Bargaining order and delays in multilateral bargaining with heterogeneous sellers

We consider a complete-information multilateral bargaining game in which a single buyer negotiates with two heterogeneous sellers selling perfect complementary units. While bilateral negotiations take place through a sequence of offers and counteroffers, the bargaining order is exogenously given. We solve for the conditions under which (a) the buyer prefers to negotiate with the lower-valuation seller first and (b) efficient (inefficient) outcomes emerge for the two bargaining orders. We find that the buyer prefers to negotiate with the lower-valuation seller first whenever the players are relatively impatient or the sellers are sufficiently heterogeneous. We show that there exists a unique efficient outcome when the buyer negotiates first with the lower-valuation seller and the sellers are sufficiently heterogeneous; however, significant delay in reaching agreements may arise when they are not. In case the buyer bargains with the higher-valuation seller first, an inefficient outcome is shown to exist even when players are extremely impatient.     

Negotiation,preferences over agreements,and the core

We analyze conditions under which negotiated agreements are efficient from the point of view of every possible coalition of negotiators. The negotiators have lexicographic preferences over agreements they reach. Their utility is the first criterion. The coalition reaching an agreement is the second criterion. In the analyzed non-cooperative discrete time bargaining game Γ the players bargain about the choice from the sets of utility vectors feasible for coalitions in a given NTU game (N, V). If Γ has a Markov perfect equilibrium, then the set of equilibrium utility vectors in Markov perfect equilibria in it equals the core of (N, V). I thank an anonymous referee, an anonymous Associate Editor, and the Editor for their comments that helped me to improve the paper. The research reported in this paper was supported by the Grant VEGA 1/1223/04 of the Ministry of Education of the Slovak Republic.     

Kalai-Smorodinsky Bargaining Solution Equilibria

Multicriteria games describe strategic interactions in which players, having more than one criterion to take into account, don’t have an a-priori opinion on the relative importance of all these criteria. Roemer (Econ. Bull. 3:1–13, 2005) introduces an organizational interpretation of the concept of equilibrium: each player can be viewed as running a bargaining game among criteria. In this paper, we analyze the bargaining problem within each player by considering the Kalai-Smorodinsky bargaining solution (see Kalai and Smorodinsky in Econometrica 43:513–518, 1975). We provide existence results for the so called Kalai-Smorodinsky bargaining solution equilibria for a general class of disagreement points which properly includes the one considered by Roemer (Econ. Bull. 3:1–13, 2005). Moreover we look at the refinement power of this equilibrium concept and show that it is an effective selection device even when combined with classical refinement concepts based on stability with respect to perturbations; in particular, we consider the extension to multicriteria games of the Selten’s trembling hand perfect equilibrium concept (see Selten in Int. J. Game Theory 4:25–55, 1975) and prove that perfect Kalai-Smorodinsky bargaining solution equilibria exist and properly refine both the perfect equilibria and the Kalai-Smorodinsky bargaining solution equilibria.     

Relative impatience determines preference between contract bargaining and repeated bargaining

In this paper we consider the effect of the `impatience ratio' I (of the worker discount factor to the firm discount factor) on the preferences of the players between two bargaining schemes in an asymmetric information wage bargaining context. The firm has private information about the worker's value and the worker makes wage demands. In the contact bargaining scheme, a wage demand which is accepted in one period is binding for all future periods (and hence the bargaining ends after acceptance of a wage demand). In the repeated bargaining scheme, the parties continue to bargain irrespective of whether the worker has been hired or not, and any accepted wage demand is only valid for the period in which it was accepted. We establish the following results under the assumption that the worker's value is uniformly distributed on an interval: When the firm is more patient than the worker (I<1) both parties prefer contract bargaining, and when the worker is more patient than the firm (I >1) both prefer repeated bargaining. For any value of I, the preferred type of bargaining gives the lower unemployment.  The work of Bae has already shown that when players are equally patient (I=1) the players are indifferent between the two schemes, regardless of the distribution of the worker's value. This paper shows that Bae's indifference result (Bae, 1991) cannot be extended to unequally patient players. Received: December 1996/Final version: October 1998     

三方相互体谅讨价还价模型

对于较重要的三方合作利益分配博弈问题,将相互体谅作为基本遵循并基于适应性预期模型,建立三方相互体谅讨价还价模型.运用这个模型,可以把三方相互体谅讨价还价达成均衡,归结为单调有界数列收敛与常系数非齐次线性差分方程组求解的结合运用.三方相互体谅讨价还价属于不完全信息动态合作博弈.对模型求解,一般地揭示三方相互体谅讨价还价达成均衡的过程与结果,阐明讨价还价有满足初始条件的唯一解和充要条件,以及充要条件的经济含义和数学意义.三方相互体谅讨价还价模型是对于两方相互体谅讨价还价模型的拓展,可以为建立更多方相互体谅讨价还价模型提供借鉴.     

Sequential share bargaining

This paper presents a new extension of the Rubinstein-St?hl bargaining model to the case with n players, called sequential share bargaining. The bargaining protocol is natural and has as its main feature that the players’ shares in the surplus are determined sequentially rather than simultaneously. The protocol also assumes orderly voting, a restriction on the order in which players respond to a proposal. The bargaining protocol requires unanimous agreement for proposals to be implemented. Unlike all existing bargaining protocols with unanimous agreement, the resulting game has unique subgame perfect equilibrium utilities for any value of the discount factor. The result builds on the analysis of so-called one-dimensional bargaining problems. We show that also one-dimensional bargaining problems have unique subgame perfect equilibrium utilities for any value of the discount factor.     

Flexibility of disagreement actions in negotiations

Negotiation games in which two players engage in disagreement games to determine their payoffs during the bargaining generally admit multiple perfect equilibrium outcomes, including inefficient ones. This paper shows that if the two players cannot change their disagreement actions as frequent as they bargain, then the set of perfect equilibrium payoffs shrinks. As disagreement actions become completely inflexible relative to the bargaining frequency, the set of equilibrium payoffs shrinks to Nash's (1953) bargaining solution with strategic threat. Received January 1999/Final version March 2001     

考虑损失厌恶行为与谈判破裂的Rubinstein谈判博弈研究

本文考虑具有损失厌恶行为与破裂风险的Rubinstein谈判博弈。首先构建子博弈完美均衡,并证明子博弈完美均衡的存在性及唯一性。然后分析子博弈完美均衡的性质,结果表明:参与人受益于对手的损失厌恶行为,而因自身具有损失厌恶行为遭受损失;谈判破裂概率对均衡结果的影响取决于贴现因子与参与人的损失厌恶系数;当谈判破裂的概率趋于零时,极限均衡结果收敛于经典的Rubinstein谈判博弈结果。最后建立了与非对称Nash谈判解的关系,其中参与人的议价能力与自身的损失厌恶水平呈负相关性,与对手的损失厌恶水平呈正相关性;参与人的议价能力依赖于谈判破裂概率与出价时间间隔的比值。     

Equilibrium computation of the Hart and Mas-Colell bargaining model

The 8-th problem raised by [Hart, S., Mas-Colell, A., 2010. Bargaining and cooperation in strategic form games. Journal of the European Economics Association 8 (1), 7–33], is solved. To be specific, I show that the set of SP equilibria can be determined by a finite number of systems of linear inequalities, which are efficiently solvable when there are two players. This is more or less surprising because the Hart and Mas-Colell bargaining model and the SP equilibrium both seem to be rather complicated, and it is well known that an arbitrary Nash equilibrium is hard to compute, even when there are only two players. Using this algorithm, it is shown that players of Prisoners’ Dilemma can cooperate to some extent in the Hart and Mas-Colell bargaining, and full cooperation is attainable as ρ$\rho$, a parameter of this model, approaches to 1. Quantitative efficiency, i.e. price of anarchy, is also analyzed.     

The Kalai-Smorodinsky bargaining solution with loss aversion

We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z.We show that n-player bargaining problems have a unique self-supporting outcome under the Kalai-Smorodinsky solution. For all possible loss aversion coefficients we determine the bargaining solutions that give exactly these outcomes, and characterize them by the standard axioms of Scale Invariance, Individual Monotonicity, and Strong Individual Rationality, and a new axiom called Proportional Concession Invariance (PCI). A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome does not change this outcome.     

Spanning subgraphs of graphs partitioned into two isomorphic pieces

A graph has the neighbor‐closed‐co‐neighbor, or ncc property, if for each of its vertices x, the subgraph induced by the neighbor set of x is isomorphic to the subgraph induced by the closed non‐neighbor set of x. As proved by Bonato and Nowakowski [ 5 ], graphs with the ncc property are characterized by the existence of perfect matchings satisfying certain local conditions. In the present article, we investigate the spanning subgraphs of ncc graphs, which we name sub‐ncc. Several equivalent characterizations of finite sub‐ncc graphs are given, along with a polynomial time algorithm for their recognition. The infinite sub‐ncc graphs are characterized, and we demonstrate the existence of a countable universal sub‐ncc graph satisfying a strong symmetry condition called pseudo‐homogeneity. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Quota games with a continuum of players

In this paper the notion ofm-quota game with a continuum of players is defined and the theory of bargaining sets is generalized to this new class of games. We discuss only the bargaining setM ₀ and our results are similar to those obtained in the finite case. Our main result is that for maximal coalition structures the stable payoff functions are exactly those in which almost every non-weak player receives no more than his quota and the weak players receive zero.     

Repeated bargaining with reference-dependent preferences

We consider a two-player bargaining model in which one or both players have reference-dependent preferences, but are otherwise perfectly rational. Our behavioural assumption is that players with reference-dependent preferences prefer impasse to consuming strictly less than their current reference points. Reference points adjust each period according to some exogenously specified law of motion. When reference points do not adjust following disagreement, we show that disagreement does not arise in equilibrium, but they do influence the division of the pie. In contrast, when reference points adjust downwards following disagreement, disagreements arise and players may try to manipulate the reference point of their opponent. When reference points adjust downwards following a rejection, for a particular Markov equilibrium, we show that the set of feasible allocations can be divided into agreement and disagreement regions. In particular, there are thresholds such that if one (or more) player’s reference point is above the threshold, disagreement necessarily arises.     

Commitments to threat strategies in Nash bargaining

Nash's two-person bargaining model consists of two stages: threat strategies and utility demands are chosen in the first and second stages respectively. Here players commit to play the threat strategies chosen in the first stage for the case where disagreement occurs in the second stage. Whether a player commits or not to play a threat strategy, however, is voluntary in principle. This leads to some possible extensions of Nash's model so that players' commitment choices are taken into account. In this paper, we consider three extensions. In the two person case, these three extensions give essentially the same result as that given by Nash. This is not the case for more than two players; the result depends upon an extension. In one extension, Nash's result always holds for more than two players. In the other two extensions, however, we give a three person example where not all players choose commitments in equilibrium.     

The price of fairness with the extended Perles–Maschler solution

In Nash bargaining problem, due to fairness concerns of players, instead of maximizing the sum of utilities of all players, an implementable solution should satisfy some axioms or characterizations. Such a solution can result in the so-called price of fairness, because of the reduction in the sum of utilities of all players. An important issue is to quantify the system efficiency loss under axiomatic solutions through the price of fairness. Based on Perles–Maschler solution of two-player Nash bargaining problem, this paper deals with the extended Perles–Maschler solution of multi-player Nash bargaining problem. We give lower bounds of three measures of the system efficiency for this solution, and show that the lower bounds are asymptotically tight.     

Efficiency by trust in fairness? multiperiod ultimatum bargaining experiments with an increasing cake

Previous ultimatum bargaining experiments have shown that bargainers face the conflict whether to exploit bargaining power or to comply with basic norms of distributive justice. In multiperiod ultimatum bargaining for an increasing cake, trust in fairness can enable cooperation and thus more efficient results but is also open to opportunistic exploitation. In such a game the two players take turns in being the one who suggests an agreement and decides whether this is the final proposal, whereas his partner can only accept this proposal or reject it. While the game theoretic solution implies an immediate agreement assigning nearly all the cake to the demanding player, efficiency requires to postpone the agreement to the last possible round. Our 2 × 2-factorial design varies the number of possible bargaining periods and the cake increase, allowing us to explore several hypotheses.     

三方相互威慑讨价还价模型

研究了三方相互威慑讨价还价问题,从合作博弈的角度建立了三方相互威慑讨价还价模型.并且通过对均衡存在性和冲突可能性的分析,给出了三方相互威慑问题冲突发生的条件以及存在均衡状态时的均衡解.最后通过实验验证了本文分析的正确性.     

Alliance or no alliance—Bargaining power in competing reverse supply chains

This work investigates how bargaining power affects negotiations between manufacturers and reverse logistics providers in reverse supply chains under government intervention using a novel three-stage reverse supply chain model for two scenarios, a reverse logistics provider alliance and no reverse logistics provider alliance. Utilizing the asymmetric Nash bargaining game, this work seeks equilibrium negotiation solutions. Analytical results indicate that the reverse logistics provider alliance increases the bargaining power of reverse logistics providers when negotiating with a manufacturer for a profitable recycled-component supply contract; however, manufacturer profits are often reduced. Particularly in the case of an recycled-component vender-dominated market, a reverse logistics alliance with extreme bargaining power may cause a counter-profit effect that results in the decreases of profits for all players involved, including buyers (i.e., manufacturers) and allied recycled-component venders (i.e., reverse logistics providers). Additional managerial insights are provided for discussion.     

Interactive multiobjective programming in airline network design for international airline code-share alliance

This study presents an interactive airline network design procedure to facilitate bargaining interactions necessitated by international code-share alliance agreements. Code sharing involves partner airlines individually maximizing their own profits, while mutually considering overall profitability, traffic gains, and quality benefits for the markets in which they cooperate with their partners. This study uses a reference point method to solve the interactive multiobjective programming model, to support the bargaining interactions between two partner-airlines in an alliance negotiation. The impact of the code-share alliance network on market demand, alliance partners’ costs and profits, and levels of service are also discussed. A case study demonstrates the feasibility of applying the proposed models and elucidates how interactive multiobjective programming models may be applied to determine flight frequencies for airline code-share alliance networks. The results of this study provide ways by which alliance airlines can evaluate iteratively the output and profits of the alliance members under code-share alliance agreements.     

Bargaining in endogenous trading networks

We study a model in which heterogeneous agents first form a trading network where linking costs are positive but infinitesimally small. Then, a seller and a buyer are randomly selected among the agents to bargain through a chain of intermediaries. We determine both the trading path and the allocation of the surplus among the seller, the buyer and the intermediaries at equilibrium. We show that, under the initiator bargaining protocol, a trading network is pairwise stable if it is a core–periphery network where the core consists of all impatient agents who are linked to each other and the periphery consists of all patient agents who have a single link towards an impatient agent. Once agents do not know the impatience of other agents, each bilateral bargaining session may involve delay. Then, core–periphery networks may not be pairwise stable because agents may prefer to add links for reducing the length of trading paths and so avoiding costly delays in reaching a global agreement.