### Abstract

For each cooperative n-person game v and each hε{1, 2 . . . . . n}, let ν_{h} be the average worth of coalitions of size h and ν^{i}_{h} the average worth of coalitions of size h which do not contain player iε N. The paper introduces the notion of a proportional average worth game (or PAW-game), i.e., the zero-normalized game v for which there exist numbers c_{h}εℝ such that ν_{h}- ν_{h}^{i}=c_{h} (ν_{n-1}-v _{n-1}^{i}) for all hε{2, 3 . . . . , n-1}, and iε N. The notion of average worth is used to prove a formula for the Shapley value of a PAW-game. It is shown that the Shapley value, the value representing the center of the imputation set, the egalitarian nonseparable contribution value and the egalitarian non-average contribution value of a PAW-game are collinear. The class of PAW-games contains strictly the class of k-coalitional games possessing the collinearity property discussed by Driessen and Funaki (1991). Finally, it is illustrated that the unanimity games and the landlord games are PAW-games.

Original language | English |
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Pages (from-to) | 97-105 |

Number of pages | 9 |

Journal | OR Spectrum |

Volume | 18 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1996 Jan 1 |

Externally published | Yes |

### Keywords

- Cooperative game
- Egalitarian division rules
- PAW-game
- Shapley value
- k-Coalitional game

### ASJC Scopus subject areas

- Management Science and Operations Research

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## Cite this

*OR Spectrum*,

*18*(2), 97-105. https://doi.org/10.1007/BF01539733