On the Interpretation of Nonsymmetric Bargaining Solutions and Their Extension to Nonexpected Utility Preferences

In a recent paper, Rubinstein, Safra, and Thomson reexamine the foundations of the axiomatic bargaining theory as formulated by Nash. In it they reinterpret the Nash bargaining solution and extend it to a family of nonexpected utility preferences, providing an interpretation of this solution concept and an axiomatization by means of a suitable translation of Nash axioms. This paper extends the work of these authors to the family, characterized by Kalai, of nonsymmetric solutions that appear from the Nash bargaining solution when the symmetry axiom is dropped. This extension requires an axiom relating the solution outcomes of isomorphic games. Journal of Economic Literature Classification Number: C78.     

On weighted Kalai–Samet solutions for non-transferable utility coalitional form games

We investigate the implications of the axiom of coalitional concavity for non-transferable utility coalitional form games. This axiom says that if the feasible set of some coalition is uncertain whereas the feasible sets of other coalitions are known, then all players in the coalition with the uncertain feasible set should (weakly) benefit from reaching a compromise before the uncertainty is resolved. By imposing this axiom, in addition to other minor axioms, we characterize the weighted Kalai–Samet [Econometrica 53 (1985) 307] solutions: these solutions coincide with the weighted egalitarian solutions on the domain of bargaining problems, and with the weighted Shapley values on the domain of transferable utility coalitional form games.     

THE BEHAVIOUR OF SOLUTIONS TO BARGAINING PROBLEMS ON THE BASIS OF SOLIDARITY*

The main purpose of this paper is to provide a systematic study of the solidarity‐type axioms for classical convex bargaining problems. As a consequence, we present alternative characterizations for some well‐known solutions in the literature. Instead of using the Monotonicity axiom, the paper provides characterizations of the egalitarian solution and the Kalai–Somorodinsky solution using slightly weaker versions of Nash's original IIA for convex bargaining problems with a fixed population.     

Wage bargaining and employment revisited: separability and efficiency in collective bargaining*

We analyse the two-dimensional Nash bargaining solution (NBS) by deploying the standard labour market negotiations model of McDonald and Solow. We show that the two-dimensional bargaining problem can be decomposed into two one-dimensional problems, such that the two solutions together replicate the solution of the two-dimensional problem if the NBS is applied. The axiom of “independence of irrelevant alternatives” is shown to be crucial for this type of decomposability. This result has significant implications for actual negotiations because it allows for the decomposition of a multi-dimensional bargaining problem into one-dimensional problems – and thus helps to facilitate real-world negotiations.     

Proportional concessions and the leximin solution

In this note we introduce a new axiom for bargaining solutions, named Proportional Concession Monotonicity (PCM), which imposes that no player benefits when all players collectively make proportional concessions with respect to their respective utopia values. We reconsider the leximin solution (Imai, 1983), and obtain an alternative characterization on the basis of PCM.     

Fair Bargains: Distributive Justice and Nash Bargaining Theory

The Suppes–Sen dominance relation is a weak criterion of impartiality in distributive justice. I propose its application to Nash bargaining theory. The Nash Bargaining Solution (NBS) is characterized by replacing the controversial Independence of Irrelevant Alternatives axiom with an axiom embodying the Suppes–Sen principle. This allows a clear interpretation of the NBS as a fair arbitration scheme. In addition, the proposed characterization is more robust than the standard one with respect to variations in the domain of bargaining problems.     

Symmetry-based compromise and the Nash solution to convex bargaining problems

A new axiom, symmetry-based compromise, is introduced to study solutions to convex bargaining problems. It is shown that any efficient solution that is also scale invariant and satisfies symmetry-based compromise must be Nash.     

Characterizations of bargaining solutions in production economies with unequal skills

In production economies with unequal skills, this paper characterizes bargaining solutions by using axioms on allocation rules rather than axioms on classical bargaining solutions. We introduce a new axiom, consistency w.r.t. technological innovations, so that the non-welfaristic characterizations of bargaining solutions in the production economies are provided. By the characterizations, we can classify the three bargaining solutions (the Nash, the Kalai-Smorodinsky, and the Egalitarian solutions) from the viewpoint of responsibility and compensation discussed by Dworkin.     

A class of solutions to bargaining problems

A reformation of Nash's axiom of Independence of Irrelevant Alternatives is proposed, generalizing an idea due to Roth. It involves the concept of a reference function g, whose purpose is to summarize the main features of the bargaining problem and so facilitate the evaluation of the relative bargaining positions of the players. It is with respect to g that the Independence condition is reformulated. In conjunction with the other axioms used by Nash, it is shown that if g satisfies certain natural properties, the bargaining problem admits of a unique solution. Several examples of reference functions are shown to satisfy these conditions. An existence problem discovered by Roth for a particular choice of g is shown to be resolved if solutions are permitted to be multi-valued, the axioms being correspondingly reformulated.     

A note on bargaining over a finite number of feasible agreements

Summary In this note we show that theuniqueness of the subgame perfect equilibrium of Rubinstein's (1982) bargaining theory does not hold if the number of feasible agreements isfinite. It will be shown thatany Pareto-efficient agreement (belonging to thefinite set of feasible agreements) can be supported as a subgame perfect equilibrium of the Rubinstein alternating-offers bargaining game, provided the length of a single bargaining period is sufficiently small.     

Nonconvex n-person bargaining: efficient maxmin solutions

Summary. This paper provides an axiomatic characterization of a family of so-called efficient maxmin solutions which can be seen as generalizations of the Kalai-Smorodinsky solution to nonconvex n-person bargaining problems. Moreover, it is shown that even though there are several efficient maxmin solutions for some bargaining problems, there is typically a unique efficient maxmin solution. Received: February 15, 2001; revised version: November 14, 2001     

The Biodiversity Bargaining Problem

We employ cooperative bargaining theory and Nash’s ‘rational threats’ idea to cast light on the biodiversity bargaining problem. The problem of global environmental negotiations is argued to be of the nature of a bargaining problem, in which bargainers must agree on the distribution of cooperative surplus in order to move to the bargaining frontier. We discuss the importance of both efficiency (bargaining frontier) and fairness (recognition of characteristics of bargainers) in the choice of the appropriate contract. We show that the incremental cost contract, used to resolve the biodiversity bargaining problem, is of the form of an extreme point contract that fails to recognise the contributions of the South to the production of cooperative surplus. A rational response to such a contract is the use of threats of biodiversity destruction. Contracts must evince both efficiency and fairness in order to represent lasting solutions.     

The Choquet Bargaining Solutions

We axiomatically investigate the problem of rationalizing bargaining solutions by social welfare functions that are linear in every rank-ordered subset of Rⁿ₊ . Such functions, the so-called Choquet integrals, have been widely used in the theories of collective and individual choice. We refer to bargaining solutions that can be rationalized by Choquet integrals as Choquet bargaining solutions. Our main result is a complete characterization of Choquet bargaining solutions. As a corollary of our main result, we also obtain a characterization of the generalized Gini bargaining solutions introduced by Blackorby et al. (1994, Econometrica62, 1161–1178). Journal of Economic Literature Classification Numbers: D71, C78.     

Comparing Bargaining Solutions in the Shadow of Conflict: How Norms against Threats Can Have Real Effects

In many economic environments agents make costly and irreversible investments (in “guns”) that may enhance their respective threat payoffs but also shrink the utility possibilities set. In such settings, with variable threats and a variable utility possibilities set, it becomes possible to rank different bargaining solutions in terms of efficiency. We compare bargaining solutions within a class in which the influence of the threat point on the bargaining outcome varies across solutions. Under symmetry, we find that the solution in which the threat point is least influential—the equal sacrifice solution—Pareto-dominates the other solutions. Since the equal sacrifice solution puts the least weight on the threat point, norms against threats (that can be seen in many seemingly rhetorical pronouncements in adversarial relations) can mitigate some of the costs of conflict and therefore have efficiency-enhancing effects. Journal of Economic Literature Classification Numbers: C72, C78, D30, D70, D72, K42.     

Independence of the status quo? A weak and a strong impossibility result for social decisions by bargaining

There are several ethical objections to the dependence on the status quo when bargaining mechanisms are applied to problems of collective choice and, in this context, there is also some discussion about how to define the status quo. Can one think of bargaining mechanisms defined for some set of decision problems or of bargaining solutions for special problems that are independent of the status quo? Under which conditions do they exist? What are the crucial properties of the classical bargaining solutions or of bargaining mechanisms on economic environments implying dependence? These questions are answered by two impossibility results. It turns out that the only crucial assumption we search for is the axiom of “Weak Individual Rationality”. We also point out the consequences of our results for the discussion mentioned above.     

Finite Horizon Bargaining and the Consistent Field

We explore the relationships between noncooperative bargaining games and the consistent value for nontransferable utility (NTU) cooperative games. A dynamic approach to the consistent value is introduced: the consistent vector field. The main contribution is to establish that theconsistent fieldis intimately related to the concept ofsubgame perfectionfor finite horizon noncooperative bargaining games, as the horizon goes to infinity and the cost of delay goes to zero. We then show that in the general NTU case—unlike the transferable utility and pure bargaining cases—the finite horizon subgame perfect equilibria need not approach the consistent value.Journal of Economic LiteratureClassification Numbers: C71, C72.     

Sustainability,Optimality, and Development Policy

Considering sustainability a matter of intergenerational welfare equity, this paper examines whether an optimal development path can also be sustainable. It argues that the general “zero‐net‐aggregate‐investment” condition for an optimal development path to be sustainable in the sense of the maximin criterion of intergenerational justice is too demanding to be practical, especially in the context of developing countries. It further argues that while the maximin criterion of sustainability may be appealing to the rich advanced industrial countries, for the poor developing countries it implies equalization of poverty across generations, and as such is too costly a moral obligation to be acceptable. The paper suggests that a compromise development policy that follows the optimal growth approach but adopts certain measures to mitigate both the intergenerational and intragenerational welfare inequalities may be more appropriate for these countries. Some of the principal elements of such a policy are highlighted.     

A Dynamic Characterization of Rawls's Maximin Principle: Theory and Implications

The purpose of this paper is to provide a reinterpretation of the maximin principle in Rawls's theory of justice. As we discuss the issue of social justice, we differentiate the point in time at which rectifying practices are discussed and the time intervals in which these practices are implemented; we explore the information content for people behind the veil of ignorance; and we distinguish the means and ends for social justice. With these points clarified, we interpret the veil of ignorance in Rawls's theory as a form of Knightian uncertainty, and show that Rawls's maximin criterion would be chosen by rational individuals behind the veil of ignorance.     

Multilateral Bargaining Problems

In many situations in economics and political science there are gains from forming coalitions but conflict over which coalition to form and how to distribute the gains. This paper presents an approach to suchmultilateral bargaining problems. Asolutionto a multilateral bargaining problem specifies an agreement for each coalition that is consistent with the bargaining process in every coalition. We establish the existence of such solutions, show that they are determined by reservation prices, and characterize these reservation prices as the payoffs ofsubgame perfect equilibrium outcomesof a non-cooperative bargaining model.Journal of Economic LiteratureClassification Numbers: C71, C72, C78.     

Simple Flow Games

This paper considers simple flow situations. We introduce the minimum cut solution and characterize it in terms of one-person efficiency, consistency, and converse consistency. Furthermore, we give several relations between the core of a simple flow game and four other solution concepts: the minimum cut solution, the least core, the kernel, and the bargaining set.Journal of Economic LiteratureClassification Numbers: C71.