Abstract
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.
| 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
Collinearity between the Shapley value and the egalitarian division rules for cooperative games. / Dragan, Irinel; Driessen, Theo; Funaki, Yukihiko.
In: OR Spectrum, Vol. 18, No. 2, 1996, p. 97-105.Research output: Contribution to journal › Article
}
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 -