On a class of vertices of the core

Michel Grabisch, Peter Sudhölter

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

It is known that for supermodular TU-games, the vertices of the core are the marginal vectors, and this result remains true for games where the set of feasible coalitions is a distributive lattice. Such games are induced by a hierarchy (partial order) on players. We propose a larger class of vertices for games on distributive lattices, called min–max vertices, obtained by minimizing or maximizing in a given order the coordinates of a core element. We give a simple formula which does not need to solve an optimization problem to compute these vertices, valid for connected hierarchies and for the general case under some restrictions. We find under which conditions two different orders induce the same vertex for every game, and show that there exist balanced games whose core has vertices which are not min–max vertices if and only if n>4.

Original languageEnglish
JournalGames and Economic Behavior
Volume108
Issue numberMarch
Pages (from-to)541-557
ISSN0899-8256
DOIs
Publication statusPublished - 1 Mar 2018

Fingerprint

Optimization problem
Partial order
TU game
Balanced games

Keywords

  • Core
  • Game with precedence constraints
  • Restricted cooperation
  • TU games
  • Vertex

Cite this

Grabisch, Michel ; Sudhölter, Peter. / On a class of vertices of the core. In: Games and Economic Behavior. 2018 ; Vol. 108, No. March. pp. 541-557.
@article{fe64954e041e437e9e7be0800ee90f18,
title = "On a class of vertices of the core",
abstract = "It is known that for supermodular TU-games, the vertices of the core are the marginal vectors, and this result remains true for games where the set of feasible coalitions is a distributive lattice. Such games are induced by a hierarchy (partial order) on players. We propose a larger class of vertices for games on distributive lattices, called min–max vertices, obtained by minimizing or maximizing in a given order the coordinates of a core element. We give a simple formula which does not need to solve an optimization problem to compute these vertices, valid for connected hierarchies and for the general case under some restrictions. We find under which conditions two different orders induce the same vertex for every game, and show that there exist balanced games whose core has vertices which are not min–max vertices if and only if n>4.",
keywords = "Core, Game with precedence constraints, Restricted cooperation, TU games, Vertex",
author = "Michel Grabisch and Peter Sudh{\"o}lter",
year = "2018",
month = "3",
day = "1",
doi = "10.1016/j.geb.2017.09.001",
language = "English",
volume = "108",
pages = "541--557",
journal = "Games and Economic Behavior",
issn = "0899-8256",
publisher = "Heinemann",
number = "March",

}

On a class of vertices of the core. / Grabisch, Michel; Sudhölter, Peter.

In: Games and Economic Behavior, Vol. 108, No. March, 01.03.2018, p. 541-557.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - On a class of vertices of the core

AU - Grabisch, Michel

AU - Sudhölter, Peter

PY - 2018/3/1

Y1 - 2018/3/1

N2 - It is known that for supermodular TU-games, the vertices of the core are the marginal vectors, and this result remains true for games where the set of feasible coalitions is a distributive lattice. Such games are induced by a hierarchy (partial order) on players. We propose a larger class of vertices for games on distributive lattices, called min–max vertices, obtained by minimizing or maximizing in a given order the coordinates of a core element. We give a simple formula which does not need to solve an optimization problem to compute these vertices, valid for connected hierarchies and for the general case under some restrictions. We find under which conditions two different orders induce the same vertex for every game, and show that there exist balanced games whose core has vertices which are not min–max vertices if and only if n>4.

AB - It is known that for supermodular TU-games, the vertices of the core are the marginal vectors, and this result remains true for games where the set of feasible coalitions is a distributive lattice. Such games are induced by a hierarchy (partial order) on players. We propose a larger class of vertices for games on distributive lattices, called min–max vertices, obtained by minimizing or maximizing in a given order the coordinates of a core element. We give a simple formula which does not need to solve an optimization problem to compute these vertices, valid for connected hierarchies and for the general case under some restrictions. We find under which conditions two different orders induce the same vertex for every game, and show that there exist balanced games whose core has vertices which are not min–max vertices if and only if n>4.

KW - Core

KW - Game with precedence constraints

KW - Restricted cooperation

KW - TU games

KW - Vertex

U2 - 10.1016/j.geb.2017.09.001

DO - 10.1016/j.geb.2017.09.001

M3 - Journal article

VL - 108

SP - 541

EP - 557

JO - Games and Economic Behavior

JF - Games and Economic Behavior

SN - 0899-8256

IS - March

ER -