scholarly journals Associated Consistency Characterization of Two Linear Values for Tu Games by Matrix Approach

2012 ◽  
Author(s):  
Genjiu Xu ◽  
René van den Brink ◽  
Gerard van der Laan ◽  
Hao Sun
Keyword(s):  
Matrix Approach ◽  
Tu Games ◽  
Associated Consistency

scholarly journals Associated consistency characterization of two linear values for TU games by matrix approach

2015 ◽  
Vol 471 ◽  
pp. 224-240 ◽  
Author(s):  
Genjiu Xu ◽  
René van den Brink ◽  
Gerard van der Laan ◽  
Hao Sun
Keyword(s):  
Matrix Approach ◽  
Tu Games ◽  
Associated Consistency

THE EXTENSION OF DUTTA–RAY'S SOLUTION TO CONVEX NTU GAMES

2010 ◽  
Vol 12 (04) ◽  
pp. 339-361
Author(s):  
ELENA YANOVSKAYA
Keyword(s):  
Axiomatic Characterization ◽  
Convexity Property ◽  
Tu Games ◽  
Ntu Games ◽  
The Core ◽  
Egalitarian Solution

The egalitarian solution for the class of convex TU games was defined by Dutta and Ray [1989] and axiomatized by Dutta 1990. An extension of this solution — the egalitarian split-off set (ESOS) — to the class of non-levelled NTU games is proposed. On the class of TU games it coincides with the egalitarian split-off set [Branzei et al. 2006]. The proposed extension is axiomatized as the maximal (w.r.t. inclusion) solution satisfying consistency à la Hart–Mas-Colell and agreeing with the solution of constrained egalitarianism for arbitrary two-person games. For ordinal convex NTU games the ESOS turns out to be single-valued and contained in the core. The totally cardinal convexity property of NTU games is defined. For the class of ordinal and total cardinal convex NTU games an axiomatic characterization of the Dutta–Ray solution with the help of Peleg consistency is given.

A MATRIX APPROACH TO THE ASSOCIATED CONSISTENCY WITH AN APPLICATION TO THE SHAPLEY VALUE

2010 ◽  
Vol 12 (02) ◽  
pp. 175-187 ◽  
Author(s):  
GÉRARD HAMIACHE
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Shapley Value ◽  
Matrix Approach ◽  
Present Contribution ◽  
Matrix Operations ◽  
The Shapley Value ◽  
Associated Consistency

In an article by Hamiache (IJGT, 2001) an axiomatization of the Shapley value has been proposed. Three axioms were called on, inessential game, continuity and associated consistency. This present article proposes a new proof, based on elementary linear algebra. Games are represented by vectors. Associated games are the results of matrix operations. The eigenvalues of the involved matrices are computed and it is shown that they are diagonalizable. The present contribution offers a powerful tool allowing further generalizations of the Shapley value, which were difficult to consider on the basis of the previous proof.

scholarly journals -Pascal and -Wronskian Matrices with Implications to -Appell Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Thomas Ernst
Keyword(s):  
Difference Equation ◽  
Recent Article ◽  
Matrix Multiplication ◽  
Euler Polynomials ◽  
Matrix Approach ◽  
Appell Polynomials ◽  
Sum Of Products ◽  
Functional Matrix

We introduce a -deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for -Appell polynomials, and in particular for -Bernoulli and -Euler polynomials. Furthermore, the --polynomial, anticipated by Ward, can be expressed as a sum of products of -Bernoulli and -Euler polynomials. The pseudo -Appell polynomials, which are first presented in this paper, enable multiple -analogues of the Yang and Youn formulas. The generalized -Pascal functional matrix, the -Wronskian vector of a function, and the vector of -Appell polynomials together with the -deformed matrix multiplication from the authors recent article are the main ingredients in the process. Beyond these results, we give a characterization of -Appell numbers, improving on Al-Salam 1967. Finally, we find a -difference equation for the -Appell polynomial of degree .

POTENTIAL IN MULTI-CHOICE COOPERATIVE TU GAMES

2008 ◽  
Vol 25 (05) ◽  
pp. 591-611 ◽  
Author(s):  
YAN-AN HWANG ◽  
YU-HSIEN LIAO
Keyword(s):  
Transferable Utility ◽  
Tu Games ◽  
Cooperative Tu Games ◽  
All Solutions ◽  
The Family ◽  
Balanced Contributions ◽  
Associated Consistency

This paper is devoted to the study of solutions for multi-choice transferable-utility (TU) games which admit a potential, such as the potential associated with a solution in the context of multi-choice TU games. Several axiomatizations of the family of all solutions that admit a potential are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the weighted associated consistent value proposed in this paper to an appropriately modified game. We also characterize the weighted associated consistent value by means of the weighted balanced contributions and the associated consistency.

scholarly journals Characterizations of Three Linear Values for TU Games by Associated Consistency: Simple Proofs Using the Jordan Normal Form

2016 ◽  
Vol 18 (01) ◽  
pp. 1650003 ◽  
Author(s):  
Sylvain Béal ◽  
Eric Rémila ◽  
Philippe Solal
Keyword(s):  
Normal Form ◽  
Cooperative Games ◽  
Matrix Analysis ◽  
Transferable Utility ◽  
Jordan Normal Form ◽  
Linear Transformations ◽  
Tu Games ◽  
The Matrix ◽  
Matrix Expression ◽  
Associated Consistency

This paper studies values for cooperative games with transferable utility. Numerous such values can be characterized by axioms of [Formula: see text]-associated consistency, which require that a value is invariant under some parametrized linear transformation [Formula: see text] on the vector space of cooperative games with transferable utility. Xu et al. [(2008) Linear Algebr. Appl. 428, 1571–1586; (2009) Linear Algebr. Appl. 430, 2896–2897] Xu et al. [(2013) Linear Algebr. Appl. 439, 2205–2215], Hamiache [(2010) Int. Game Theor. Rev. 12, 175–187] and more recently Xu et al. [(2015) Linear Algebr. Appl. 471, 224–240] follow this approach by using a matrix analysis. The main drawback of these articles is the heaviness of the proofs to show that the matrix expression of the linear transformations is diagonalizable. By contrast, we provide quick proofs by relying on the Jordan normal form of the previous matrix.

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