425 research outputs found
A note on a core catcher of a cooperative game
No full textIn 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
A Comment on Dehez and Tellone, “Data games sharing public goods with exclusion”
No full textThis comment shows that the data cost game introduced in Dehez and Tellone (Journal of Public Economic Theory, 2013) coincides with the nonadditive component of the library cost game studied in Driessen, Khmelnitskaya, and Sales (TOP, 2012) where the core, nucleolus, and Shapley value were also investigated
A bankruptcy problem and an information trading problem: Applictions to k-convex games
No full textThe paper is devoted to two real problems that generate a cooperative game model with a so-calledk-convex characteristic function when certain conditions are fulfilled. Both a bankruptcy problem and an information trading problem are modelled as a cooperative game by constructing the corresponding bankruptcy game as well as the information market game. Firstly, it is established that the bankruptcy game is ak-convexn-person game where 1 ≤k ≤n − 2 if and only if the estate is sufficient to meet the claims of creditors in any (n − k)-person coalition. Secondly, it is shown that thek-convexity property for the information market game is equivalent to the nonexistence of profits with respect to a restricted class of submarkets
A comment on P. Dehez and D. Tellone "Data games: sharing goods with exclusion", published in JPET 15 (2013): 654-673
No full textWe show that the mathematical model of the data cost game introduced in Dehez and Tellone (JPET, 2013) coincides with the model of the library cost game studied in Driessen, Khmelnitskaya and Sales (TOP, 2012) where its core, nucleolus and Shapley value were also investigated
Consistency à la Hart and Mas-Colell of efficient, linear, and symmetric values for TU-games
No full textBy Hart and Mas-Colell's axiomatization, it is known that the Shapley value for TU-games is fully characterized by its 1-standardness for two-person games and its consistency property with respect to a particular reduced game. In the framework of values for TU-games, this paper establishes a similar axiomatization for almost every value that is supposed to be efficient, linear, and symmetric (like the Shapley value). For that purpose, we introduce a general type of reduced game that takes into account the (probabilities of) two events that a removed player joins or does not join a proposed coalition in the reduced game. Similar to Hart and Mas-Colell's reduced game (in which the player joins the coalition with probability one), the general model of the reduced game involves the value itself. According to this unified approach, almost every efficient, linear, and symmetric value is consistent with respect to an appropriately chosen reduced game. The relevant reduced game varies whenever the efficient, linear, and symmetric value varies, nevertheless we present an operational criterion how to determine the appropriate reduced game (by solving an associated system of linear equations in a recursive manner). The second result states that the resulting consistency property, together with some kind of standardness for two-person games, fully characterize the given value. This paper extends the main result from a 1998 article of Driessen, Radzik and Wanink on the consistency (or reduced game) property for values that are supposed to have a weighted potential representation (since the consistency theory developed avoids the concept called a weighted potential function). Finally, the consistency theory is illustrated in the context of several known values, among which the least square values (including the Shapley value)
The Shapley value for games on matroids: the dynamic model
No full textAccording 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
1-convex transferable utility games, a reappraisal
No full text1-convex games have been introduced by Theo Driessen in his 1985 PhD dissertation. They form an interesting class of games for at least one reason: the core of a 1-convex n-player game is a regular simplex of dimension n – 1 or a single point. As a consequence, its nucleolus is the center of gravity of the core. We recall and extend the results obtained by Driessen and provide examples and applications
1-convex transferable utility games, a reappraisal
No full text1-convex games have been introduced by Theo Driessen in his 1985 PhD dissertation. They form an interesting class of games for at least one reason: the core of a 1-convex n-player game is a regular simplex of dimension n – 1 or a single point. As a consequence, its nucleolus is the center of gravity of the core. We recall and extend the results obtained by Driessen and provide examples and applications
Matrix analysis for associated consistency in cooperative game theory
No full textHamiache's recent axiomatization of the well-known Shapley value for TU games states that the Shapley value is the unique solution verifying the following three axioms: the inessential game property, continuity and associated consistency. Driessen extended Hamiache's axiomatization to the enlarged class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. In this paper, we introduce the notion of row (resp. column)-coalitional matrix in the framework of cooperative game theory. Particularly, both the Shapley value and the associated game are represented algebraically by their coalitional matrices called the Shapley standard matrix and the associated transformation matrix respectively. We develop a matrix approach for Hamiache's axiomatization of the Shapley value. The associated consistency for the Shapley value is formulated as the matrix equality The diagonalization procedure of and the inessential property for coalitional matrices are fundamental tools to prove the convergence of the sequence of repeated associated games as well as its limit game to be inessential. In addition, a similar matrix approach is applicable to study Driessen's axiomatization of a certain class of linear values. Matrix analysis is adopted throughout both the mathematical developments and the proofs. In summary, it is illustrated that matrix analysis is a new and powerful technique for research in the field of cooperative game theory
Generalized concavity in cooperative game theory: characterizations in terms of the core
No full text- …
