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.     

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     

Contract law and the value of a game

For the special case of games with linearly transferable utility, a treatment preserving the main features of the controversial treatment in the author’s doctoral dissertation is newly derived from a model for negotiation and play that is more elaborate than most such models. The crucial point of the derivation is that the author’s special bargaining theory is not needed; the usual Zeuthen-Nash-Harsanyi bargaining theory gives the same result. The main novelty in the model that makes this possible is the replacement of customary informal uses of ‘enforceable agreements’ by explicit contract law. The problems of contract law for cooperative games seem to be very complex, and the present work makes only a bare beginning on them. A characteristic function and value are derived.     

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.     

A note on efficient signaling of bargaining power

Strategic delay and restricted offers are two modes of signaling bargaining power in alternating offers bargaining games. This paper shows that when both modes are available, the best signaling strategy of the “strong” type of the informed player consists of a pure strategic delay followed by an offer on the whole pie. There is no signaling motivation for issue-by-issue bargaining when the issues are perfectly substitutable. Received: July 1996/Final version: August 1999     

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.     

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.     

Random marginal and random removal values

We propose two variations of the non-cooperative bargaining model for games in coalitional form, introduced by Hart and Mas-Colell (Econometrica 64:357–380, 1996a). These strategic games implement, in the limit, two new NTU-values: the random marginal and the random removal values. Their main characteristic is that they always select a unique payoff allocation in NTU-games. The random marginal value coincides with the Consistent NTU-value (Maschler and Owen in Int J Game Theory 18:389–407, 1989) for hyperplane games, and with the Shapley value for TU games (Shapley in In: Contributions to the theory of Games II. Princeton University Press, Princeton, pp 307–317, 1953). The random removal value coincides with the solidarity value (Nowak and Radzik in Int J Game Theory 23:43–48, 1994) in TU-games. In large games we show that, in the special class of market games, the random marginal value coincides with the Shapley NTU-value (Shapley in In: La Décision. Editions du CNRS, Paris, 1969), and that the random removal value coincides with the equal split value.      

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     

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.     

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.     

Bargaining sets in cooperative games with limited communication

Given a connected undirected graph ϕ with vertex set N, cooperative games (N, v) are considered in which players can cooperate only when the corresponding vertices form a connected subgraph in the graph ϕ. For such games, two generalizations of the bargaining set M ₁ ⁱ , which was introduced by Aumann and Maschler, are investigated.     

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.     

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 consistency principle and an axiomatization of the α-core

This paper examines the α-core of strategic games by means of the consistency principle. I provide a new definition of a reduced game for strategic games. And I define consistency (CONS) and two forms of converse consistency (COCONS and COCONS^*) under this definition of reduced games. Then I axiomatize the α-core for families of strategic games with bounded payoff functions by the axioms CONS, COCONS^*, weak Pareto optimality (WPO) and one person rationality (OPR). Furthermore, I show that these four axioms are logically independent. In proving this, I also axiomatize the α-individually rational solution by CONS, COCONS and OPR for the same families of games. Here the α-individually rational solution is a natural extension of the classical `maximin' solution. Received: June 1998/Final version: 6 July 2001     

The core and related solution concepts for infinite assignment games

Assignment problems where both sets of agents that have to be matched are countably infinite, the so-called infinite assignment problems, are studied as well as the related cooperative assignment games. Further, several solution concepts for these assignment games are studied. The first one is the utopia payoff for games with an infinite value. In this solution each player receives the maximal amount he can think of with respect to the underlying assignment problem. This solution is contained in the core of the game. Second, we study two solutions for assignment games with a finite value. Our main result is the existence of core-elements of these games, although they are hard to calculate. Therefore another solution, the f-strong ε-core is studied. This particular solution takes into account that due to organisational limitations it seems reasonable that only finite groups of agents will eventually protest against unfair proposals of profit distributions. The f-strong ε-core is shown to be nonempty. These authors’ research is partially supported by the Generalitat Valenciana (Grant number GV-CTIDIA-2002-32) and by the Government of Spain (through a joint research grant Universidad Miguel Hernández — Università degli Studi di Genova HI2002-0032).     

The multichoice coalition value

In this paper we define a solution for multichoice games which is a generalization of the Owen coalition value (Lecture Notes in Economics and Mathematical Systems: Essays in Honor of Oskar Morgenstern, Springer, New York, pp. 76–88, 1977) for transferable utility cooperative games and the Egalitarian solution (Peters and Zanks, Ann. Oper. Res. 137, 399–409, 2005) for multichoice games. We also prove that this solution can be seen as a generalization of the configuration value and the dual configuration value (Albizuri et al., Games Econ. Behav. 57, 1–17, 2006) for transferable utility cooperative games.     

Nonzero-sum differential games with bargaining solutions

Nash's bargaining solution for finite games is extended to differential games with nonzero-sum integral payoffs. Sufficient conditions for the optimality of a strategy pair are established. An example is given.