1-concave basis for TU games and the library game

Driessen, Theo; Khmelnitskaya, Anna Borisovna; Sales, J.

doi:10.1007/s11750-010-0157-5

Cited by 10 publications

(14 citation statements)

References 8 publications

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“…Finally, an alternative proof of the main Theorem 3 is treated in [11] in terms of Dutch soccer teams and their potential fans. Three other applications of oneconcavity or one-convexity, called library game, coinsurance game, and the dual game of the Stackelberg oligopoly game, respectively, can be found in [10,12,13]. The nucleolus for 2convex games is treated in [14].…”

Section: Discussionmentioning

confidence: 99%

“…Corollary 10. As shown in [10], the well-known Shapley cost allocation charged to the agents of any -person complementary unanimity cost game ⟨ , ⟩ amounts Sh ( , ) = 1 ∀ ∈ \ , = 0, every -person airport cost game with stand-alone costs ({ }), ∈ , ordered as an increasing sequence, is associated with a bicycle cost game with adapted stand-alone costs ({ }) − ({ }), ∈ , to be ordered as an increasing sequence. In this setting, (21) is a direct consequence of (20) applied to this latter bicycle cost game.…”

Section: Corollarymentioning

confidence: 91%

“…The proposed new basis has been introduced and developed in [10] as a subclass of 1-concave -person games, which are called complementary unanimity cost games. Definition 9 (see [10] with adapted notation). With every coalition ⊆ , ̸ = , ̸ = 0, there is associated complementary unanimity cost game ⟨ , ⟩ given by…”

Section: Corollarymentioning

confidence: 99%

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

Hou

Driessen

2014

Journal of Applied Mathematics

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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.

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Section: Discussionmentioning

confidence: 99%

Section: Corollarymentioning

confidence: 91%

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

Hou

Driessen

2014

Journal of Applied Mathematics

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“…The first practical example of a 1-concave game, the so-called library cost game, and the 1-concave complementary unanimity basis for the entire space of TU games were introduced in Driessen et al (2010). A co-insurance game under the hypothesis of Theorem 1 provides a new practical example of a 1-convex game.…”

Section: Discussionmentioning

confidence: 99%

On 1-convexity and nucleolus of co-insurance games

Driessen

Fragnelli

Katsev

et al. 2011

Insurance: Mathematics and Economics

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JEL classification: C71MSC: 91A12 91A40 91B30 a b s t r a c t The insurance situation in which an enormous risk is insured by a number of insurance companies is modeled through a cooperative TU game, the so-called co-insurance game, first introduced in Fragnelli and Marina (2004). In this paper we present certain conditions on the parameters of the model that guarantee the 1-convexity property of co-insurance games which in turn ensures the nonemptiness of the core and the linearity of the nucleolus as a function of the variable premium. Further we reveal conditions when a co-insurance game is representable in the form of a veto-removed game and present an efficient final algorithm for computing the nucleolus of a veto-removed game.

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“…Note that U T (N ) = 0 for all T N , except T = ∅. As is shown in [10], the well known Shapley value cost allocation charged to the players of any n-person complementary unanimity cost game (N, U T ) agrees with the semi-egalitarian rule such that members of T receive less the unitary amount, that is,…”

Section: The Shapley Value Of the Sharing Car Pooling Cost Game And Tmentioning

confidence: 61%

Models, theory and applications in cooperative game theory

Hou

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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.

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1-concave basis for TU games and the library game

Cited by 10 publications

References 8 publications

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

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

On 1-convexity and nucleolus of co-insurance games

Models, theory and applications in cooperative game theory

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