M. Maschler

Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts

Two solution concepts for cooperative games in characteristic-function form, the kernel and the nucleolus, are studied in their relationship to a number of other concepts, most notably the core. The unifying technical idea is to analyze the behavior of the strong e-core as e varies. One of the central results is that the portion of the prekernel that falls within the core, or any other strong e-core, depends only on the latters geometrical shape. The prekernel is closely related to the kernel and often coincides with it, but has a simpler definition and simpler analytic properties. A notion of “quasi-zero-monotonicity” is developed to aid in enlarging the class of games in which kernel considerations can be replaced by prekernel considerations. The nucleolus is approached through a new, geometrical definition, equivalent to Schmeidlers original definition but providing very elementary proofs of existence, unicity, and other properties. Finally, the intuitive interpretations of the two solution concepts are clarified: the kernel as a kind of multi-bilateral bargaining equilibrium without interpersonal utility comparisons, in which each pair of players bisects an interval which is either the battleground over which they can push each other aided by their best allies if they are strong or the no-mans-land that lies between them if they are weak; the nucleolus as the result of an arbitrators desire to minimize the dissatisfaction of the most dissatisfied coalition.

THE KERNEL AND BARGAINING SET FOR CONVEX GAMES

It is shown that for convex games the bargaining setℳ1(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.

The consistent Shapley value for hyperplane games

A new value is defined for n-person hyperplane games, i.e., non-sidepayment cooperative games, such that for each coalition, the Pareto optimal set is linear. This is a generalization of the Shapley value for side-payment games.It is shown that this value is consistent in the sense that the payoff in a given game is related to payoffs in reduced games (obtained by excluding some players) in such a way that corrections demanded by coalitions of a fixed size are cancelled out. Moreover, this is the only consistent value which satisfies Pareto optimality (for the grand coalition), symmetry and covariancy with respect to utility changes of scales. It can be reached by players who start from an arbitrary Pareto optimal payoff vector and make successive adjustments.

The super-additive solution for the Nash bargaining game

The feasible set in a Nash bargaining game is a set in the utility space of the players. As such, its points often represent expectations on uncertain events. If this is the case, the feasible set changes in time as uncertainties resolve. Thus, if time for reaching agreement is not fixed in advance, one has to take into account options for delaying an agreement. This paper studies such games and develops a solution concept which has the property that its followers will always prefer to reach an immediate agreement, rather than wait until a new feasible set arises.

The Structure of the Kernel of a Cooperative Game

Abstract : A study of the kernel of a cooperative game. In this paper the authors derive a procedure for the players which, if abided by, leads to an outcome in the kernel. Moreover, each outcome in the kernel can be reached by this procedure. The procedure consists of a set of three rules and involves the formation of intermediate coalitions which play intermediate games, after which the members of each intermediate coalition play a reduced game to decide the share of their spoils. The procedure is further analyzed in the case of monotonic games and in the case of simple games, and the results that are obtained reduce considerably the amount of computation which is needed to compute the kernels of such games. In particular, they compute the kernel of the 7-person projective game (for the grand coalition), which is a star consisting of seven straight-line segments connecting the center to the points of the main simple solution. Finally, conditions under which modifications of the characteristic function do not change its kernel are presented. (Author)

The kernel/nucleolus of a standard tree game

In this paper we characterize the nucleolus (which coincides with the kernel) of a tree enterprise. We also provide a new algorithm to compute it, which sheds light on its structure. We show that in particular cases, including a chain enterprise one can compute the nucleolus in O(n) operations, wheren is the number of vertices in the tree.

The general nucleolus and the reduced game property

The nucleolus of a TU game is a solution concept whose main attraction is that it always resides in any nonempty ɛ-core. In this paper we generalize the nucleolus to an arbitrary pair (Π,F), where Π is a topological space andF is a finite set of real continuous functions whose domain is Π. For such pairs we also introduce the “least core” concept. We then characterize the nucleolus forclasses of such pairs by means of a set of axioms, one of which requires that it resides in the least core. It turns out that different classes require different axiomatic characterizations.One of the classes consists of TU-games in which several coalitions may be nonpermissible and, moreover, the space of imputations is required to be a certain “generalized” core. We call these gamestruncated games. For the class of truncated games, one of the axioms is a new kind ofreduced game property, in which consistency is achieved even if some coalitions leave the game, being promised the nucleolus payoffs. Finally, we extend Kohlbergs characterization of the nucleolus to the class of truncated games.

Stable Sets and Stable Points of Set-Valued Dynamic Systems with Applications to Game Theory

Let X be a metric space. A dynamic system on X is a set-valued function

A characterization, existence proof and dimension bounds for the kernel of a game

varphi

Minimal domains and their Bergman kernel function

from X to X which satisfies