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     

A cardinal bargaining set for games without side payments

A bargaining set for games without side payments, based on cardinal considerations is introduced. It is proved that it is never empty for a fairly general class of games without side payments, that it coincides with the classical bargaining set if the game has side payments and that it is a subset of an ordinal bargaining set described inAsscher [1975a].     

An impossibility result concerningn-person bargaining games

In this note we show that a solution proposed byRaiffa for two-person bargaining games, which has recently been axiomatized byKalai/Smorodinsky, does not generalize in a straightforward manner to generaln-person bargaining games. Specifically, the solution is not Pareto optimal on the class of alln-person bargaining games, and no solution which is can possess the other properties which characterizeRaiffa's solution in the two-person case.     

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.     

The reactive bargaining set for cooperative games

We introduce a new bargaining set for cooperative games in characteristic function form, and investigate its structure and properties. We prove that the new bargaining set is not empty. In fact, we show that it contains the kernel and is contained in the classical bargaining set ${\mathcal{M}^i_1}$ , and we further prove that it consists of the unique symmetric vector for the class of simple majority games.     

The reactive bargaining set of some flow games and of superadditive simple games

We prove that the reactive bargaining set coincides with the core of simple flow games, and it essentially coincides with the kernel of simple superadditive games.     

Stabilizing network bargaining games by blocking players

Cooperative matching games (Shapley and Shubik) and Network bargaining games (Kleinberg and Tardos) are games described by an undirected graph, where the vertices represent players. An important role in such games is played by stable graphs, that are graphs whose set of inessential vertices (those that are exposed by at least one maximum matching) are pairwise non adjacent. In fact, stable graphs characterize instances of such games that admit the existence of stable outcomes. In this paper, we focus on stabilizing instances of the above games by blocking as few players as possible. Formally, given a graph G we want to find a minimum cardinality set of vertices such that its removal from G yields a stable graph. We give a combinatorial polynomial-time algorithm for this problem, and develop approximation algorithms for some NP-hard weighted variants, where each vertex has an associated non-negative weight. Our approximation algorithms are LP-based, and we show that our analysis are almost tight by giving suitable lower bounds on the integrality gap of the used LP relaxations.     

Aubin cores and bargaining sets for convex cooperative fuzzy games

In this paper, we deal with Aubin cores and bargaining sets in convex cooperative fuzzy games. We first give a simple and direct proof to the well-known result (proved by Branzei et al. (Fuzzy Sets Syst 139:267–281, 2003)) that for a convex cooperative fuzzy game v, its Aubin core C(v) coincides with its crisp core C ^cr (v). We then introduce the concept of bargaining sets for cooperative fuzzy games and prove that for a continuous convex cooperative fuzzy game v, its bargaining set coincides with its Aubin core, which extends a well-known result by Maschler et al. for classical cooperative games to cooperative fuzzy games. We also show that some results proved by Shapley (Int J Game Theory 1:11–26, 1971) for classical decomposable convex cooperative games can be extended to convex cooperative fuzzy games.     

Non-existence of super-additive solutions for 3-person games

The super-additive solution for 2-person Nash bargaining games (with constant threat) was defined axiomatically inPerles/Maschler [1981]. That paper contains also a study of its basic properties. In this paper we show that the axioms are incompatible even for 3-person unanimity games. This raises the problem of finding a satisfactory generalization of this solution concept to multi-person games.     

Asymptotic stability and other properties of trajectories and transfer sequences leading to the bargaining sets

The foundation of a dynamic theory for the bargaining sets started withStearns, when he constructed transfer sequences which always converge to appropriate bargaining sets. A continuous analogue was developed byBillera, where sequences where replaced by solutions of systems of differential equations. In this paper we show that the nucleolus is locally asymptotically stable both with respect toStearns' sequences andBillera's solutions if and only if it is an isolated point of the appropriate bargaining set. No other point of the bargaining set can be locally asymptotically stable. Furthermore, it is always stable in these processes. As by-products of the study we derive the results ofBillera andStearns in a different fashion. We also show that along the non-trivial trajectories and sequences, the vector of the excesses of the payoffs, arranged in a non-increasing order, always decreases lexicographically, thus each bargaining set can be viewed as resulting from a certain monotone process operating on the payoff vectors.     

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     

Bargaining sets in exchange economies with indivisibilities and money

This paper introduces and investigates the semireactive bargaining set (Sudhölter and Potters (2001)) and the reactive bargaining set (Granot (1994)), which are originally solution concepts for TU-games, in economies in which agents exchange indivisible goods and one perfectly divisible good (money). Under the assumptions that the preferences of the agents are quasi-linear and the endowments satisfy the Total Abundance condition, a condition on the amounts of money agents initially have, it is shown that the (semi)reactive bargaining set is nonempty. In addition, we prove that in such an economy the (semi)reactive bargaining set and the (strong) core coincide if and only if the (semi)reactive bargaining set and the core of the underlying TU-game coincide.The authors are grateful to Peter Sudhölter for his helpful suggestions that greatly improved the paper     

Paths leading to the Nash set for nonsmooth games

Maschler, Owen and Peleg (1988) constructed a dynamic system for modelling a possible negotiation process for players facing a smooth n-person pure bargaining game, and showed that all paths of this system lead to the Nash point. They also considered the non-convex case, and found in this case that the limiting points of solutions of the dynamic system belong to the Nash set. Here we extend the model to i) general convex pure bargaining games, and to ii) games generated by “divide the cake” problems. In each of these cases we construct a dynamic system consisting of a differential inclusion (generalizing the Maschler-Owen-Peleg system of differential equations), prove existence of solutions, and show that the solutions converge to the Nash point (or Nash set). The main technical point is proving existence, as the system is neither convex valued nor continuous. The intuition underlying the dynamics is the same as (in the convex case) or analogous to (in the division game) that of Maschler, Owen, and Peleg. Received August 1997/Final version May 1998     

Cooperative games with imcomplete information

A bargaining solution concept which generalizes the Nash bargaining solution and the Shapley NTU value is defined for cooperative games with incomplete information. These bargaining solutions are efficient and equitable when interpersonal comparisons are made in terms of certainvirtual utility scales. A player's virtual utility differs from his real utility by exaggerating the difference from the preferences of false types that jeopardize his true type. In any incentive-efficient mechanism, the players always maximize their total virtual utility ex post. Conditionally-transferable virtual utility is the strongest possible transferability assumption for games with incomplete information.     

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。     

A bargaining set for monotonic simple games based on external and internal stability

A new bargaining set based on notions of both internal and external stability is developed in the context of endogenous coalition formation. It allows one to make an explicit distinction between within-group and outside-group deviation options. This type of distinction is not present in current bargaining sets. For the class of monotonic proper simple games, the outcomes in the bargaining set are characterized. Furthermore, it is shown that the bargaining set of any homogeneous weighted majority game contains an outcome for which the underlying coalition structure consists of a minimal winning coalition and its complement.     

Bargaining set theory and stability in coalition governments

Various bargaining set theories are compared as predictors of coalition government portfolio distribution. While the kernel and B₁-bargaining set are known to exist in voting games with side payments, it is argued that the kernel, and thus B₁, are poor predictors. The B₂-bargaining set, a subset of B₁, when it exists is shown to be a good payoff predictor in a fractionalized and depolarized parliamentary situation (Finland: 1945ndash;1971). Moreover this predictor provides some explanation for the formation of surplus (winning but not minimal) coalitions.     

An acyclic relation for comparison of bargaining powers of coalitions and its interrelationship with bargaining set

This paper proposes a method to compare bargaining power of coalitions within the framework of games in coalition form with transferable utility. The method is expressed by a relation on the set of all coalitions in a game, the relation which is defined based on the players’ bargaining power. It is shown in this paper that the newly defined relation satisfies acyclicity. Also, it is verified in this paper that the set of all individually rational payoff configurations under which all coalitions have the equal bargaining power coincides with the bargaining set. Some examples demonstrate how the newly proposed method works.     

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.