A note on the nucleolus for 2-convex TU games

Journal of Applied Mathematics

Driessen

2014

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The main goal is to reveal the 1-concavity property for a subclass of cost games called data cost games. The motivation for the study of the 1-concavity property is the appealing theoretical results for both the core and the nucleolus, in particular their geometrical characterization as well as their additivity property. The characteristic cost function of the original data cost game assigns to every coalition the additive cost of reproducing the data the coalition does not own. The underlying data and cost sharing situation is composed of three components, namely, the player set, the collection of data sets for individuals, and the additive cost function on the whole data set. The proof of 1-concavity is direct, but robust to a suitable generalization of the characteristic cost function. As an adjunct, the 1-concavity property is shown for the subclass of so-called “bicycle” cost games, inclusive of the data cost games in which the individual data sets are nested in a decreasing order.

Section: Discussionmentioning

confidence: 99%

Data Cost Games as an Application of 1-Concavity in Cooperative Game Theory

Journal of Applied Mathematics

Driessen

2014

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“…In words, the TU game v is said to be 1-convex if its corresponding (nonnegative) gap function g v attains its minimum at the grand coalition. Clearly, the class of compromise stable TU games contains the subclass of 1-convex n-person games [3,4], as well as the 2-convex n-person games [3,5] and the big boss and clan games [2,8,9].…”

Section: Compromise Stable Tu Gamesmentioning

confidence: 99%

“…The indirect function is also a helpful tool for the determination of the nucleolus for the subclasses of big boss games as well as 1-convex and 2-convex n-person games [5].…”

Section: Casementioning

confidence: 99%

Determining the Nucleolus of Compromise Stable Games

Driessen

2015

Bull. Aust. Math. Soc.

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The main goal is to illustrate that the so-called indirect function of a cooperative game in characteristic function form is applicable to determine the nucleolus for a subclass of coalitional games called compromise stable transferable utility (TU) games. In accordance with the Fenchel-Moreau theory on conjugate functions, the indirect function is known as the dual representation of the characteristic function of the coalitional game. The key feature of a compromise stable TU game is the coincidence of its core with a box prescribed by certain upper and lower core bounds. For the purpose of the determination of the nucleolus, we benefit from the interrelationship between the indirect function and the prekernel of coalitional TU games. The class of compromise stable TU games contains the subclasses of clan games, big boss games and 1-and 2-convex n-person TU games. As an adjunct, this paper reports the indirect function of clan games for the purpose of determining their nucleolus.2010 Mathematics subject classification: primary 91A12.

“…Three other applications of one-concavity or one-convexity, called library game, co-insurance game, and the dual game of the Stackelberg oligopoly game respectively, can be found in [10], [13] and [12]. The nucleolus for 2-convex games is treated in [15]. The search for other appealing classes of cost games satisfying the 1-concavity property is still going on.…”

Section: Discussionmentioning

confidence: 99%

Models, theory and applications in cooperative game theory

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This thesis consists of an introductory chapter (Chapter 1) followed by nine research chapters (Chapters 2-10), each of which is written as a self-contained journal paper, except that all references are gathered at the end of the thesis. These nine chapters are based on the nine papers that are listed below and have been submitted to journals for publication. The paper that forms the basis for Chapter 5 has been published in the International Journal of Game Theory, the papers underlying Chapter 2 and Chapter 8 are both accepted by the Journal Applied Mathematics and the paper underlying Chapter 10 is accepted by the Journal International Game Theory Review. The other papers are in different stages of the refereeing process. Chapters 2, 3 and 4 deal with game models, Chapters 5, 6 and 7 contain theoretical contributions to Cooperative Game Theory, while Chapters 8, 9 and 10 can be understood as the application of Game Theory. This explains the title of the thesis. Since the thesis has been written as a collection of more or less independent papers, the reader will find a certain amount of repetition of relevant concepts, definitions and background. The author apologizes for any inconvenience.