Cooperative Games In Characteristic Function Form Research Articles

Allocation of Cost Savings in a Three-Level Supply Chain with Demand Information Sharing: A Cooperative-Game Approach

We analyze the problem of allocating cost savings from sharing demand information in a three-level supply chain with a manufacturer, a distributor, and a retailer. To find a unique allocation scheme, we use concepts from cooperative game theory. First, we analytically compute the expected cost incurred by the manufacturer and then use simulation to obtain expected costs for the distributor and the retailer. We construct a three-person cooperative game in characteristic-function form and derive necessary conditions for the stability of each of five possible coalitions. To divide the cost savings between two members, or among three supply chain members, we use various allocation schemes. We present numerical analyses to investigate the impacts of the demand autocorrelation coefficient, ρ, and the unit holding and shortage costs on the allocation scheme.

Nucleolus as a Selector of Subcore

In this paper TU-cooperative games are considered from the linear programming point of view. Using the linear programming approach for construction of subcore and grand subcore, which are multiple selectors of the core, we represent them in analytical forms. The investigation of subcore's properties illustrates the possibility to obtain more simple conditions to check whether an arbitrary one point solution lies in the core and subcore. Two solution concepts for cooperative games in characteristic-function form, the nucleolus and Shapley value, are studied in their relationship to the grand subcore in symmetrical n-person TU-cooperative games. Under additional restrictions for the structure of optimal solutions’ set of considered linear programming problem sufficient conditions have been obtained for the nucleolus to be a selector of the grand subcore.

Totally balanced games arising from controlled programming problems

A cooperative game in characteristic-function form is obtained by allowing a number of individuals to esercise partial control over the constraints of a (generally nonlinear) mathematical programming problem, either directly or through committee voting. Conditions are imposed on the functions defining the programming problem and the control system which suffice to make the game totally balanced. This assures a nonempty core and hence a stable allocation of the full value of the programming problem among the controlling palyers. In the linear case the core is closely related to the solutions of the dual problem. Applications are made to a variety of economic models, including the transferable utility trading economies of Shapley and Shubik and a multishipper one-commodity transshipment model with convex cost functions and concave revenue functions. Dropping the assumption of transferable utility leads to a class of controlled multiobjective or ‘Pareto programming’ problems, which again yield totally balanced games.

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 ϵ-core as ϵ varies. One of the central results is that the portion of the prekernel that falls within the core, or any other strong ϵ-core, depends only on the latter's 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 Schmeidler's 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-man's-land that lies between them (if they are weak); the nucleolus as the result of an arbitrator's desire to minimize the dissatisfaction of the most dissatisfied coalition.