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.     

Dynamic approach to solving bargaining games

The article examines the solution of bargaining games on the basis of the previously proposed paradigm of stationary strategies in supergames. The value of axiomatic bargaining theory for choosing the unique solution in agreement games is considered. The “splitting the dollar” game for two and three players is discussed as an application. __________ Translated from Prikladnaya Matematika i Informatika, No. 22, pp. 91–104, 2005.     

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     

An equitable solution for multicriteria bargaining games

In this paper we study bargaining models where the agents consider several criteria to evaluate the results of the negotiation process. We propose a new solution concept for multicriteria bargaining games based on the distance to a utopian minimum level vector. This solution is a particular case of the class of the generalized leximin solutions and can be characterized as the solution of a finite sequence of minimax programming problems.     

A note on nonzero-sum differential games with bargaining solution

The extension of Nash's bargaining solution to differential games is discussed. It is shown that a closed-loop solution verifies very stringent necessary conditions and that an open-loop solution can present serious weakness from a normative point of view.This research has been supported by the Canada Council (S73-0935) and the Ministère de l'Education du Québec (DGES).     

On the non-emptiness of the one-core and the bargaining set of committee games

We study the committee decision making process using game theory. Shenoy  [15] introduced two solution concepts: the one-core and the bargaining set, and showed that the one-core of a simple committee game is nonempty if there are at most four players. We extend this result by proving that whether the committee is simple or not, as far as there are less than five players, the one-core is nonempty. This result also holds for the bargaining set.     

The bargaining set of four-person balanced games

It is well known that in three-person transferable-utility cooperative games the bargaining set ℳⁱ ₁ and the core coincide for any coalition structure, provided the latter solution is not empty. In contrast, five-person totally-balanced games are discussed in the literature in which the bargaining set ℳⁱ ₁ (for the grand coalition) is larger then the core. This paper answers the equivalence question in the remaining four-person case. We prove that in any four-person game and for arbitrary coalition structure, whenever the core is not empty, it coincides with the bargaining set ℳⁱ ₁. Our discussion employs a generalization of balancedness to games with coalition structures. Received: August 2001/Revised version: April 2002     

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.     

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.     

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.     

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 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].     

The reactive bargaining set: Structure, dynamics and extension to NTU games

The reactive bargaining set (Granot [1994]) is the set of outcomes for which no justified objection exists. Here, in a justified objection the objector first watches how the target tries to act (if he has such an option), and then reacts by making a profit and ruining the target's attempt to maintain his share. In this paper we explore properties of the reactive bargaining set, set up the system of inequalities that defines it, and construct a dynamic system in the sense of Stearns' transfer scheme that leads the players to this set. We also extend the definition of the reactive bargaining set toNTU games in a way that keeps it nonempty. To shed light on its nature and its relative ease of computation, we compute the reactive bargaining set for games that played important role in the game theory literature.     

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.     

On bargaining position descriptions in non-transferable utility games

We present a generalization to the Harsanyi solution for non-transferable utility (NTU) games based on non-symmetry among the players. Our notion of non-symmetry is presented by a configuration of weights which correspond to players' relative bargaining power in various coalitions. We show not only that our solution (i.e., the bargaining position solution) generalizes the Harsanyi solution, (and thus also the Shapley value), but also that almost all the non-symmetric generalizations of the Shapley value for transferable utility games known in the literature are in fact bargaining position solutions. We also show that the non-symmetric Nash solution for the bargaining problem is also a special case of our general solution. We use our general representation of non-symmetry to make a detailed comparison of all the recent extensions of the Shapley value using both a direct and an axiomatic approach.     

The core and the bargaining set in glove-market games

In a glove-market game, the worth of a coalition is defined as the minimum, over all commodities in the market, of the total quantity that the coalition owns of each commodity. We identify a subclass of these games for which the core and the bargaining set coincide with the set of competitive equilibrium outcomes. We present examples showing that these solution concepts differ outside that subclass. We also illustrate a peculiar behavior of the bargaining set with respect to replication of a glove-market. These examples provide a simple economic setting in which the merits of the various solution concepts may be discussed and compared.Correspondence with Michael Maschler is gratefully acknowledged.This authors work was partly done while visiting the Department of Mathematics, Massachusetts Institute of Technology. Research supported by the Fund for the Promotion of Research at the Technion.     

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.     

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.     

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.