A note on a core catcher of a cooperative game

In Driessen (1986) it is shown that for games satisfying a certain condition the core of the game is included in the convex hull of the set of certain marginal worth vectors of the game, while it is conjectured that the inclusion holds without any condition on the game. In this note it is proved that the inclusion holds for all games

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

A multiplicative potential approach to solutions for cooperative TU-games

Concerning the solution theory for cooperative games with transferable utility, it is well-known that the Shapley value is the most appealing representative of the family of (not necessarily efficient) game-theoretic solutions with an additive potential representation. This paper introduces a new solution concept, called Multiplicativily Proportional ($MP$) value, that can be regarded as the counterpart of the Shapley value if the additive potential approach to the solution theory is replaced by a multiplicative potential approach in that the difference of two potential evaluations is replaced by its quotient. One out of two main equivalence theorems states that every solution with a multiplicative potential representation is equivalent to this specifically chosen efficient value in that the solution of the initial game coincides with the $MP$ value of an auxiliary game. The associated potential function turns out to be of a multiplicative form (instead of an additive form) with reference to the worth of all the coalitions. The second equivalence theorem presents four additional characterizations of solutions that admit a multiplicative potential representation, e.g., preservation of discrete ratios or path independence

A potential approach to solutions for set games

Concerning the solution theory for set games, the paper introduces a new solution by allocating, to any player, the items (taken from an universe) that are attainable for the player, but can not be blocked (by any coalition not containing the player). The resulting value turns out to be an utmost important concept for set games to characterize the family of set game solutions that possess a so-called potential representation (similar to the potential approaches applied in both physics and cooperative game theory). An axiomatization of the new value, called Driessen--Sun value, is given by three properties, namely one type of an efficiency property, the substitution property and one type of a monotonocity property

A uniform approach to semi-marginalistic values for set games

Concerning the solution theory for set games, the paper focuses on a family of solutions, each of which allocates to any player some type of marginalistic contribution with respect to any coalition containing the player. Here the marginalistic contribution may be interpreted as an individual one, or a coalitionally one. For any value of the relevant family, an axiomatization is given by three properties, namely one type of an efficiency property, the substitution property and one type of a monotonocity property. We present two proof techniques, each of which is based on the decomposition of any arbitrary set game into a union of either simple set games or elementary set games, the solutions of which are much easier to determine. A simple respectively elementary set game is associated with an arbitrary, but fixed item of the universe respectively coalition

Matrix approach to the Shapley value and dual similar associated consistency

Replacing associated consistency in Hamiache's axiom system by dual similar associated consistency, we axiomatize the Shapley value as the unique value verifying the inessential game property, continuity and dual similar associated consistency. Continuing the matrix analysis for Hamiache's axiomatization of the Shapley value, we construct the dual similar associated game and introduce the dual similar associated transformation matrix $M_\lambda^{DSh}$ as well. In the game theoretic framework we show that the dual game of the dual similar associated game is Hamiache's associated game of the dual game. For the purpose of matrix analysis, we derive the similarity relationship $M_\lambda^{DSh}=QM_\lambda Q^{-1}$ between the dual similar associated transformation matrix $M_\lambda^{DSh}$ and associated transformation matrix $M_\lambda$ for Hamiache's associated game, where the transformation matrix $Q$ represents the duality operator on games. This similarity of matrices transfers associated consistency into dual similar associated consistency, and also implies the inessential property for the limit game of the convergent sequence of repeated dual similar associated games. We conclude this paper with three tables summarizing all matrix results

Ultrafast all-optical wavelength conversion in silicon waveguides using femtosecond pump-probe pulses

Experimental results on ultrafast all-optical wavelength conversion in silicon-on-insulator waveguides are presented. Red and blue shifts of 10nm have been observed in femtosecond pump-probe experiments. Alloptical switching and the importance of waveguide dispersion are discussed

Microwave-induced nonequilibrium temperature in a suspended carbon nanotube

Antenna-coupled suspended single carbon nanotubes exposed to 108 GHz microwave radiation are shown to be selectively heated with respect to their metal contacts. This leads to an increase in the conductance as well as to the development of a power-dependent DC voltage. The increased conductance stems from the temperature dependence of tunneling into a one-dimensional electron system. The DC voltage is interpreted as a thermovoltage, due to the increased temperature of the electron liquid compared to the equilibrium temperature in the leads