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

Consistency and potentials in cooperative TU-games: Sobolev's reduced game revived

It was a quarter of a century ago that Sobolev proved the reduced game (otherwise called consistency) property for the much-discussed Shapley value of cooperative TU-games. The purpose of this paper is to extend Sobolev's result in two ways. On the one hand the unified approach applies to the enlarged class consisting of game-theoretic solutions that possess a so-called potential representation; on the other Sobolev's reduced game is strongly adapted in order to establish the consistency property for solutions that admit a potential. Actually, Sobolev's explicit description of the reduced game is now replaced by a similar, but implicit definition of the modified reduced game; the characteristic function of which is implicitly determined by a bijective mapping on the universal game space (induced by the solution in question). The resulting consistency property solves an outstanding open problem for a wide class of game-theoretic solutions. As usual, the consistency together with some kind of standardness for two-person games fully characterize the solution. A detailed exposition of the developed theory is given in the event of dealing with so-called semivalues of cooperative TU-games and the Shapley and Banzhaf values in particular

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

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

Two extensions of the Shapley value for cooperative games

Two extensions of the Shapley value are given. First we consider a probabilistic framework in which certain consistent allocation rules such as the Shapley value are characterized. The second generalization of the Shapley value is an extension to the structure of posets by means of a recursive form. In the latter setting, the Shapley value for quasi-concave games is shown to be a core-allocation. \u

Ring-resonator-based wavelength filters

Microring resonators (MR) represent a class of filters with characteristics very similar to those of Fabry–Perot filters. However, they offer the advantage that the injected and reflected signals are separated in individual waveguides, and in addition, their design does not require any facets or gratings and is thus particularly simple. MRs evolved from the fields of fibre optic ring resonators and micron scale droplets. Their inherently small size (with typical diameters in the range between several to tens of micrometres), their filter characteristics and their potential for being used in complex and flexible configurations make these devices particularly attractive for integrated optics or VLSI photonics applications.\ud MRs for filter applications, delay lines, as add/drop multiplexers, and modulators will be covered in detail in this chapter, while other applications such as in optical sensing, in spectroscopy or for coherent light generation (MR lasers) are outside the scope of this chapter.\ud This chapter focuses primarily on 4-port microrings, while 2-port devices will play a minor role here and are covered in more detail in Chap. 9. The present chapter starts with design considerations, the functional behaviour, and key characteristics of a single microring resonator and continues with the design of cascaded MRs allowing the implementation of higher order filters. Finally, complex devices like add-drop filters, tuneable dispersion compensators, all-optical wavelength converters, and tuneable cross-connects are treated.\u

(Average-) convexity of common pool and oligopoly TU-games

The paper studies both the convexity and average-convexity properties for a particular class of cooperative TU-games called common pool games. The common pool situation involves a cost function as well as a (weakly decreasing) average joint production function. Firstly, it is shown that, if the relevant cost function is a linear function, then the common pool games are convex games. The convexity, however, fails whenever cost functions are arbitrary. We present sufficient conditions involving the cost functions (like weakly decreasing marginal costs as well as weakly decreasing average costs) and the average joint production function in order to guarantee the convexity of the common pool game. A similar approach is effective to investigate a relaxation of the convexity property known as the average-convexity property for a cooperative game. An example illustrates that oligopoly games are a special case of common pool games whenever the average joint production function represents an inverse demand function