Two simple characterizations of the Nash bargaining solution

We provide two alternative characterizations of the Nash bargaining solution. We introduce new simple axioms, strong undominatedness by the disagreement point, and egalitarian Pareto optimality. First, we prove that the Nash solution is characterized by symmetry, scale invariance, independence of irrelevant alternatives, and strong undominatedness by the disagreement point. Second, we replace the independence of irrelevant alternatives axiom with the sandwich axiom (Rachmilevitch in Theory Decis 80:427–442, 2016) and egalitarian Pareto optimality. We then demonstrate that the Nash solution is characterized by symmetry, scale invariance, strong undominatedness by the disagreement point, the sandwich axiom, and egalitarian Pareto optimality.     

Randomized dictatorship and the Kalai–Smorodinsky bargaining solution

“Randomized dictatorship,” one of the simplest ways to solve bargaining situations, works as follows: a fair coin toss determines the “dictator”—the player to be given his first-best payoff. The two major bargaining solutions, that of Nash (Econometrica 18:155–162, 1950) and that of Kalai and Smorodinsky (Econometrica, 43:513–518, 1975), Pareto-dominate this process (in the ex ante sense). However, whereas the existing literature offers axiomatizations of the Nash solution in which this ex ante domination plays a central role (Moulin, Le choix social utilitariste, Ecole Polytechnique Discussion Paper, 1983 ; de Clippel, Social Choice and Welfare, 29:201–210, 2007), it does not provide an analogous result for Kalai–Smorodinsky. This paper fills in this gap: a characterization of the latter is obtained by combining the aforementioned domination with three additional axioms: Pareto optimality, individual monotonicity, and a weakened version of the Perles–Maschler (International Journal of Game Theory, 10:163–193, 1981) super additivity axiom.     

Uncertainty,the bargaining problem,and the Nash-Zeuthen solution

The Zeuthen bargaining model occupies a prominent place among those theories of the bargaining process that have been formulated and expounded by economists. Its solution to the bargaining problem is essentially economic, since invariant utility functions based on economic factors alone determine the outcome. However, this paper shows that a necessary condition for reaching the Zeuthen solution (shown by Harsanyi to be mathematically equivalent to the game-theoretic solution of Nash's theory) is that bargainers initially take up positions on opposite sides of the outcome that maximizes their utility product. Whether utility functions are mutually known or unknown, inherent in the bargaining situation itself is the requirement that bargainers be at least initially uncertain as to each other's subsequent concession behaviour. With uncertainty, von Neumann-Morgenstern rationality implies that each bargainer would make an initial demand that maximizes the expected gain from holding fast. Therefore, even if Zeuthen's concession criterion should subsequently dictate concession behaviour, expected utility maximization within the context of subjective uncertainty may well yield initial demands that are inconsistent with reaching the Nash-Zeuthen solution. Finally, a general methodological conclusion that emerges from the analysis is that, since the bargaining process necessarily proceeds from a context of subjective uncertainty, greater emphasis needs to be placed on its role as a device for affecting expectations.     

Risk aversion and the relationship between Nash's solution and subgame perfect equilibrium of sequential bargaining

This article presents some new, intuitive derivations of several results in the bargaining literature. These new derivations clarify the relationships among these results and allow them to be understood in a unified way. These results concern the way in which the risk posture of the bargainers affects the outcome of bargaining as predicted by Nash's (axiomatic) solution of a static bargaining model (Nash, 1950) and by the subgame perfect equilibrium of the infinite horizon sequential bargaining game analyzed by Rubinstein (1982). The analogous, experimentally testable predictions for finite horizon sequential bargaining games are also presented.     

On arbitration schemes for a wealth distribution problem

This paper develops an arbitration scheme for resolving a distribution of wealth problem by applying Nash's assumptions to marginal rather than to total utilities. The problem considered is that of distributing a fixed amount of wealth between two claimants, and the paper compares properties of this problem's Nash solution with those of the marginal utility solution. The Nash solution is shown to emphasize application of symmetry considerations to the status quo ante, the marginal utility solution their application to the players' post arbitration positions (as measured by functions that fully describe the players' utilities but that are independent of positive linear transformations). It is argued that while the Nash assumptions are appropriate for many arbitration problems in which a solution reflects the players' status quo ante positions, the marginal utility assumptions are useful when it is desired that a solution attempt to minimize post-arbitration differences between the players' positions. The latter seems to be preferable in contexts where an arbitrator weights the solution's effects on out-comes more heavily than he weights considerations of the status quo ante. Examples are situations such as those involving income redistribution, where attempts to reduce inequality may guide the redistribution decisions. The effects on each type of solution of changes in the status quo ante are also investigated and related to the risk preference properties of the players' utilities.     

Simple Characterizations of the Nash and Kalai/smorodinsky Solutions

In this study we introduce two new properties, the Midpoint Outcome on a Linear Frontier (MOLF) and Balanced Focal Point (BFP) properties, to replace the Weak Pareto Optimality (WPO), Symmetry (SYM) and Independence of Equivalent Utility Representations (IEUR) properties in the axiomatic characterizations of the two most prominent solution concepts, namely the Nash and Kalai/Smorodinsky solutions, respectively.     

An extension of the Nash bargaining problem and the Nash social welfare function

This paper provides an extension of Nash's bargaining model. The extension permits the domain of a solution function to contain non-convex sets. First, it is shown that the proposed extension is a mathematical generalization of Nash's bargaining problem. Next, some relations between the extension and the Nash social welfare function are developed. The formal equivalence of the two constructs is demonstrated, and the intuitive significance of the results is discussed.I am indebted to S. Ishikawa for helpful comments on an earlier draft and to K. Nakamura for helpful discussions. Any errors that remain are of course my responsibility.     

Nash Bargaining Theory,Nonconvex Problems and Social Welfare Orderings

In this paper we deal with the extension of Nash bargaining theory to nonconvex problems. By focussing on the Social Welfare Ordering associated with a bargaining solution, we characterize the symmetric Nash Bargaining Solution (NBS). Moreover, we obtain a unified method of proof of recent characterization results for the asymmetric single-valued NBS and the symmetric multivalued NBS, as well as their extensions to different domains.     

Bargaining with Incomplete information an axiomatic approach

Within this paper we consider a model of Nash bargaining with incomplete information. In particular, we focus on fee games, which are a natural generalization of side payment games in the context of incomplete information. For a specific class of fee games we provide two axiomatic approaches in order to establish the Expected Contract Value, which is a version of the Nash bargaining solution.     

Nonlinear social welfare functions

It is argued that Bayesian decision theory is a solution of an important philosophical problem, viz. the problem of how to define rational behavior under risk and uncertainty. The author has shown in earlier papers that if we take the Bayesian rationality postulates seriously, and take an individualistic point of view about social welfare, then our social welfare function must be a linear function of individual utilities: indeed, it must be their arithmetic mean. The present paper criticizes Diamond's and Sen's contention that one of the Bayesian postulates (viz. the sure-thing principle) does not apply to social decisions, even though it may apply to individual decisions. It also criticizes Sen's proposal of making social welfare a nonlinear function of individual utilities. The social welfare function proposed by the author depends on interpersonal utility comparisons. The use of such comparisons is defended. It is also argued that anybody who feels that the utilitarian (i.e., linear) form of the social welfare function is not egalitarian enough, should reject the author's individualism axiom, instead of trying to reject the Bayesian rationality axioms. However, this would be equivalent to giving egalitarian considerations a priority in many cases over humanitarian considerations. Finally, the paper discusses the reasons why even full agreement on the mathematical form of the social welfare function would not give rise to a Utopian state of moral consensus: moral controversies arising from disagreements about what predictions to make about future empirical facts would still remain.     

A note on the two person exchange game

The two person exchange game is analysed using core theory, the Nash solution and the competitive equilibrium. It is shown for specific utility functions that relative to the other trader, the less wealthy trader does better under the Nash solution than he would under the competitive solution. Conversely, the trader with the lower initial utility level does better under the competitive solution than he would under the Nash solution. If traders have disparate initial endowments, it is argued that outcomes should be restricted to a reduced core containing the Nash and competitive solutions as endpoints.     

Fairness motivation in bargaining: a matter of principle

In this paper, we study the role of fairness motivation in bargaining. We show that bargaining between two strongly fairness motivated individuals who have different views about what represents a fair division may end in disagreement. Further, by applying the Nash bargaining solution, we study the influence of fairness motivation on the bargaining outcome when an agreement is reached. In particular, we show that the bargaining outcome is sensitive to the fairness motivation of the two individuals, unless they both consider an equal division fair. We argue that our results accommodate existing experimental and field data on bargaining.     

Rawlsian Nash solutions

We investigate the conditions which guarantee the choice of Rawlsian solutions by players of unequal maximin positions bargaining under Nash rules. The conditions, viz., subsymmetry and the location of the Pareto set, relate to the structure of the utility set,S.     

Alternating-Offer Bargaining and Common Knowledge of Rationality

This paper reconsiders Rubinstein's alternating-offer bargaining game with complete information. We define rationalizability and trembling- hand rationalizability (THR) for multi-stage games with observed actions. We show that rationalizability does not exclude perpetual disagreement or delay, but that THR implies a unique solution. Moreover, this unique solution is the unique subgame perfect equilibrium (SPE). Also, we reconsider an extension of Rubinstein's game where a smallest money unit is introduced: THR rules out the non-uniqueness of SPE in some particular case. Finally, we investigate the assumption of boundedly rational players. Perpetual disagreement is excluded, but not delay. Furthermore, we cannot use the asymmetric Nash bargaining solution as an approximation of the alternating-offer bargaining model once the players are boundedly rational ones.     

Distributive justice of bargaining and risk sensitivity

In this paper we will point out some possibilities and limitations of the discussion of distributive justice by bargaining in the classical bargaining models.We start by considering a kind of bargaining situation where two persons with different risk aversions have to distribute a given quantity of a certain good. Then we define a model in which two bargaining situations are compared. In both situations two persons divide a quantity of a certain good; in the second situation one of the persons, say person 2, is replaced by a more risk averse person. From a well-known theorem of Kihlstrom, Roth and Schmeidler it follows that in the Nash solution, the Kalai-Smorodinsky solution and the Maschler-Perles solution person 1 prefers the situation with the more risk averse opponent.In both classes of problems the judgement of distributive justice is impossible because of an informational poverty of the classical bargaining model. We propose to integrate changes in the economic situation of the persons into the model.Therefore, in a third step, we compare two distributive situations, where differences in the situations are implied by changes in the initial endowments of the persons. Under the assumption that each person has a decreasing local risk aversion, we show that every reallocation of the initial endowments is enlarged or at least preserved by risk sensitive bargaining solutions. This fact has some significance for the discussion of distributive justice in social decision making by bargaining.     

Loss Aversion and Bargaining

We consider bargaining situations where two players evaluate outcomes with reference-dependent utility functions, analyzing the effect of differing levels of loss aversion on bargaining outcomes. We find that as with risk aversion, increasing loss aversion for a player leads to worse outcomes for that player in bargaining situations. An extension of Nash's axioms is used to define a solution for bargaining problems with exogenous reference points. Using this solution concept we endogenize the reference points into the model and find a unique solution giving reference points and outcomes that satisfy two reasonable properties, which we predict would be observed in a steady state. The resulting solution also emerges in two other approaches, a strategic (non-cooperative) approach using Rubinstein's (1982) alternating offers model and a dynamic approach in which we find that even under weak assumptions, outcomes and reference points converge to the steady state solution from any non-equilibrium state.     

On the evaluation of solution concepts

It is proposed that solution concepts for games should be evaluated in a way that is analogous to the way a logic is evaluated by a model theory for the language. A solution concept defines a set of strategy profiles, as a logic defines a set of theorems. A model theoretic analysis for a game defines a class of models, which are abstract representations of particular plays of the game. Given an appropriate definition of a model, one can show that various solution concepts are characterized by intuitively natural classes of models in the same sense that the set of theorems of a logic is characterized by a class of models of the language. Sketches of characterization results of this kind are given for rationalizability, Nash equilibrium, and for a refinement of rationalizability —strong rationalizability — that has some features of an equilibrium concept. It is shown that strong rationalizability is equivalent to Nash equilibrium in perfect information games. Extensions of the model theoretic framework that represent belief revision and that permit the characterization of other solution concepts are explored informally.     

What price compromise?

This paper identifies, and tests experimentally, a prediction of the Nash bargaining axioms that may appear counterintuitive. The context is a simple bargaining problem in which two players have to agree a choice from three alternatives. One alternative favours one player and a second favours the other. The third is an apparently reasonable compromise, but is in fact precluded as an agreed choice by the Nash axioms. Experimental results show that agreement on this third alternative occurs rather often. Our subjects’ behaviour could be interpreted as the paying of an irrationally high price, according to the Nash axioms, in order to reach a compromise agreement.     

Finite Horizon Bargaining With Outside Options And Threat Points

We characterize equilibrium behavior in a finite horizon multiple-pie alternating offer bargaining game in which both agents have outside options and threat points. In contrast to the infinite horizon case the strength of the threat to delay agreement is non-stationary and decreases over time. Typically the delay threat determines equilibrium proposals in early periods, while the threat to opt out characterizes those in later ones. Owing to this non-stationarity both threats may appear in the equilibrium shares immediately agreed upon in the first period. When the threat to opt out is empty for both agents, the outcome corresponds exactly with the (generalized) Nash bargaining solution. The latter observation may prove useful for designing experiments that are meant to test economic models that include a bargaining stage.JEL CLASSIFICATION CODES: C78, C91.The useful remarks of an anonymous referee greatly improved the presentation of the material and are gratefully acknowledged. The usual disclaimer applies.     

Bargaining Solutions as Social Compromises

A bargaining solution is a social compromise if it is metrically rationalizable, i.e., if it has an optimum (depending on the situation, smallest or largest) distance from some reference point. We explore the workability and the limits of metric rationalization in bargaining theory where compromising is a core issue. We demonstrate that many well-known bargaining solutions are social compromises with respect to reasonable metrics. In the metric approach, bargaining solutions can be grounded in axioms on how society measures differences between utility allocations. Using this approach, we provide an axiomatic characterization for the class of social compromises that are based on p-norms and for the attending bargaining solutions. We further show that bargaining solutions which satisfy Pareto Optimality and Individual Rationality can always be metrically rationalized.