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.     

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.     

Risk aversion in n-person bargaining

Relations between risk properties of Pareto optimal n-person bargaining solutions are studied. The (weak) worse alternative property, risk sensitivity property, and risk profit opportunity property are considered, in particular relations between these properties, and between these properties and other properties such as independence of irrelevant alternatives and individual monotonicity. A distinction is made between bargaining games where all Pareto optimal outcomes are riskless, and bargaining games where Pareto optimal outcomes may represent the utilities of lotteries between riskless alternatives. In the first mentioned case, more general results can be obtained.     

Mechanism robustness in multilateral bargaining

This paper discusses the relationship between coalitional stability and the robustness of bargaining outcomes to the bargaining procedure. We consider a class of bargaining procedures described by extensive form games, where payoff opportunities are given by a characteristic function (cooperative) game. The extensive form games differ on the probability distribution assigned to chance moves which determine the order in which players take actions. One way to define mechanism robustness is in terms of the property of no first mover advantage. An equilibrium is mechanism robust if for each member the expected payoff before and after being called to propose is the same. Alternatively one can define mechanism robustness as a property of equilibrium outcomes. An outcome is said to be mechanism robust if it is supported by some equilibrium in all the extensive form games (mechanisms) within our class. We show that both definitions of mechanism robustness provide an interesting characterization of the core of the underlying cooperative game.     

Experiments on bilateral bargaining in markets

We present experimental data on a simple market game. Several solution concepts from cooperative game theory are applied to predict the observed payoff distributions. Notably, a recently introduced solution concept meant to capture the influence of outside options on the payoff distribution within groups fares better than most other solution concepts under consideration. Our results shed some light on the effects of scarcity relations on markets on bargaining outcomes within negotiating dyads.     

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.     

Delay in a bargaining game with contracts

In a multilateral bargaining game where a proposer and responders can set up a “principal–agent” relationship by means of binding cash-offer contracts, we show that there is a Markov SPE with a delay in reaching an agreement. We also show that all the individually rational and efficient payoffs can be supported by SPE.      

Effects of fixed costs in two-person sequential bargaining

Rubinstein (1982) considered the problem of dividing a given surplus between two players sequentially, and then proposed a model in which the two players alternately make and respond to each other's offers through time. He further characterized the perfect equilibrium outcomes, which depend on the players time preferences and order of moves. Using both equal and unequal bargaining cost conditions and an unlimited number of rounds, two experiments were designed to compare the perfect equilibrium model to alternative models based on norms of fairness. We report analyses of final agreements, first offers, and number of bargaining rounds, which provide limited support to the perfect equilibrium model, and then conclude by recommending a shift in focus from model testing to specification of the conditions favoring one model over another.     

对美国农业市场上合作性谈判协会的考察

一、引言合作性谈判协会是一些美国农业市场的一种制度特征。这种协会以为农民社员提供广泛的多样化服务为特征,其中主要是与各种市场中间商的价格合同谈判。从发展历程来看,     

Risk aversion in bargaining: An experimental study

This paper reports the results of three experiments designed to test the predictions of the principal game-theoretic models of bargaining concerning the influence of risk aversion on bargaining outcomes. These models predict that risk aversion will be disadvantageous in bargaining except in situations in which potential agreements are lotteries with a positive probability of being worse than disagreement. The experimental results support the models' predictions. However, in the range of payoffs studied here, the effects due to risk aversion may be smaller than some of the focal point effects observed in previous experiments. Implications for further theoretical and experimental work are considered.     

On the first-offer dilemma in bargaining and negotiations

Theory and Decision - In bargaining and negotiations, should one make the first offer or wait for the opponent to do so? Practitioners support the idea that moving first in bargaining is a mistake,...     

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.     

Subgame perfect equilibrium in a bargaining model with deterministic procedures

Two players, A and B, bargain to divide a perfectly divisible pie. In a bargaining model with constant discount factors, \(\delta _A\) and \(\delta _B\), we extend Rubinstein (Econometrica 50:97–110, 1982)’s alternating offers procedure to more general deterministic procedures, so that any player in any period can be the proposer. We show that each bargaining game with a deterministic procedure has a unique subgame perfect equilibrium (SPE) payoff outcome, which is efficient. Conversely, each efficient division of the pie can be supported as an SPE outcome by some procedure if \(\delta _A+\delta _B\ge 1\), while almost no division can ever be supported in SPE if \(\delta _A+\delta _B < 1\).     

Collective bargaining levels,employment and wage inequality in Spain

After the increase in inequalities following the Great Recession, studies on wage bargaining systems have increasingly focused on wage inequality. This research examines wage inequality associated with collective bargaining levels in Spain, based on matched employer–employee microdata and quantile regression methods. These methods are applied across the wage distribution, following the method proposed by Firpo et al. (2011), to estimate wage premiums associated with agreement levels and to decompose the wage differentials observed at different points of the wage distribution. From the evidence obtained it can be concluded that, although the higher wages found in firm-level agreements are explained by the better observed characteristics of firms and workers covered by these collective agreements, there remains a positive wage premium. Although this premium is seen throughout the wage distribution, it favours mostly workers in the middle and upper-middle end. This slightly increases wage inequality in comparison with sectoral agreements. In contrast, workers without collective bargaining coverage generally suffer a wage penalty. This penalty is only observed on the left of the wage distribution. It becomes a significant wage premium in the upper end of the distribution, which implies a significant increase in wage inequality. In short, the evidence of this research suggests that reducing the coverage of collective bargaining could be associated with a significant increase in wage inequality. A better policy option for countries with a predominant sectoral model, such as Spain, would be to move towards an organized decentralization model. This would cause significant gains in employment as suggested by OECD (2019) and only a slight increase in wage inequality.     

Interpersonal comparisons of utility in bargaining: evidence from a transcontinental ultimatum game

This article presents the experimental results of a “Transcontinental Ultimatum Game” implemented between India and France. We use a standard ultimatum game, but in one treatment, Indian subjects made offers to French subjects (ItoF treatment) and, in another treatment, French subjects made offers to Indian subjects (FtoI treatment). We observed that FtoI treatment bargaining mostly ended up with unequal splits of money in favor of French, while nearly equal splits were the most frequent outcome in ItoF treatment interactions. The experimental results are organized through a standard social reference model, modified for taking into account the different marginal value of money for bargainers. In our model, bargaining is driven by relative standing comparisons between players, occurring in terms of real earnings (that is monetary earnings corrected for a purchasing power factor) obtained in the game. The norm of equity behind the equalization of real earnings is called local equity norm, and contrasted to a global equity norm which would encompass the wealth of players beyond the game. According to what we observed, no beyond-game concern seems to be relevantly endorsed by subjects.     

The role of risk preferences in bargaining when acceptance of a proposal requires less than unanimous approval

The role of risk preferences in determining the outcome to bargaining is examined for the case in which acceptance of a proposal requires less than unanimous approval. Using an n-agent extension of the Ståhl-Rubinstein alternating offer model, we find that risk preferences play a fundamentally different role when bargaining is settled using a nonunanimity voting rule. Risk preferences determine not only an agent's reservation price but also the likelihood that he is made part of the winning coalition. An implication of this analysis is that when the preferences of the agents are not too diverse, it is advantageous for an agent to be relatively risk-averse.The helpful comments and suggestions of Rich McLean and an anonymous referee are gratefully acknowledged. This article was presented as a paper at the 1988 Winter Econometric Society Meetings.     

社会理论的空间转向

社会理论的空间转向是社会理论的当代进展的一个重要组成部分,它沿着两条路径展开：一方面,吉登斯、布迪厄等社会理论大师在现代性架构下检视空间与社会的交互关系对于研究社会结构与社会过程的重要性;另一方面,后现代社会理论家采用一系列的地理学概念和隐喻来探索日益复杂和分化的社会世界。本文旨在系统地呈现这一转向的基本脉络。     

When you don't need to join: The effects of guaranteed payoffs on bargaining in three-person cooperative games

Theory and Decision - Two coalition formation experiments were performed in which subjects played twenty 3-person games in characteristic function form. The experiments were identical except for...