 2000

We present an alternative formulation of the two-person equal area bargaining solution based on a dynamical process describing the disagreement point set. This alternative formulation provides an interpretation of the idea of equal concessions. Furthermore, it leads to an axiomatic characterization of the solution.