A bargaining model in general equilibrium

The goal of the paper is to present a simple model of rational endogenous household formation in a general equilibrium framework in which Pareto optimality at the economy level is not necessarily obtained. The simplest example of household formation is the case in which pairs of individuals engage themselves in a bargaining process on the division of some wealth: if an agreement on the distribution is (not) reached, we can say that the household is (not) formed. The vast majority of existing bargaining models predicts agreements on an efficient outcome. A seminal paper by Crawford (Econometrica 50:607–637, 1982) describes a very simple game with incomplete information in which, even with rational agents, disagreement causes welfare losses. We embed that model in a general equilibrium framework and present some results on equilibria both in the bargaining game and the associated exchange economy. Crawford’s results support Schelling’s intuition on the reasons of disagreement: it may arise if players’ commitments are reversible. Crawford shows that high probabilities of reversibility tend to favor the bargaining impasse, in fact with low probability. We prove that even if those probabilities are arbitrarily close to zero, disagreement is an equilibrium outcome, with high probability. That conclusion seems to be an even stronger support to Schelling’s original viewpoint. In the exchange economy model with that noncooperative bargaining game as a first stage, we present significant examples of economies for which equilibria exist. Because of disagreement, Pareto suboptimal exchange economy equilibria exist for all elements in the utility function and endowment spaces and they may coexist with Pareto optimal equilibria even at the same competitive prices.     

利益相关者间的谈判与企业治理结构

企业的契约性与利益相关者理论具有一致性 ,它们都隐含了企业是一种再谈判机制。利益相关者对企业的所有权分配进行谈判 ,谈判结果决定出企业的治理结构。在这个过程中 ,各方的谈判破裂结果效用、谈判力和对谈判破裂的担心程度是关键变量。我们使用Nash谈判模型及其推广模型分析了这个过程。我们的结论是 ,企业治理结构是内生的 ,每个企业的治理结构都是特殊的个案 ,一种治理结构并不具有普适性。这个结论具有重要的立法含义。     

Collective choice with endogenous reference outcome

A collective choice problem—essentially a bargaining problem without disagreement outcome—is studied. An extended solution, which determines a solution and a reference point simultaneously, is characterized. The unique extended solution that meets the extended versions of Pareto-optimality, independence of irrelevant alternatives, symmetry, and scale invariance maximizes the Nash product with respect to both the solution and the reference point.     

N-PERSON NASH BNARGAINING WITH VARIABEL THREATS*

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash's variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large.     

Bargaining and Boldness

We study a multiperson bargaining problem with general risk preferences through the use of Shaked's game of cycling offers with exogenous breakdown. If preferences are “smooth,” then as the risk of breakdown vanishes, the limiting outcome is one in which bargainers are equally marginally bold; where a bargainer's marginal boldness measures his willingness to risk disagreement in return for a marginal improvement in his position. Under smoothness, any (ordinal-)Nash solution is an equally marginally bold outcome. However, unlike the concept of the (ordinal-)Nash solution, a unique equally marginally bold outcome exists in natural cases—in particular, if all bargainers have risk-averse preferences of the rank-dependent expected utility type. For these preferences, the equally marginally bold outcome maximizes a “bargaining power”-adjusted (asymmetric) Nash product where the degree of asymmetry is determined by the disparity in the marginal valuation of certainty among bargainers. Journal of Economic Literature Classification Numbers: C72, C78, D81.     

Bargaining with random implementation: An experimental study

We use a laboratory experiment to study bargaining with random implementation. We modify the standard Nash demand game so that incompatible demands do not necessarily lead to the disagreement outcome. Rather, with exogenous probability q, one bargainer receives his/her demand, with the other getting the remainder. We use an asymmetric bargaining set (favouring one bargainer) and disagreement payoffs of zero, and we vary q over several values.Our results mostly support game theoryʼs directional predictions. As with conventional arbitration, we observe a strong chilling effect on bargaining for q near one: extreme demands and low agreement rates. Increasing q reinforces the gameʼs built-in asymmetry – giving the favoured player an increasingly large share of payoffs – but also raising efficiency. These effects are non-uniform: over sizable ranges, increases in q have minimal effect, but for some q, small additional increases lead to sharp changes in results.     

The welfare effect of bargaining power in the licensing of a cost-reducing technology

This study considers licensing of a cost-reducing technology through bargaining between a technology-holding firm and its rival firm in a Cournot duopoly market. To consider the relative bargaining power of both firms, the asymmetric Nash bargaining solution is applied as our solution. Then, we specify the combinations of lump-sum fee and per-unit royalty that are realized through bargaining, and examine the effect on social welfare of the technology-holding firm’s bargaining power. The principal findings are as follows. Regardless of the technology-holding firm’s bargaining power, pure royalty licensing is carried out, and social welfare is non-increasing in its bargaining power. In our model, licensing through a take-it-or-leave-it offer, which is often assumed in the literature, is regarded as the case in which the technology-holding firm has full bargaining power. Thus, the result on social welfare implies that the take-it-or-leave-it offer licensing mechanism leads to the socially worst outcome.     

Special interest politics: Contribution schedules vs. Nash bargaining

The article compares two models of lobby influence on policy choice: The Grossman and Helpman (1994) contribution‐schedules model and a negotiation between the lobbies and the government summarized by a Nash‐bargaining function. The literature uses the models interchangeably because they imply the same equilibrium policy. We show that particular assumptions about bargaining power and disagreement utility in the Nash‐bargaining solution are required for the models to lead to the same equilibrium payments and utilities. This implies that the models usually imply different sets of lobbies if lobby formation is an endogenous decision, such that the equilibrium policies also differ.     

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.     

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.     

Competitive bargaining equilibrium

In a simple exchange economy we propose a bargaining procedure that leads to a Walrasian outcome as the agents become increasingly patient. The competitive outcome therefore obtains even if agents have market power and are not price-takers. Moreover, where in other bargaining protocols the final outcome depends on bargaining power or relative impatience, the outcome here is determinate and depends only on preferences and endowments. Our bargaining procedure involves bargaining over prices and maximum quantity constraints, and it guarantees convergence to a Walrasian outcome for any standard exchange economy. In contrast, without quantity constraints we show that equilibrium is generically inefficient.     

Achieving efficiency with manipulative bargainers

Two agents bargain over the allocation of a bundle of divisible commodities. After strategically reporting utility functions to a neutral arbitrator, the outcome is decided by using a bargaining solution concept chosen from a family that includes the Nash and the Raiffa–Kalai–Smorodinsky solutions. When reports are restricted to be continuous, strictly increasing and concave, it has been shown that this kind of “distortion game” leads to inefficient outcomes. We study the distortion game originated when agents are also allowed to claim non-concave utility functions. Contrasting with the previous literature, any interior equilibrium outcome is efficient and any efficient allocation can be supported as an equilibrium outcome of the distortion game. In a similar fashion to the Nash demand game we consider some uncertainty about the opponent's features to virtually implement the Nash bargaining solution.     

Modeling longevity risk transfers as Nash bargaining problems: Methodology and insights

The problem of longevity risk has recently received considerable attention. In this paper, we apply economic modeling methods to longevity risk securitization, which is now regarded by pension and insurance industries as a solution to the problem. Specifically, we model the trade of a longevity security as a two-player bargaining game, and use Nash's bargaining solution to determine the outcome of it. Our work not only offers an alternative method for pricing longevity securities, but also reveals several properties about the market for longevity securities. First, a trade would occur if the longevity security is an effective hedging instrument, and the trade would benefit all agents involved. Second, a trade of longevity risk can reduce pension plans' bankruptcy risk, safeguarding the financial security of pension plan members. Finally, compared to the competitive equilibrium, Nash's bargaining solution yields higher trading prices. Therefore, as the market becomes more competitive, pension plans may hedge longevity risk at a lower cost.     

How sensitive are bargaining outcomes to changes in disagreement payoffs?

We use a human-subjects experiment to investigate how bargaining outcomes are affected by changes in bargainers’ disagreement payoffs. Subjects bargain against changing opponents, with randomly drawn asymmetric disagreement outcomes that vary over plays of the game, and with complete information about disagreement payoffs and the cake size. We find that subjects only respond about half as much as theoretically predicted to changes in their own disagreement payoff and to changes in their opponent’s disagreement payoff. This effect is observed in a standard Nash demand game and a related unstructured bargaining game, in both early and late rounds, and is robust to moderate changes in stake sizes. We show theoretically that standard models of expected utility maximisation are unable to account for this under-responsiveness, even when generalised to allow for risk aversion. We also show that quantal-response equilibrium has, at best, mixed success in characterising our results. However, a simple model of other-regarding preferences can explain our main results.     

An Experimental Study of the Effect of Private Information in the Coase Theorem

We investigate, in an experimental setting, the effect of private information on the Coase theorem's predictions of efficiency and allocative neutrality. For a two-person bargaining game, we find significantly more inefficiency and allocative bias in the case of private information compared with the case of complete information. We also find substantial bargaining breakdown, which is not predicted by the Coase theorem. For the case of private information, we reject the Coase theorem in favor of the alternative of a generalized version of the Myerson—Satterthwaite theorem, which predicts inefficiency, allocative bias in the direction of the disagreement point, and some bargaining breakdown.     

Union–firm bargaining: Order of play and efficiency

This paper shows that a modified alternating offers Rubinstein model can provide a Pareto superior outcome in the context of the right-to-manage union–firm bargaining. Two examples of bargaining protocols that yield a superior outcome are provided. In the first example, the parties engage in a game in which the order of play is determined as part of the bargaining. We show that the game has a unique subgame perfect equilibrium in which the firm always moves first in the wage bargaining game. The equilibrium wage is, therefore, unique. In the second example, we examine a two-part-tariff alternating offers bargaining protocol, where the parties bargain over the wage and transfer payments. We show that this bargaining protocol has a Pareto efficient, unique subgame perfect equilibrium. Thus, although the parties do not bargain over the level of employment, the outcome under this protocol is, nevertheless, socially optimal.     

Extending the Nash solution to choice problems with reference points

In 1985 Aumann axiomatized the Shapley NTU value by non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives. We show that, when replacing unanimity by “unanimity for the grand coalition” and translation covariance, these axioms characterize the Nash solution on the class of n-person choice problems with reference points. A classical bargaining problem consists of a convex feasible set that contains the disagreement point here called reference point. The feasible set of a choice problem does not necessarily contain the reference point and may not be convex. However, we assume that it satisfies some standard properties. Our result is robust so that the characterization is still valid for many subclasses of choice problems, among those is the class of classical bargaining problems. Moreover, we show that each of the employed axioms – including independence of irrelevant alternatives – may be logically independent of the remaining axioms.     

One-dimensional bargaining with Markov recognition probabilities

We study a process of bargaining over alternatives represented by points in the unit interval. The paper focuses on the asymptotic behavior of the subgame perfect equilibrium in stationary strategies as the continuation probability approaches one. We give a complete characterization of the limit of the equilibrium proposals as the generalized fixed point of the decumulative distribution of the players' ideal points as induced by the recognition probabilities. In contrast to the existing literature, we find no general relationship between the limit equilibrium proposals and either the Nash bargaining solution or the median voter outcome.     

Dominance solvability of dynamic bargaining games

We formulate and study a general finite-horizon bargaining game with simultaneous moves and a disagreement outcome that need not be the worst possible result for the agents. Conditions are identified under which the game is dominance solvable in the sense that iterative deletion of weakly dominated strategies selects a unique outcome. Our analysis uses a backward induction procedure to pinpoint the latest moment at which a coalition can be found with both an incentive and the authority to force one of the available alternatives. Iterative dominance then implies that the alternative characterized in this way will be agreed upon at the outset—or, if a suitable coalition is never found, that no agreement will be reached.     

The Scope of Anonymous Voluntary Bargaining Under Asymmetric Information

We present a model of anonymous collective bargaining where individuals'preferences and information may be significantly interdependent. We showthat the bargaining outcome becomes independent of individuals'preferences and information as the bargaining group increases in size. As acorollary, we show that anonymous voluntary bargaining completely fails inlarge groups. Either the difference between the bargaining outcome and thestatus quo vanishes as the size of the group becomes larger, or, thebargaining becomes coercive and results in a violation of at least someindividuals' rights. The result provides a rationale for the inherentdifficulty of reform in the presence of asymmetric information.
"There is nothing more difficult to carry out, nor more doubtful ofsuccess, nor more dangerous to handle, than to initiate a new order ofthings."
Niccolò Machiavelli, The Prince (1532)