### 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 language | English |
---|---|

Journal | Games and Economic Behavior |

Volume | 108 |

Issue number | March |

Pages (from-to) | 541-557 |

ISSN | 0899-8256 |

DOIs | |

Publication status | Published - 1. Mar 2018 |

### Fingerprint

### Keywords

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

### Cite this

*Games and Economic Behavior*,

*108*(March), 541-557. https://doi.org/10.1016/j.geb.2017.09.001

}

*Games and Economic Behavior*, vol. 108, no. March, pp. 541-557. https://doi.org/10.1016/j.geb.2017.09.001

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

Research output: Contribution to journal › Journal article › Research › peer-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 -