Associated Games to Optimize 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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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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A NOTE ON THE MONOTONIC CORE

International Game Theory Review ◽

10.1142/s0219198909002285 ◽

2009 ◽

Vol 11 (02) ◽

pp. 229-235 ◽

Cited By ~ 1

Author(s):

JESÚS GETÁN ◽

JESÚS MONTES ◽

CARLES RAFELS

Keyword(s):

Cooperative Game ◽

Transferable Utility ◽

The Core ◽

Restricted Cooperation ◽

Population Monotonic Allocation Schemes

The monotonic core of a cooperative game with transferable utility is the set formed by all its Population Monotonic Allocation Schemes. In this paper we show that this set always coincides with the core of a certain game, with and without restricted cooperation, associated to the initial game.

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Uncoupled Aspiration Adaptation Dynamics Into the Core

German Economic Review ◽

10.1111/geer.12160 ◽

2019 ◽

Vol 20 (2) ◽

pp. 243-256 ◽

Cited By ~ 4

Author(s):

Heinrich H. Nax

Keyword(s):

Finite Time ◽

Cooperative Games ◽

Transferable Utility ◽

The Core ◽

Best Response ◽

Dynamic Blocking ◽

Evolutionary Interpretation ◽

Require Information ◽

Empty Core

Abstract Dynamics for play of transferable-utility cooperative games are proposed that require information regarding own payoff experiences and other players’ past actions, but not regarding other players’ payoffs. The proposed dynamics provide an evolutionary interpretation of the proto-dynamic ‘blocking argument’ (Edgeworth, 1881) based on the behavioral principles of ‘aspiration adaptation’ (Sauermann and Selten, 1962) instead of best response. If the game has a non-empty core, the dynamics are absorbed into the core in finite time with probability one. If the core is empty, the dynamics cycle infinitely through all coalitions.

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