Multi-agent bilateral bargaining and the Nash bargaining solution

This paper studies a bargaining model where n   players negotiate how to share a pie through (n−1)$(n-1)$ bilateral bargaining sessions. In each session, two players bargain for a partial agreement that specifies who exits and who moves on to the next session (if there is any) via the alternating-proposal framework of Rubinstein [Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–109]. We consider two bargaining procedures under which the subgame perfect equilibrium outcomes converge to the Nash [Nash, J., 1950. The bargaining problem. Econometrica 18, 155–162] bargaining solution for the corresponding bargaining problem as the players’ discount factor goes to one. Hence, the model studied here provides a non-cooperative foundation for the Nash cooperative bargaining solution in the multilateral case.     

Trading bargaining weights

We consider a model, in which two agents are engaged in two separate bargaining problems. We introduce a notion of bargaining weights (bargaining power), which is basically given by asymmetric versions of the Perles–Maschler bargaining solution. Thereby, we view bargaining power as ordinary goods that can be traded in an exchange economy. With equal initial endowment of bargaining power there exists a Walrasian equilibrium in this exchange economy such that the utility allocation in equilibrium coincides with the Perles–Maschler bargaining solution of the aggregate bargaining problem. Equilibrium prices are given by the primitives of the two bargaining problems.     

Trading bargaining weights

We consider a model, in which two agents are engaged in two separate bargaining problems. We introduce a notion of bargaining weights (bargaining power), which is basically given by asymmetric versions of the Perles–Maschler bargaining solution. Thereby, we view bargaining power as ordinary goods that can be traded in an exchange economy. With equal initial endowment of bargaining power there exists a Walrasian equilibrium in this exchange economy such that the utility allocation in equilibrium coincides with the Perles–Maschler bargaining solution of the aggregate bargaining problem. Equilibrium prices are given by the primitives of the two bargaining problems.     

Buying cooperation in an asymmetric environmental differential game

We consider a two-player asymmetric differential game of pollution control. One player is non-vulnerable to pollution, or unwilling to consider damages when choosing her production policy in a non-cooperative game. We characterize the feedback-Nash equilibrium and the cooperative solution. We establish conditions under which the vulnerable player can buy the cooperation of the non-vulnerable player to control her emissions. We further use the Nash bargaining solution to allocate the total cooperative dividend between the two players and propose a time-consistent decomposition overtime of the total payoff.     

Mediation and the Nash bargaining solution

This paper analyzes a model of bargaining in which two parties use a mediator who sequentially makes random proposals until agreement by both parties is reached. I show that as the cost of delay shrinks to zero, the subgame perfect payoff converges to the asymmetric Nash bargaining solution with weights determined by the relative discount rates of the players. I also establish conditions for the uniqueness of the subgame perfect equilibrium for arbitrary discount rates.     

Bargaining solutions with non-standard objectives

I examine the pure-strategy solutions of the sealed-bid bargaining game with incomplete information, when the buyer's and seller's objectives are other than the standard objective, namely maximization of expected profit. The motivation for this exploration lies in three problems of the standard formulation: the necessity of assuming common priors, the existence of uncountably many Nash equilibria, with no means for the players to coordinate on any one of them, and the uncertain relationship between these equilibria and observed behavior in bargaining experiments. Specifically, I consider two alternative objectives: minimization of maximum regret, and maximization of maximum profit. The solution concept here is not Nash equilibrium, but rather -individually rational strategy bundle. For that reason, I shall, where appropriate, use the word “solution” in place of “equilibrium.” Yet we find that the notion of Nash Equilibrium reappears, in a sense to be explained. In the minimax-regret case I find (in contrast to the case of expected profit) a unique solution; this solution reduces, for priors with coincident support, to the linear equilibrium of Chatterjee-Samuelson. In the maximum-profit case there are many solutions; they turn out to be slight generalizations of the one-step equilibria of Leininger-Linhart-Radner.     

A noncooperative foundation of the asymmetric Nash bargaining solution

We consider a noncooperative multilateral bargaining model with heterogeneous time preferences in which the first rejector of a proposal in the current round becomes the proposer in the next round. We show the existence of a stationary subgame perfect equilibrium (SSPE), characterize SSPEs and show the efficiency of SSPEs. We show that any sequence of SSPE payoff profiles converges to the asymmetric Nash bargaining solution weighted by the inverses of discount rates as the bargaining friction vanishes.     

Bargaining over an uncertain outcome:¶the role of beliefs

We study the Nash bargaining solution of a problem in which two agents bargain over an uncertain outcome. Under the assumptions of risk neutrality and of constant absolute risk aversion, we study the way that the solution varies, ex ante, when we vary the beliefs of one agent. Changing an agent's beliefs in a way that makes them “more distant” from the other agent's beliefs makes the second agent better off. Received: 10 May 2001 / Accepted: 22 August 2001     

Threats and bargaining in capitalism a differential game view

The paper examines the Nash bargaining equilibrium in a simple model of growth and distribution formulated as a differential game between workers, who may save or consume, and capitalists, who may consume or invest. Optimal threats announced by the social classes to affect the negotiated solution to their own advantage are also derived. It is demonstrated that the workers' threat takes the form of refusing to save while capitalists threaten to refrain from investing. The threats as well as the possible gains from cooperation determine the players' relative importance or bargaining power. It is shown that capitalists are in general in a stronger bargaining position than workers. The dependence of the bargaining power on the parameters of the model is examined.     

On Maskin monotonicity of solution based social choice rules

Howard (J Econ Theory 56:142–159, 1992) argues that the Nash bargaining solution is not Nash implementable, as it does not satisfy Maskin monotonicity. His arguments can be extended to other bargaining solutions as well. However, by defining a social choice correspondence that is based on the solution rather than on its realizations, one can overcome this shortcoming. We even show that such correspondences satisfy a stronger version of monotonicity that is even sufficient for Nash implementability.     

网络外部性、内生成本差异与企业定价策略

在产品成本内生条件下,通过建立扩展的Hotelling模型,分析网络外部性对企业定价策略的影响,结果显示:当网络外部性强度较低时,企业将选择歧视定价策略,当网络外部性强度较高时,企业将选择单一定价策略;在两种定价策略下,Nash谈判能力强的企业将获得更高的市场份额和利润;社会福利大小由网络外部性强度和企业Nash谈判能力共同决定,与产品定价策略无关,当网络外部性强度较高时,社会福利与企业间Nash谈判能力的差异正相关,反之则负相关。     

Implementing the nash extension bargaining solution for non-convex problems

Conley and Wilkie (1993) introduced an axiomatization at the Nash extension bargaining solution defined on a domain of comprehensive but not necessarily convex problems. In this paper we present a non-cooperative game which implements the Nash extension solution in subgame perfect equilibria in the limit as the discount rate applied between rounds of play vanishes.     

Cost sharing: efficiency and implementation

We study environments where a production process is jointly shared by a finite group of agents. The social decision involves the determination of input contribution and output distribution. We define a competitive solution when there is decreasing-returns-to-scale which leads to a Pareto optimal outcome. Since there is a finite number of agents, the competitive solution is prone to manipulation. We construct a mechanism for which the set of Nash equilibria coincides with the set of competitive solution outcomes. We define a marginal-cost-pricing equilibrium (MCPE) solution for environments with increasing returns to scale. These solutions are Pareto optimal under certain conditions. We construct another mechanism that realizes the MCPE.     

The <Emphasis Type="Italic">n</Emphasis>-person Kalai-Smorodinsky bargaining solution under pre-donations

This study examines the behavior of simple n-person bargaining problems under pre-donations where the Kalai-Smorodinsky (KS) solution is operant. Pre- donations are a unilateral commitment to transfer a portion of one’s utility to someone else, and are used to distort the bargaining set and thereby influence the bargaining solution. In equilibrium, these pre-donations are Pareto-improving over the undistorted solution; moreover, when the agents’ preferences are sufficiently distinct, the equilibrium solution coincides with the concessionary division rule.     

Bargaining with independence of higher or irrelevant claims

This paper studies independence of higher claims and independence of irrelevant claims on the domain of bargaining problems with claims. Independence of higher claims requires that the payoff of an agent does not depend on the higher claim of another agent. Independence of irrelevant claims states that the payoffs should not change when the claims decrease but remain higher than the payoffs. Interestingly, in conjunction with standard axioms from bargaining theory, these properties characterize a new constrained Nash solution, a constrained Kalai–Smorodinsky solution, and a constrained Kalai solution.     

Nash bargaining in ordinal environments

We analyze the implications of Nash’s (Econometrica 18:155–162, 1950) axioms in ordinal bargaining environments; there, the scale invariance axiom needs to be strenghtened to take into account all order-preserving transformations of the agents’ utilities. This axiom, called ordinal invariance, is a very demanding one. For two-agents, it is violated by every strongly individually rational bargaining rule. In general, no ordinally invariant bargaining rule satisfies the other three axioms of Nash. Parallel to Roth (J Econ Theory 16:247–251, 1977), we introduce a weaker independence of irrelevant alternatives (IIA) axiom that we argue is better suited for ordinally invariant bargaining rules. We show that the three-agent Shapley–Shubik bargaining rule uniquely satisfies ordinal invariance, Pareto optimality, symmetry, and this weaker IIA axiom. We also analyze the implications of other independence axioms.     

Consistent collusion-proofness and correlation in exchange economies

We present a feasible strategic market mechanism with finitely many agents whose Nash, semi-strong Nash and coalition-proof Nash equilibria fully implement the Walrasian equilibria. We define a strategic equilibrium concept, called correlated semi-strong equilibrium, and show that the Walrasian equilibria can be implemented by these equilibria, and also by the coalition-proof correlated equilibria of our mechanism. We show that these two concepts, suitably modified with transfers, fully implement the Pareto optimal allocations.     

Fiscal and monetary policy co-ordination: A differential game approach

In this paper the interaction between the Treasury and the central bank is examined in the case of both cooperative and non-cooperative behaviour. Differential games are used in the framework of a continuous-time econometric model of the Italian economy. The Nash and the Stackelberg non-cooperative equilibrium solutions are computed, and the case for cooperation is analysed by considering the Nash and the Kalai-Smorodinsky bargaining models. It is shown that, in the Italian case, the government has a stronger bargaining position than the central bank. A comparison is then made between the different solutions to show that the drawbacks that emerge from non-cooperation are not simply those depending on the players' payoffs. Other features are in fact considered which constitute a further argument for policy co-ordination.     

产品成本内生化的企业定价选址博弈分析

运用三阶段博弈模型研究了双寡头厂商面对需求非对称、产品成本内生时的定价选址决策问题,并分别分析了供应商Nash谈判参数的变化对生产商均衡位置以及均衡利润的影响.研究发现,与产品成本外生的情形相比,供应商的存在改变了生产商的均衡位置.当某一供应商的相对谈判能力增强时,与之形成双边垄断关系的生产商就会远离其垄断市场,均衡利润将会减少,而竞争对手则会靠近其对应的垄断市场,均衡利润将会增加.两个生产商之间的均衡距离会随供应商谈判能力的增强而缩小.     

Efficient equilibria in a differential game of capitalism

One considers a differential game of capitalism ‘à la Lancaster’. Feedback Nash equilibria and Pareto optimal solutions are characterized under the assumption that the planning horizon is infinite. It is then shown that, by combining a Pareto optimal solution with the Nash feedback equilibrium strategy pair, which plays the role of a threat, one can obtain an efficient equilibrium which is also subgame perfect in the sense of Selten. This result modifies sensibly the previous interpretation of the inherent inefficiency of capitalism.