### Abstract

Original language | English |
---|---|

Place of Publication | Tilburg |

Publisher | Department of Econometrics and Operations Research |

Number of pages | 19 |

Volume | 2013-007 |

Publication status | Published - 2013 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 2013-007 |

### Fingerprint

### Keywords

- TU game · Nucleolus · Kernel

### Cite this

*Axiomatizations Of Symmetrically Weighted Solutions*. (CentER Discussion Paper; Vol. 2013-007). Tilburg: Department of Econometrics and Operations Research.

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**Axiomatizations Of Symmetrically Weighted Solutions.** / Kleppe, J.; Reijnierse, J.H.; Sudhölter, P.

Research output: Working paper › Discussion paper › Other research output

TY - UNPB

T1 - Axiomatizations Of Symmetrically Weighted Solutions

AU - Kleppe, J.

AU - Reijnierse, J.H.

AU - Sudhölter, P.

N1 - Pagination: 19

PY - 2013

Y1 - 2013

N2 - If the excesses of the coalitions in a transferable utility game are weighted, then we show that the arising weighted modifications of the well-known (pre)nucleolus and (pre)kernel satisfy the equal treatment property if and only if the weight system is symmetric in the sense that the weight of a subcoalition of a grand coalition may only depend on the grand coalition and the size of the subcoalition. Hence, the symmetrically weighted versions of the (pre)nucleolus and the (pre)kernel are symmetric, i.e., invariant under symmetries of a game. They may, however, violate anonymity, i.e., they may depend on the names of the players. E.g., a symmetrically weighted nucleolus may assign the classical nucleolus to one game and the per capita nucleolus to another game. We generalize Sobolev’s axiomatization of the prenucleolus and its modification for the nucleolus as well as Peleg’s axiomatization of the prekernel to the symmetrically weighted versions. Only the reduced games have to be replaced by suitably modified reduced games whose definitions may depend on the weight system. Moreover, it is shown that a solution may only satisfy the mentioned sets of modified axioms if the weight system is symmetric.

AB - If the excesses of the coalitions in a transferable utility game are weighted, then we show that the arising weighted modifications of the well-known (pre)nucleolus and (pre)kernel satisfy the equal treatment property if and only if the weight system is symmetric in the sense that the weight of a subcoalition of a grand coalition may only depend on the grand coalition and the size of the subcoalition. Hence, the symmetrically weighted versions of the (pre)nucleolus and the (pre)kernel are symmetric, i.e., invariant under symmetries of a game. They may, however, violate anonymity, i.e., they may depend on the names of the players. E.g., a symmetrically weighted nucleolus may assign the classical nucleolus to one game and the per capita nucleolus to another game. We generalize Sobolev’s axiomatization of the prenucleolus and its modification for the nucleolus as well as Peleg’s axiomatization of the prekernel to the symmetrically weighted versions. Only the reduced games have to be replaced by suitably modified reduced games whose definitions may depend on the weight system. Moreover, it is shown that a solution may only satisfy the mentioned sets of modified axioms if the weight system is symmetric.

KW - TU game · Nucleolus · Kernel

M3 - Discussion paper

VL - 2013-007

T3 - CentER Discussion Paper

BT - Axiomatizations Of Symmetrically Weighted Solutions

PB - Department of Econometrics and Operations Research

CY - Tilburg

ER -