Stationary perfect equilibria of an n-person noncooperative bargaining game and cooperative solution concepts

This paper characterizes the stationary (subgame) perfect equilibria of an n-person noncooperative bargaining model with characteristic functions, and provides strategic foundations of some cooperative solution concepts such as the core, the bargaining set and the kernel. The contribution of this paper is twofold. First, we show that a linear programming formulation successfully characterizes the stationary (subgame) perfect equilibria of our bargaining game. We suggest a linear programming formulation as an algorithm for the stationary (subgame) perfect equilibria of a class of n-person noncooperative games. Second, utilizing the linear programming formulation, we show that stationary (subgame) perfect equilibria of n-person noncooperative games provide strategic foundations for the bargaining set and the kernel.     

拟凸对策的解

本文先引入拟凸对策的概念作为凸对策的推广,然后研究这种对策的各种解的性质。我们主要证得,当局中人数小于6或者对策的复盖严格凸时,谈判集与核心重合,核是单点集。另外,存在一个6人拟凸对策,其谈判集与核心不同。     

On the bargaining set, kernel and core of superadditive games

We prove that for superadditive games a necessary and sufficient condition for the bargaining set to coincide with the core is that the monotonic cover of the excess game induced by a payoff be balanced for each imputation in the bargaining set. We present some new results obtained by verifying this condition for specific classes of games. For N-zero-monotonic games we show that the same condition required at each kernel element is also necessary and sufficient for the kernel to be contained in the core. We also give examples showing that to maintain these characterizations, the respective assumptions on the games cannot be lifted. Received: March 1998/Revised version: December 1998     

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.     

NEGOTIATING OVER EXTERNALITIES

We present a simple model in which two perfectly informed, risk neutral agents will not negotiate an efficient agreement to lessen the effects of an externality and for which the outcome of negotiation depends on the legal assignment of property rights. The model permits agents to pre-commit themselves to refuse to negotiate particular agenda issues. The result is obtained because we prove that one player is always made strictly worse off from the addition of side-payments to a bargaining game. Along the way, we devise a supporting hyperplane for the n-person Nash bargaining game solution. We also display a simple game which establishes that our main result holds true—for at least some games—for an array of alternative bargaining game solutions such as that of Raiffa, Kalai, and Smorodinski.     

Bargaining and negative externalities

Two important issues in distributive bargaining theory are, first, the conditions under which a negotiation breakdown occurs, and second, what and how source of parties’ bargaining powers influences the properties of a possible agreement. Research based on classic Nash’s demand game has explored both questions by sophisticating the original game a lot. As an attempt to deal with both issues under a simpler framework, we propose a modification of the Nash demand game in which bargainers suffer negative externalities proportional to the share of the surplus captured by their rival. It is shown that the negotiator experiencing a relatively high externality level has greater bargaining power and thus, appropriates a larger proportion of the surplus at stake. However, if externality levels are sufficiently high, bargaining powers become incompatible and a negotiation breakdown emerges from the bargaining process. We compare our results with the previous literature, and argue that they can be especially relevant in negotiations held under highly polarized environments.     

多人多目标资源分配协商对策

资源分配问题,一直是决策分析理论研究的核心问题之一.从对各种经济系统进行决策分析中,可得出一类多人多目标资源分配问题的数学描述如下:     

The semireactive bargaining set of a cooperative game

The semireactive bargaining set, a solution for cooperative games, is introduced. This solution is in general a subsolution of the bargaining set and a supersolution of the reactive bargaining set. However, on various classes of transferable utility games the semireactive and the reactive bargaining set coincide. The semireactive prebargaining set on TU games can be axiomatized by one-person rationality, the reduced game property, a weak version of the converse reduced game property with respect to subgrand coalitions, and subgrand stability. Furthermore, it is shown that there is a suitable weakening of subgrand stability, which allows to characterize the prebargaining set. Replacing the reduced game by the imputation saving reduced game and employing individual rationality as an additional axiom yields characterizations of both, the bargaining set and the semireactive bargaining set. Received September 2000/Revised version June 2001     

The kernel and bargaining set for convex games

It is shown that for convex games the bargaining set? ₁ ⁽ⁱ⁾ (for the grand coalition) coincides with the core. Moreover, it is proved that the kernel (for the grand coalition) of convex games consists of a unique point which coincides with the nucleolus of the game.     

On a dynamic theory for the kernel of ann-person game

In this paper, a dynamic theory for the kernel ofn-person games given byBillera is studied. In terms of the (bargaining) trajectories associated with a game (i.e. solutions to the differential equations defined by the theory), an equivalence relation is defined. The “consistency” of these equivalence classes is examined. Then, viewing the pre-kernel as the set of equilibrium points of this system of differential equations, some topological, geometric, symmetry and stability properties of the pre-kernel are given.     

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.     

Fuzzy extensions of bargaining sets and their existence in cooperative fuzzy games

Recently, the concept of classical bargaining set given by Aumann and Maschler in 1964 has been extended to fuzzy bargaining set. In this paper, we give a modification to correct some weakness of this extension. We also extend the concept of the Mas-Colell's bargaining set (the other major type of bargaining sets) to its corresponding fuzzy bargaining set. Our main effort is to prove existence theorems for these two types of fuzzy bargaining sets. We will also give necessary and sufficient conditions for these bargaining sets to coincide with the Aubin Core in a continuous superadditive cooperative fuzzy game which has a crisp maximal coalition of maximum excess at each payoff vector. We show that both Aumann-Maschler and Mas-Colell fuzzy bargaining sets of a continuous convex cooperative fuzzy game coincide with its Aubin core.     

A simple “market value” bargaining model for weighted voting games: characterization and limit theorems

We offer a bargaining model for weighted voting games that is a close relative of the nucleolus and the kernel. We look for a set of weights that preserves winning coalitions that has the property of minimizing the difference between the weight of the smallest and the weight of the largest Minimum Winning Coalition. We claim that such a set of weights provides an a priori measure of a weighted voter’s bribeworthiness or market value. After introducing our model, we provide a characterization result for this model and show its links to other bargaining model approaches in the literature. Then we offer some limit results showing that, with certain reasonable conditions on the distributions of weights, as the size of the voting body increases, the values of bribeworthiness we calculate will approach both the weights themselves and the Banzhaf scores for the weighted voting game. We also show that, even for relatively small groups using weighted voting, such as the membership of the European Council of Ministers (and its predecessors) 1958–2003, similarities among the usual a priori power scores, bribeworthiness/market value, and the weights themselves, will be quite strong.     

Solutions for some bargaining games under the Harsanyi-Selten solution theory,part II: Analysis of specific bargaining games

Part II of the paper (for Part I see Harsanyi (1982)) describes the actual solutions the Harsanyi-Selten solution theory provides for some important classes of bargaining games, such as unanimity games; trade between one seller and several potential buyers; and two-person bargaining games with incomplete information on one side or on both sides. It also discusses some concepts and theorems useful in computing the solution; and explains how our concept of risk dominance enables us to analyze game situations in terms of some intuitively very compelling probabilistic (subjective-probability) considerations disallowed by classical game theory.     

Nash bargaining solution under externalities

We define a Nash bargaining solution (NBS) of partition function games. Based on a partition function game, we define an extensive game, which is a propose–respond sequential bargaining game where the rejecter of a proposal exits from the game with some positive probability. We show that the NBS is supported as the expected payoff profile of any stationary subgame perfect equilibrium (SSPE) of the extensive game such that in any subgame, a coalition of all active players forms immediately. We provide a necessary and sufficient condition for such an SSPE to exist. Moreover, we consider extensions to the cases of nontransferable utilities, time discounting and multiple-coalition formation.     

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.     

Solution concepts ofk-convexn-person games

For any positive integersk andn, the subclass ofk-convexn-person games is considered. In casek=n, we are dealing with convexn-person games. Three characterizations ofk-convexn-person games, formulated in terms of the core and certain adapted marginal worth vectors, are given. Further it is shown that fork-convexn-person games the intersection of the (pre)kernel with the core consists of a unique point (namely the nucleolus), but that the (pre)kernel may contain points outside the core. For certain 1-convex and 2-convexn-person games the part of the bargaining set outside the core is even disconnected with the core. The Shapley value of ank-convexn-person game can be expressed in terms of the extreme points of the core and a correction-vector whenever the game satisfies a certain symmetric condition. Finally, theτ-value of ank-convexn-person game is given.     

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.     

Noncooperative formation of coalitions in hedonic games

We study a bargaining procedure of coalition formation in the class of hedonic games, where players’ preferences depend solely on the coalition they belong to. We provide an example of nonexistence of a pure strategy stationary perfect equilibrium, and a necessary and sufficient condition for existence. We show that when the game is totally stable (the game and all its restrictions have a nonempty core), there always exists a no-delay equilibrium generating core outcomes. Other equilibria exhibiting delay or resulting in unstable outcomes can also exist. If the core of the hedonic game and its restrictions always consist of a single point, we show that the bargaining game admits a unique stationary perfect equilibrium, resulting in the immediate formation of the core coalition structure.     

A characterization of a limit solution for finite horizon bargaining problems

We investigate a two-person random proposer bargaining game with a deadline. A bounded time interval is divided into bargaining periods of equal length and we study the limit of the subgame perfect equilibrium outcomes as the number of bargaining periods goes to infinity while the deadline is kept fixed. This limit is close to the discrete Raiffa solution when the time horizon is very short. If the deadline goes to infinity the limit outcome converges to the time preference Nash solution. Regarding this limit as a bargaining solution under deadline, we provide an axiomatic characterization.