Automorphisms and weighted values

The notion of automorphism is an essential tool to capture the meaning of any mathematical structure. We apply this idea to cooperative games and obtain two interesting characterizations of the automorphisms of such a game: the one, in the complete case, as the permutations of players which preserve the (classical) Shapley value; the other, for the general case, as the permutations preservingall weighted Shapley values.Carreras' research supported by project UPV 036321-HA186/92 of the Basque Country University. Owen's research supported by the National Science Foundation, Grant DMS-9116416. This collaboration was made possible by the Polytechnic University of Catalonia, which sponsored Owen's visit to Terrassa.     

Dividends and weighted values in games with externalities

In this paper, we provide further support for the family of average values for games with externalities defined in Macho-Stadler et al. (J Econ Theory 135:339–356, 2007). This is a large family of values that includes several recent proposals. We show that they can be constructed through the sharing of appropriately defined dividends. Furthermore, we show the flexibility of this approach by using it to generate non-symmetric values.     

Mixing weighted values of non-atomic games

Weighted values of non-atomic games were introduced by Hart and Monderer (1997). They study these values by using two approaches: the potential approach and the asymptotic approach. In this study we develop the random order approach (the mixing value, Aumann and Shapley, 1974) to weighted values and prove that these values coincide with the asymptotic weighted values of Hart and Monderer in pNA. Received April 1995/Revised version May 1996/ Final version October 1997     

Small functions and weighted sharing three values

This article studies the problem of the uniqueness of meromorphic functions that weighted sharing three values which improve some results given by Yi [Theorem 4, Yi, H.X., 1995, Unicity theorems for meromorphic functions that share three values. Kodai Mathematical Journal, 18, 300-314] and Ueda [Ueda, H., 1983, Unicity theorems for meromorphic or entire functions II. Kodai Mathematical Journal, 6, 26-36] and other authors. An application of these new results, if f and g are two distinct nonconstant meromorphic functions sharing 0, 1 and CM, and a is a nonconstant rational function, then N2)(r,1/(g-a))= [image omitted] An example shows that the latter result is not true for some transcendental small functions of f and g.     

On the symmetric and weighted shapley values

We present new axiomatic characterizations of the symmetric Shapley value and of weighted Shapley values for transferable utility coalitional form games without imposing the axiom ofadditivity (Shapley [1953a,b]). Our main condition iscoalitional strategic equivalence, introduced by Chun [1989]. We show thatcoalitional strategic equivalence, together withefficiency, andsymmetry, characterizes the symmetric Shapley value, and this axiom, together withefficiency, positivity, homogeneity, andpartnership, characterizes weighted Shapley values.     

Axiomatic characterizations of the weighted solidarity values

We define and characterize the class of all weighted solidarity values. Our first characterization employs the classical axioms determining the solidarity value (except symmetry), that is, efficiency, additivity and the A-null player axiom, and two new axioms called proportionality and strong individual rationality. In our second axiomatization, the additivity and the A-null player axioms are replaced by a new axiom called average marginality.     

Operads of compatible structures and weighted partitions

In this paper we describe operads encoding two different kinds of compatibility of algebraic structures. We show that there exist decompositions of these in terms of black and white products and we prove that they are Koszul for a large class of algebraic structures by using the poset method of B. Vallette. In particular we show that this is true for the operads of compatible Lie, associative and pre-Lie algebras.     

Monge extensions of cooperation and communication structures

Cooperation structures without any a priori assumptions on the combinatorial structure of feasible coalitions are studied and a general theory for marginal values, cores and convexity is established. The theory is based on the notion of a Monge extension of a general characteristic function, which is equivalent to the Lovász extension in the special situation of a classical cooperative game. It is shown that convexity of a cooperation structure is tantamount to the equality of the associated core and Weber set. Extending Myerson’s graph model for game theoretic communication, general communication structures are introduced and it is shown that a notion of supermodularity exists for this class that characterizes convexity and properly extends Shapley’s convexity model for classical cooperative games.     

Optimum parameters of hierarchical control structures

The use of multilevel hierarchical structures to effectively conduct inspections and control corruption under the conditions of a shortage of authorized representatives is proposed. Game theory models of such structures are considered and the problem of their organization is reviewed. The optimum strategy for checking in a hierarchy is determined. The results from numerical modeling for the example of taxation of small businesses are given.     

Characteristic vertices of weighted trees via perron values

We consider a weighted tree T with algebraic connectivity μ, and characteristic vertex v. We show that μ and its associated eigenvectors can be described in terms of the Perron value and vector of a nonnegative matrix which can be computed from the branches of T at v. The machinery of Perron-Frobenius theory can then be used to characterize Type I and Type II trees in terms of these Perron values, and to show that if we construct a weighted tree by taking two weighted trees and identifying a vertex of one with a vertex of the other, then any characteristic vertex of the new tree lies on the path joining the characteristic vertices of the two old trees.     

Meromorphic functions with weighted sharing of three values

In this article we study the problem of uniqueness of meromorphic functions with weighted sharing of three values. The results in this article improve some theorems of M. Ozawa, H. Ueda, G. Brosch, H. Yi, I. Lahiri and other authors. Examples are provided to show that the conditions of our results are sharp.     

Cooperative game theory approach to allocating benefits of horizontal cooperation

Logistics costs in general, and transportation costs in particular, represent a large fraction of the operating costs of many companies. One way to try to reduce these costs is through horizontal cooperation among shippers. Thus, when the transportation needs of two or more companies are merged, their collective transportation requirements can be met at lower cost. The attainable cost savings are due to economies of scale, which translate into cheaper rates due to increased negotiation power, use of larger vehicles and bundling of shipments. In this paper, a linear model is presented and used to study the cost savings that different companies may achieve when they merge their transportation requirements. On the one hand, solving this optimization model for different collaboration scenarios allows testing and quantifying the synergies among different potential partners, thus identifying the most profitable collaboration opportunities. On the other, the problem of allocating the joint cost savings of the cooperation is tackled using cooperative game theory. The proposed approach is illustrated with an example in which different cooperative game solution concepts are compared. Extensive numerical experiments have also been carried out to gain insight into the properties of the corresponding cost savings game and the behavior of the different solution concepts.     

Further results on weighted sharing of values for meromorphic functions

In this paper, we deal with the problem of uniqueness and weighted sharing of two meromorphic functions with their first derivatives sharing a nonzero polynomial with the same multiplicities. The result in this paper improve those given by K. Tohge, Xiao-Min Li & Hong-Xun Yi, I. Lahiri & P. Sahoo.     

Strategically supported cooperation in dynamic games with coalition structures

The problem of strategic stability of long-range cooperative agreements in dynamic games with coalition structures is investigated. Based on imputation distribution procedures, a general theoretical framework of the differential game with a coalition structure is proposed. A few assumptions about the deviation instant for a coalition are made concerning the behavior of a group of many individuals in certain dynamic environments.From these, the time-consistent cooperative agreement can be strategically supported by ε-Nash or strong ε-Nash equilibria. While in games in the extensive form with perfect information, it is somewhat surprising that without the assumptions of deviation instant for a coalition, Nash or strong Nash equilibria can be constructed.     

Some exact values of the Harborth constant and its plus-minus weighted analogue

The Harborth constant of a finite abelian group is the smallest integer ${\ell}$ such that each subset of G of cardinality ${\ell}$ has a subset of cardinality equal to the exponent of the group whose elements sum to the neutral element of the group. The plus-minus weighted analogue of this constant is defined in the same way except that instead of considering the sum of all elements of the subset, one can choose to add either the element or its inverse. We determine these constants for certain groups, mainly groups that are the direct sum of a cyclic group and a group of order 2. Moreover, we contrast these results with existing results and conjectures on these problems.     

Analysis of functions that are hierarchical as algebraic structures

The mathematical analysis of a connection can be implemented according to different approaches in order to characterize the system structure R.     

On the characteristics of meromorphic functions with three weighted sharing values

In this paper, we deal with the relation between the characteristic function of two nonconstant meromorphic functions with three weighted sharing values, which improves a result given by H.X. Yi and Y.H. Li. From this we establish a theorem which improves a result given by P. Li and C.C. Yang.