Theo Driessen

Coincidence of and collinearity between game theoretic solutions

SummaryThe first part is the study of several conditions which are sufficient for the coincidence of the prenucleolus concept and the egalitarian nonseparable contribution (ENSC-) method. The main sufficient condition for the coincidence involved requires that the maximal excesses at the ENSC-solution are determined by the (n-1)-person coalitions in then-person game. The second part is the study of both a new type of games, the so-calledk-coalitionaln-person games, and the interrelationship between solutions on the class of those games. The main results state that the Shapley value of ak-coalitionaln-person game can be written as a convex or affine combination of the ENSC-solution and the centre of the imputation set.ZusammenfassungIm ersten Teil der Arbeit werden verschiedene hinreichende Bedingungen für die Koinzidenz des Prenukleolus-Lösungskonzepts und der ENSC-Regel vorgestellt. Es wird dabei gezeigt, daß der Prenukleolus mit der ENSC-Lösung zusammenfällt, falls die maximalen Exzesse der ENSC-Lösung durch die (n-1)-Personen Koalitionen desn-Personenspiels bestimmt werden. Im zweiten Teil der Arbeit untersuchen wir eine Klasse von Spielen, die sogenanntenk-Koalitions-n-Personenspiele und untersuchen die Zusammenhänge zwischen den Lösungskonzepten für diesen speziellen Typ von Spielen. Es stellt sich heraus, daß der Shapleywert einesk-Koalitions-n-Personenspiels beschrieben werden kann als eine Linearkombination der ENSC-Lösung und des Schwerpunktes der Auszahlungsmenge.

A note on the inclusion of the kernal in the core of the bilateral assignment game

Abstract. In this note we prove that the kernel of a bilateral assignment game is always included in the core. This solves an outstanding open problem for bilateral assignment games.

The irreducible Core of a minimum cost spanning tree game

It is a known result that for a minimum cost spanning tree (mcst) game a Core allocation can be deduced directly from a mcst in the underlying network. To determine this Core allocation one only needs to determine a mcst in the network and it is not necessary to calculate the coalition values of the corresponding mcst game. In this paper we will deduce other Core allocations directly from the network, without determining the corresponding mcst game itself: we use an idea of Bird (cf. [4]) to present two procedures that determine a part of the Core (called the Irreducible Core) from the network.

The Shapley value for games on matroids: The dynamic model

Abstract. According to the work of Faigle [3] a static Shapley value for games on matroids has been introduced in Bilbao, Driessen, Jiménez-Losada and Lebrón [1]. In this paper we present a dynamic Shapley value by using a dynamic model which is based on a recursive sequence of static models. In this new model for games on matroids, our main result is that there exists a unique value satisfying analogous axioms to the classical Shapley value. Moreover, we obtain a recursive formula to calculate this dynamic Shapley value. Finally, we prove that its components are probabilistic values.

A Weighted Pseudo-potential Approach to Values for TU-games

This paper provides a twofold generalization of the well-known characterization of the Shapley value for TU-games as the discrete gradient of a so-called potential function. On the one hand the potential approach is extended to the so-called weighted pseudo-potential approach in the sense that the extended representation may incorporate, besides a fraction of the discrete gradient, a fraction of the underlying pseudo-potential function itself, as well as a fraction of the average of all the components of the gradient. On the other hand the paper fully characterizes the class of values for TU-games that admit a weighted pseudo-potential representation. Besides two individual constraints, these values have to be efficient, symmetric, and linear. The theory developed is illustrated by several examples of such values and their weighted pseudo-potential representations are discussed.

On a family of values for TU-games generalizing the Shapley value

In this paper we study a family of efficient, symmetric and linear values for TU-games, described by some formula generalizing the Shapley value. These values appear to have surprising properties described in terms of the axioms: Fair treatment, monotonicity and two types of acceptability. The results obtained are discussed in the context of the Shapley value, the solidarity value, the least square prenucleolus and the consensus value.

A regularised one-dimensional drop formation and coalescence model using a total variation diminishing (TVD) scheme on a single Eulerian grid

The breakup of an axisymmetric viscous jet is considered in the lubrication approximation. The discretised equations are solved on a fixed equidistant one-dimensional Eulerian grid. The governing equations are implemented in a conservative second order accurate total variation diminishing (TVD) scheme, preventing the numerical diffusivity. Singularities that occur at pinchoff and coalescence are regularised by a small modification on the surface tension. The modification is of the order of the spatial step Δx. This regularisation ensures that the solution of the presented numerical model converges to the exact solution of the breakup of a jet in the lubrication approximation. The results of the presented numerical model agree quantitatively with the analytical solution of the Rayleigh–Plateau instability, and with experimental results on the final stage of the Rayleigh–Plateau instability.

A cooperative game of information trading: The core, the nucleolus and the kernel

A certain trade of the information about a technological innovation between the initial owner of the information andn identical producers is studied by means of a cooperative game theoretic approach. The information trading situation is modelled as a cooperative (n+1)-person game with side payments. The symmetrical strong ɛ-cores (including the core), the nucleolus and the kernel of the cooperative game model are determined. Interpretations of these game theoretic solutions and their implications for the information trading problem are given.

Note On linear consistency of anonymous values for TU-games

Abstract. In the framework of values for TU-games, it is shown that a particular type of consistency, called linear consistency, together with some kind of standardness for two-person games, imply efficiency, anonymity, linearity, as well as uniqueness of the value. Among others, this uniform treatment generalizes Sobolevs axiomatization of the Shapley value.

Associated consistency and values for TU games

In the framework of the solution theory for cooperative transferable utility games, Hamiache axiomatized the well-known Shapley value as the unique one-point solution verifying the inessential game property, continuity, and associated consistency. The purpose of this paper is to extend Hamiache’s axiomatization to the class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. For this enlarged class of values, explicit relationships to the Shapley value are exploited in order to axiomatize such values with reference to a slightly adapted inessential game property, continuity, and a similar associated consistency. The latter axiom requires that the solutions of the initial game and its associated game (with the same player set, but a different characteristic function) coincide.