In this paper we present the Subtraction Algorithm that computes for every classical minimum cost spanning tree game a population monotonic allocation scheme. As a basis for this algorithm serves a decomposition theorem that shows that every minimum cost spanning tree game can be written as nonnegative combination of minimum cost spanning tree games corresponding to 0-1 cost functions. It turns out that the Subtraction Algorithm is closely related to the famous algorithm of Kruskal for the determination of minimum cost spanning trees. For variants of the classical minimum cost spanning tree games we show that population monotonic allocation schemes do not necessarily exist.
Different axiomatic systems for the Shapley value can be found in the literature. For games with a coalition structure, the Shapley value also has been axiomatized in several ways. In this paper, we discuss a generalization of the Shapley value to the class of partition function form games. The concepts and axioms, related to the Shapley value, have been extended and a characterization for the Shapley value has been provided. Finally, an application of the Shapley value is given.
Abstract:We introduce the class of Obligation rules for minimum cost spanning tree situations. The main result of this paper is that such rules are cost monotonic and induce also population monotonic allocation schemes. Another characteristic of Obligation rules is that they assign to a minimum cost spanning tree situation a vector of cost contributions which can be obtained as product of a double stochastic matrix with the cost vector of edges in the optimal tree provided by the Kruskal algorithm. It turns out that the Potters value (P -value) is an element of this class.
This study considers a supply chain that consists of n retailers, each of them facing a newsvendor problem, m warehouses and a supplier. The retailers are supplied with a single product via some warehouses. In these warehouses, the ordered amounts of goods of these retailers become available after some lead time. At the time that the goods arrive at the warehouses, demand realizations are known by the retailers. The retailers can increase their expected joint profits by coordinating their orders and making allocations after demand realization. For this setting, we consider an associated cooperative game between the retailers. We show that this associated cooperative game has a nonempty core. Finally, we study a variant of this game, where the retailers are allowed to leave unsold goods at the warehouses.
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