### 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 |
---|---|

Pages (from-to) | 97-105 |

Number of pages | 9 |

Journal | OR Spectrum |

Volume | 18 |

Issue number | 2 |

Publication status | Published - 1996 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Management Science and Operations Research

### Cite this

*OR Spectrum*,

*18*(2), 97-105.

**Collinearity between the Shapley value and the egalitarian division rules for cooperative games.** / Dragan, Irinel; Driessen, Theo; Funaki, Yukihiko.

Research output: Contribution to journal › Article

*OR Spectrum*, vol. 18, no. 2, pp. 97-105.

}

TY - JOUR

T1 - Collinearity between the Shapley value and the egalitarian division rules for cooperative games

AU - Dragan, Irinel

AU - Driessen, Theo

AU - Funaki, Yukihiko

PY - 1996

Y1 - 1996

N2 - For each cooperative n-person game v and each hε{1, 2 . . . . . n}, let νh be the average worth of coalitions of size h and νi 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 chεℝ such that νh- νh i=ch (νn-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.

AB - For each cooperative n-person game v and each hε{1, 2 . . . . . n}, let νh be the average worth of coalitions of size h and νi 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 chεℝ such that νh- νh i=ch (νn-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.

KW - Cooperative game

KW - Egalitarian division rules

KW - k-Coalitional game

KW - PAW-game

KW - Shapley value

UR - http://www.scopus.com/inward/record.url?scp=33746676171&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33746676171&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33746676171

VL - 18

SP - 97

EP - 105

JO - Operations-Research-Spektrum

JF - Operations-Research-Spektrum

SN - 0171-6468

IS - 2

ER -