On the Number of Extreme Points of the Core of a Transferable Utility Game

This paper deals in a unified way with the solution concepts for transferable utility games known as the Centre of the Imputation Set value (CIS-value), the Egalitarian Non-Pairwise-Averaged Contribution value (ENPAC-value) and the Egalitarian Non-Separable Contribution value (ENSC-value). These solutions are regarded as the egalitarian division of the surplus of the overall profits after each participant is conceded to get his individual contribution specified in a respective manner. We offer two interesting individual contributions (lower- and upper-k-averaged contribution) based on coalitions of size k(k ∈ {1,…,n-1}) and introduce a new solution concept called the Egalitarian Non-k-Averaged Contribution value ( EN k AC -value). CIS-, ENPAC- and ENSC-value are the same as EN 1 AC -, EN n-2 AC - and EN n-1 AC -value respectively. It turns out that the lower- and the upper-k-averaged contribution form a lower- and an upper-bound of the Core respectively. The Shapley value is the centre of gravity of n-1 points; EN 1 AC -,…, EN n-1 AC -value. EN k AC -value of the dual game is equal to EN n-k AC -value of the original game. We provide a sufficient condition on the transferable utility game to guarantee that the EN k AC -value coincides with the well-known solution called prenucleolus. The condition requires that the largest excesses at the EN k AC -value are attained at the k-person coalitions, whereas the excesses of k-person coalitions at the EN k AC -value do not differ.

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Associated Games to Optimize the Core of a Transferable Utility Game

Journal of Optimization Theory and Applications ◽

10.1007/s10957-019-01494-y ◽

2019 ◽

Vol 182 (2) ◽

pp. 816-836 ◽

Cited By ~ 1

Author(s):

Qianqian Kong ◽

Hao Sun ◽

Genjiu Xu ◽

Dongshuang Hou

Keyword(s):

Transferable Utility ◽

The Core ◽

Transferable Utility Game

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THE POSITIVE PREKERNEL OF A COOPERATIVE GAME

International Game Theory Review ◽

10.1142/s0219198900000196 ◽

2000 ◽

Vol 02 (04) ◽

pp. 287-305 ◽

Cited By ~ 5

Author(s):

PETER SUDHÖLTER ◽

BEZALEL PELEG

Keyword(s):

Cooperative Game ◽

Weak Version ◽

Transferable Utility ◽

Bargaining Set ◽

The Core ◽

Reduced Game ◽

Positive Kernel ◽

Game Property ◽

Transferable Utility Games

The positive prekernel, a solution of cooperative transferable utility games, is introduced. We show that this solution inherits many properties of the prekernel and of the core, which are both sub-solutions. It coincides with its individually rational variant, the positive kernel, when applied to any zero-monotonic game. The positive (pre)kernel is a sub-solution of the reactive (pre)bargaining set. We prove that the positive prekernel on the set of games with players belonging to a universe of at least three possible members can be axiomatized by non-emptiness, anonymity, reasonableness, the weak reduced game property, the converse reduced game property, and a weak version of unanimity for two-person games.

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Hart–Mas-Colell consistency and the core in convex games

International Journal of Game Theory ◽

10.1007/s00182-021-00798-6 ◽

2021 ◽

Author(s):

Bas Dietzenbacher ◽

Peter Sudhölter

Keyword(s):

Shapley Value ◽

Cooperative Games ◽

Individual Rationality ◽

Transferable Utility ◽

Convex Games ◽

The Core ◽

The Shapley Value ◽

Scale Covariance

AbstractThis paper formally introduces Hart–Mas-Colell consistency for general (possibly multi-valued) solutions for cooperative games with transferable utility. This notion is used to axiomatically characterize the core on the domain of convex games. Moreover, we characterize all nonempty solutions satisfying individual rationality, anonymity, scale covariance, superadditivity, weak Hart–Mas-Colell consistency, and converse Hart–Mas-Colell consistency. This family consists of (a) the Shapley value, (b) all homothetic images of the core with the Shapley value as center of homothety and with positive ratios of homothety not larger than one, and (c) their relative interiors.

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Multiplayer Allocations in the Presence of Diminishing Marginal Contributions: Cooperative Game Analysis and Applications in Management Science

Management Science ◽

10.1287/mnsc.2020.3709 ◽

2020 ◽

Author(s):

Mingming Leng ◽

Chunlin Luo ◽

Liping Liang

Keyword(s):

Cooperative Game ◽

Shapley Value ◽

Extreme Points ◽

Game Analysis ◽

The Core ◽

The Shapley Value ◽

Code Sharing ◽

Marginal Contributions ◽

Least Core ◽

Necessary And Sufficient

We use cooperative game theory to investigate multiplayer allocation problems under the almost diminishing marginal contributions (ADMC) property. This property indicates that a player’s marginal contribution to a non-empty coalition decreases as the size of the coalition increases. We develop ADMC games for such problems and derive a necessary and sufficient condition for the non-emptiness of the core. When the core is non-empty, at least one extreme point exists, and the maximum number of extreme points is the total number of players. The Shapley value may not be in the core, which depends on the gap of each coalition. A player can receive a higher allocation based on the Shapley value in the core than based on the nucleolus, if the gap of the player is no greater than the gap of the complementary coalition. We also investigate the least core value for ADMC games with an empty core. To illustrate the applications of our results, we analyze a code-sharing game, a group buying game, and a scheduling profit game. This paper was accepted by Chung Piaw Teo, optimization.

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PROJECT MANAGEMENT GAMES

International Game Theory Review ◽

10.1142/s0219198911003003 ◽

2011 ◽

Vol 13 (03) ◽

pp. 281-300 ◽

Cited By ~ 3

Author(s):

IMMA CURIEL

Keyword(s):

Project Management ◽

Completion Time ◽

Cooperative Game Theory ◽

Sufficient Conditions ◽

Extreme Points ◽

Convex Games ◽

The Core ◽

Management Games ◽

Big Boss Games ◽

Necessary And Sufficient

This paper studies situations in which companies can cooperate in order to decrease the earliest completion time of a project that consists of several tasks. This is beneficial for the client who wants the project to be completed as early as possible. The client is willing to pay more for an earlier completion time. The total payoff must be allocated among the companies that cooperate. Cooperative game theory is used to model this situation. Conditions for the core of the game to be nonempty are derived. We study a class of project management games for which necessary and sufficient conditions for the nonemptiness of the core can be derived. We will show that a subset of the set of balanced project management games can be partitioned into a class of 1-convex games and a class of big boss games. Expressions for the extreme points of the core, the τ-value, the nucleolus, and the Shapley-value of games in these two classes are derived.

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