On the implementation of the L-Nash bargaining solution in two-person bargaining games

The “Nash program” initiated by Nash (Econometrica 21:128–140, 1953) is a research agenda aiming at representing every axiomatically determined cooperative solution to a game as a Nash outcome of a reasonable noncooperative bargaining game. The L-Nash solution first defined by Forgó (Interactive Decisions. Lecture Notes in Economics and Mathematical Systems, vol 229. Springer, Berlin, pp 1–15, 1983) is obtained as the limiting point of the Nash bargaining solution when the disagreement point goes to negative infinity in a fixed direction. In Forgó and Szidarovszky (Eur J Oper Res 147:108–116, 2003), the L-Nash solution was related to the solution of multiciteria decision making and two different axiomatizations of the L-Nash solution were also given in this context. In this paper, finite bounds are established for the penalty of disagreement in certain special two-person bargaining problems, making it possible to apply all the implementation models designed for Nash bargaining problems with a finite disagreement point to obtain the L-Nash solution as well. For another set of problems where this method does not work, a version of Rubinstein’s alternative offer game (Econometrica 50:97–109, 1982) is shown to asymptotically implement the L-Nash solution. If penalty is internalized as a decision variable of one of the players, then a modification of Howard’s game (J Econ Theory 56:142–159, 1992) also implements the L-Nash solution.     

The partition bargaining problem

Consider the problem of partitioning n items among d players where the utility of each player for bundles of items is additive; so, player r has utility for item i and the utility of that player for a bundle of items is the sum of the 's over the items i in his/her bundle. Each partition S of the items is then associated with a d-dimensional utility vector V^S whose coordinates are the utilities that the players assign to the bundles they get under S. Also, lotteries over partitions are associated with the corresponding expected utility vectors. We model the problem as a Nash bargaining game over the set of lotteries over partitions and provide methods for computing the corresponding Nash solution, to prescribed accuracy, with effort that is polynomial in n. In particular, we show that points in the pareto-optimal set of the corresponding bargaining set correspond to lotteries over partitions under which each item, with the possible exception of at most d(d-1)/2 items, is assigned in the same way.     

Optimal exercise of jointly held real options: A Nash bargaining approach with value diversion

This paper provides a two-stage decision framework in which two or more parties exercise a jointly held real option. We show that a single party’s timing decision is always socially efficient if it precedes bargaining on the terms of sharing. However, if the sharing rule is agreed before the exercise timing decision is made, then socially optimal timing is attained only if there is a cash payment element in the division of surplus. If the party that chooses the exercise timing can divert value from the project, then the first-best outcome may not be possible at all and the second-best outcome may be implemented using a contract that is generally not optimal in the former cases. Our framework contributes to the understanding of a range of empirical regularities in corporate and entrepreneurial finance.     

Sufficient conditions for Nash bargaining in differential games

Sufficient conditions for Nash bargaining in differential games are given. These conditions are compared with the sufficient conditions given by Liu (Ref. 1).     

A deductive approach to friendship networks

Based on a simple actor model, I derive theory concerning satisfied networks of close, reciprocal friendships. By satisfied networks I mean networks of actors who would be unhappier with more or fewer close friends. Deductions concern characteristics of such networks, such as that a satisfied network component (connected subnetwork) containing an actor with only one friend must be a dyad. I show that the model of satisfied friendship networks fits eight empirical friendship networks significantly better than a random model. Finally, I show the theory's usefulness by making five additional predictions, including two involving intervention.     

Solutions for some bargaining games under the Harsanyi-Selten solution theory,part I: Theoretical preliminaries

Part I of this paper discusses the problem of how to model bargaining behavior, and outlines a few basic ideas of the Harsanyi-Selten solution theory. In particular, we discuss removal of imperfect equilibrium points from the game by using the uniformly perturbed game form. We also describe definition of the solution in terms of payoff-dominance and risk-dominance relations, and in terms of the net strategic distances, between the primitive equilibrium points. Part II of the paper will discuss the actual solutions our theory provides for some important classes of bargaining games.     

Nash equilibria in optimal reinsurance bargaining

We introduce a strategic behavior in reinsurance bilateral transactions, where agents choose the risk preferences they will appear to have in the transaction. Within a wide class of risk measures, we identify agents’ strategic choices to a range of risk aversion coefficients. It is shown that at the strictly beneficial Nash equilibria, agents appear homogeneous with respect to their risk preferences. While the game does not cause any loss of total welfare gain, its allocation between agents is heavily affected by the agents’ strategic behavior. This allocation is reflected in the reinsurance premium, while the insurance indemnity remains the same in all strictly beneficial Nash equilibria. Furthermore, the effect of agents’ bargaining power vanishes through the game procedure and the agent who gets more welfare gain is the one who has an advantage in choosing the common risk aversion at the equilibrium.     

On the sensitivity matrix of the Nash bargaining solution

In this note we provide a characterization of a subclass of bargaining problems for which the Nash solution has the property of disagreement point monotonicity. While the original d-monotonicity axiom and its stronger notion, strong d-monotonicity, were introduced and discussed by Thomson (J Econ Theory, 42: 50–58, 1987), this paper introduces local strong d-monotonicity and derives a necessary and sufficient condition for the Nash solution to be locally strongly d-monotonic. This characterization is given by using the sensitivity matrix of the Nash bargaining solution w.r.t. the disagreement point d. Moverover, we present a sufficient condition for the Nash solution to be strong d-monotonic.     

Inventory,channel coordination and bargaining in a manufacturer-retailer system

Substantial research literature has been developed over the years on the subject of inventory. The more recent literature has examined the fundamental relationships between inventory control and price theory. A significant portion of this literature assumes the ultimate consumer demand as a constant and characterizes the relationship between a manufacturer and a retailer as a leader-follower problem. A primary assumption in these studies is that the manufacturer, as the leader, exerts almost complete control over the behavior of the retailer. However, in practice, the retailer does exert some control over the manufacturer. This paper develops a framework that integrates inventory control with constant demand and the economic relationship between consumer demand and retail price. Within this framework, the impact of order quantity, wholesale price and retail price on the behavior of both the manufacturer and the retailer is investigated. Furthermore, this paper explores the issues and conclusions that results from coordinating the relationship between the manufacturer and the retailer. Our analyses demonstrate that channel coordination can be achieved by utilizing well-known bargaining models. A numerical example is provided to illustrate our theoretical findings.     

A dynamic solution in N-person cooperative game theory

We formulate a cooperative game as an extended form game in which each player in turn proposes payoffs to a coalition over M steps. Payoffs at time t are discounted by a penalty function f(t). If all players in a coalition agree to their payoffs, they receive them. Under a convergence hypothesis verified by computer for three players in many cases, we compute the payoffs resulting from a coalition pattern and give necessary conditions for particular patterns. The resulting solution is related to the Nash bargaining solution and the competitive solution.     

A bargaining game model for measuring performance of two-stage network structures

Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs), where the internal structures of DMUs are treated as a black-box. Recently DEA has been extended to examine the efficiency of DMUs that have two-stage network structures or processes, where all the outputs from the first stage are intermediate measures that make up the inputs to the second stage. The resulting two-stage DEA model not only provides an overall efficiency score for the entire process, but also yields an efficiency score for each of the individual stages. The current paper develops a Nash bargaining game model to measure the performance of DMUs that have a two-stage structure. Under Nash bargaining theory, the two stages are viewed as players and the DEA efficiency model is a cooperative game model. It is shown that when only one intermediate measure exists between the two stages, our newly developed Nash bargaining game approach yields the same results as applying the standard DEA approach to each stage separately. Two real world data sets are used to demonstrate our bargaining game model.     

Nash refinement of equilibria

A method for choosing equilibria in strategic form games is proposed and axiomatically characterized. The method as well as the axioms are inspired by the Nash bargaining theory. The method can be applied to existing refinements of Nash equilibrium (e.g., perfect equilibrium) and also to other equilibrium concepts, like correlated equilibrium.The authors thank the reviewers for their comments, which led to an improvement of the paper.     

A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals

In this paper, we study Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of doubly controlled backward stochastic differential equations. Our results extend former ones by Buckdahn et al. (2004) [3] and are based on a backward stochastic differential equation approach.     

A characterization of stochastically stable networks

Jackson and Watts (J Econ Theory 71: 44–74, 2002) have examined the dynamic formation and stochastic evolution of networks. We provide a refinement of pairwise stability, p-pairwise stability, which allows us to characterize the stochastically stable networks without requiring the “tree construction” and the computation of resistance that may be quite complex. When a -pairwise stable network exists, it is unique and it coincides with the unique stochastically stable network. To solve the inexistence problem of p-pairwise stable networks, we define its set-valued extension with the notion of p-pairwise stable set. The -pairwise stable set exists and is unique. Any stochastically stable networks is included in the -pairwise stable set. Thus, any network outside the -pairwise stable set must be considered as a non-robust network. We also show that the -pairwise stable set can contain no pairwise stable network and we provide examples where a set of networks is more “stable” than a pairwise stable network.     

Asymptotic inequality of the market system

Do the rich get richer under the market system? We give models supporting such an idea on the basis of: (1) the utility function of the rich person increases at a slower rate and so he bargains for more, and (2) the poor person discounts the future at a higher rate than the rich person.     

Scale changes and shared information in bargaining: An experimental study

This paper reports one of a series of experiments designed to test aspects of various game-theoretic models of bargaining. The results of this experiment consolidate those of previous experiments, which are first reviewed. The principal new result of this experiment is the observation, under conditions of partial information, of systematic violations of the axiom of independence of equivalent utility representations. The adequacy of various theories of bargaining to describe the observed data is also tested and discussed.     

Imitation,coordination and the emergence of Nash equilibrium

We consider a learning dynamic in which players imitate and better reply. Sufficient conditions are provided for Nash equilibrium play to emerge over time. The role of imitation in the learning dynamic is discussed through a series of examples. Most interestingly we demonstrate how imitation can ‘help’ the emergence of Nash equilibrium where ‘more rational’ methods do not.     

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。     

Intergenerational risk sharing in closing pension funds

We model intergenerational risk sharing in closing funded pension plans. Specifically, we consider a setting in which in each period, the pension fund’s investment and indexation policy is the outcome of a bargaining process between representatives of the then living generations. Because some generations might be under- or overrepresented in the board, we use the asymmetric Nash bargaining solution to allow for differences in bargaining powers. In a numerical study, we compare the welfare that the generations derive from the outcome of this repeated bargaining to the welfare that they would derive if a social planner’s optimal policy would instead be implemented. We find that as compared to the social optimum, older generations benefit substantially from the repeated bargaining, even if all generations are equally well-represented in the board. If older generations are relatively over-represented, as is sometimes argued, these effects are attenuated.