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.     

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.     

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.     

Now or Later?--Endogenous Timing of Threats

Whereas early threats are chosen before bargaining, late threats are determined after bargaining ended in conflict. Instead of exogenously imposing the timing of threats, these are derived endogenously as in indirect evolution or endogenous timing. Based on a duopoly market, we first derive the equilibrium for all possible constellations regarding the timing of threats. Our analysis surprisingly justifies the early timing of threats as suggested by Nash (1953).     

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.     

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.     

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.     

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.     

A Note on Implementation of Bargaining Solutions

Miyagawa (Games and Economics Behavior 41(2), 292–308 [2002]) provides a simple extensive game form that implements a large class of two-agent bargaining solutions in subgame-perfect equilibrium. This class includes all of the Nash, Kalai–Smorodinsky, and relative utilitarian solutions. This note extends Miyagawa’s result to multi-agent bargaining problems.     

On arguments from self-interest for the Nash solution and the Kalai egalitarian solution to the bargaining problem

I argue in this paper that there are two considerations which govern the dynamics of a two-person bargaining game, viz. relative proportionate utility loss from conceding to one's opponent's proposal and relative non-proportionate utility loss from not conceding to one's opponent's proposal, if she were not to concede as well. The first consideration can adequately be captured by the information contained in vNM utilities. The second requires measures of utility which allow for an interpersonal comparison of utility differences. These considerations respectively provide for a justification of the Nash solution and the Kalai egalitarian solution. However, none of these solutions taken by themselves can provide for a full story of bargaining, since, if within a context of bargaining one such consideration is overriding, the solution which does not match this consideration will yield unreasonable results. I systematically present arguments to the effect that each justification from self-interest for respectively the Nash and the Kalai egalitarian solution is vulnerable to this kind of objection. I suggest that the search for an integrative model may be a promising line of research.I am grateful to Jon Elster, A. Hylland, F. Spinnewijn, J. Verhoeven and the members of the research group for theoretical sociology in the K.U.L. for their comments on an earlier version of this paper. I also thank the Ministry for Foreign Affairs (Norway) for its financial support and the members of the department of sociology in the University of Oslo for their hospitality and interest in my work.     

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.     

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.     

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.     

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.     

Simultaneous moves multi-person continuous time concession game

The alternating offers strategic bargaining approach (Rubinstein, 1982) suggest a unique perfect equilibrium. In this equilibrium the bargaining ends immediately and the outcome is Pareto optimal. In this paper we consider a bargaining game in which players move simultaneously. Since we allow players only to lower their demands over time we denote this game as a concession game. We demonstrate that in this game there is a perfect equilibrium in which the bargaining lasts long enough so that the pie shrinks to zero. We show that we can generalize this game to a multi-player concession game in a straightforward manner and so avoiding the difficulties of generalizing the alternating offers game to the multi-player case.     

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.     

集体协商：“民工荒”背景下农民工权益保障的现实路径

缓解当前的＂民工荒＂,首先需要加强对农民工合法权益的保护,笔者认为将集体协商制度的覆盖面落实到农民工群体是一条可行路径。为此,首先需要培育农民工劳动力市场,将市场调节作为集体协商运作的经济基础;其次加强政府对农民工集体协商的宏观指导,为农民工参与集体协商制度提供政策、法规保障;第三要培育规范的谈判主体,使谈判双方主体的地位对等。     

Two-Speed Evolution of Strategies and Preferences In Symmetric Games

Agents in a large population are randomly matched to play a certain game, payoffs in which represent fitness. Agents may have preferences that are different from fitness. They learn strategies according to their preferences, and evolution changes the preference distribution in the population according to fitness. When agents know the preferences of the opponent in a match, only efficient symmetric strategy profiles of the fitness game can be stable. When agents do not know the preferences of the opponent, only Nash equilibria of the fitness game can be stable. For 2 × 2 symmetric games I characterize preferences that are stable.Jel Codes: C72, A13     

On Hobbes’s state of nature and game theory

Hobbes’s state of nature is often analyzed in two-person two-action non-cooperative games. By definition, this literature only focuses on duels. Yet, if we consider general games, i.e., with more than two agents, analyzing Hobbes’s state of nature in terms of duel is not completely satisfactory, since it is a very specific interpretation of the war of all against all. Therefore, we propose a definition of the state of nature for games with an arbitrary number of players. We show that this definition coincides with the strategy profile considered as the state of nature in two-person games. Furthermore, we study what we call rational states of nature (that is, strategy profiles which are both states of nature and Nash equilibria). We show that in rational states of nature, the utility level of any agent is equal to his maximin payoff. We also show that rational states of nature always exist in inessential games. Finally, we prove the existence of states of nature in a class of (not necessarily inessential) symmetric games.     

On the interaction between heterogeneity and decay in two-way flow models

In this article, we examine the role played by heterogeneity in the popular ??connections model?? of Jackson and Wolinsky (J Econ Theory 71(1):355?C365, 1996). We prove that when heterogeneity with respect to the values of resources, or the information decay parameter, depends on the identity of the player who forms the link, and the player who receives the link, all networks can be supported as Nash. Moreover, we show that Nash networks may not always exist. Interestingly, in the absence of decay, neither result occurs in a model with value heterogeneity. Finally, we show that when reducing heterogeneity, all networks cannot be supported as Nash, and the non-existence problem also disappears.