An optimal bound to access the core in TU-games

doi:10.1016/j.geb.2013.02.008

Games and Economic Behavior

Volume 80, July 2013, Pages 1-9

https://doi.org/10.1016/j.geb.2013.02.008 Get rights and content

Highlights

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The core of any n-player TU-game can be accessed with at most $n - 1$ blocks.
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This bound is shown to be optimal.
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The use of Davis–Maschler reduced games is important.
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Exact coalitions have a key role in our procedure.
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Our procedure generalizes Shapley (1971)ʼs result for convex games.

Abstract

We show that the core of any n-player TU-game with a non-empty core can be accessed with at most $n - 1$ blocks. It turns out that this bound is optimal in the sense there are TU-games for which the number of blocks required to access the core is exactly $n - 1$ .

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There are more references available in the full text version of this article.

Cited by (6)

Stability properties of the core in a generalized assignment problem
2021, Games and Economic Behavior
We show that the core of a generalized assignment problem satisfies two types of stability properties. First, the core is the unique stable set defined using the weak domination relation when outcomes are restricted to individually rational and pairwise feasible ones. Second, the core is the unique stable set with respect to a sequential domination relation that is defined by a sequence of weak domination relations that satisfy outsider independence. An equivalent way of stating this result is that the core satisfies the property commonly stated as the existence of a path to stability. These results add to the importance of the core in an assignment problem where agents' preferences may not be quasilinear.
The equivalence of the minimal dominant set and the myopic stable set for coalition function form games
2021, Games and Economic Behavior
Citation Excerpt :
The latter results prove the existence of an upper bound on the number of steps needed. This bound has since been drastically lowered (Yang, 2010, 2011; Béal et al., 2012, 2013a,b). It is natural to ask where do these coalitional improvements lead in case the core is (possibly) empty.
In cooperative games, the coalition structure core is, despite its potential emptiness, one of the most popular solutions. While it is a fundamentally static concept, the consideration of a sequential extension of the underlying dominance correspondence gave rise to a selection of non-empty generalizations. Among these, the payoff-equivalence minimal dominant set and the myopic stable set are defined by a similar set of conditions. We identify some problems with the payoff-equivalence minimal dominant set and propose an appropriate reformulation called the minimal dominant set. We show that replacing asymptotic external stability by sequential weak dominance leaves the myopic stable set unaffected. The myopic stable set is therefore equivalent to the minimal dominant set.
On the characterizations of viable proposals
2020, Theory and Decision
The Equivalence of the Minimal Dominant Set and the Myopic Stable Set for Coalition Function Form Games
2020, SSRN
On the Characterizations of Viable Proposals
2019, SSRN
Accessibility and stability of the coalition structure core
2013, Mathematical Methods of Operations Research

^☆: We would like to thank two anonymous reviewers for valuable remarks. We are grateful to László Á. Kóczy, Miklós Pintér and Tamás Solymosi for fruitful discussions. We have benefited from comments at the 6th seminar day “Ordered Structures in GAmes and Decision” (OSGAD). Financial support by the National Agency for Research (ANR) — research programs “Models of Influence and Network Theory” (MINT) ANR 09.BLANC-0321.03 and “Mathématiques de la décision pour lʼingénierie physique et sociale” (MODMAD) — is gratefully acknowledged.

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An optimal bound to access the core in TU-games☆

Highlights

Abstract

Discrete Appl. Math.

J. Econ. Theory

J. Math. Econ.

J. Math. Anal. Appl.

Games Econ. Behav.

Games Econ. Behav.

The kernel of a cooperative game

Nav. Res. Logist. Quart.

Extendability and von Neuman–Morgenstern stability of the core

Int. J. Game Theory