Abstract
For each cooperativen-person gamev and eachh∈{1, 2, ⋯,n}, letv h be the average worth of coalitions of sizeh andv i h the average worth of coalitions of sizeh which do not contain playeri∈N. The paper introduces the notion of a proportional average worth game (or PAW-game), i.e., the zero-normalized gamev for which there exist numbersc h ∈ℝ such thatv h −v i h =c h (v n−1−v −1/i n ) for allh∈{2, 3, ⋯,n−1}, andi∈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 non-separable contribution value and the egalitarian non-average contribution value of a PAW-game are collinear. The class of PAW-games contains strictly the class ofk-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.
Zusammenfassung
Seiv ein kooperativesn-Personenspiel und seih∈{1, 2, ⋯,n}. Mitv h bezeichnen wir die mittlere Auszahlung aller Koalitionen der Größeh und mitv i h die mittlere Auszahlung aller Koalitionen der Größeh, die den Spieleri∈N nicht enthalten. In dieser Arbeit, führen wir den Begriff des Spieles mit proportionaler mittlerer Auszahlung (oder PMA-Spiel) ein. Diese sind null-reduzierte Spielev, für die Zahlenc h ∈ℝ existieren, sodaß die Beziehungv h −v i h =c h (v n−1−v −1/i n ) für jedesh∈{2, 3, ⋯,n−1 undi∈N gilt. Der Begriff der mittleren Auszahlung wird dann benutzt, um eine Formel für den Shapley-Wert der PMA-Spiele abzuleiten. Wir zeigen, daß der Shapley-Wert, und die durch das Zentrum der Imputationsmenge, die gleichmäßigen nicht-separablen Beiträge, bzw. gleichmäßigen nicht-gemittelten Beiträge definierten Werte der PMA-Spiele kollinear sind. Die Klasse aller PMA-Spiele enthält im strengen Sinne die Klasse allerk-Koalitionsspiele, die die Kollinearitätseigenschaft haben (Driessen und Funaki, 1991). Schließlich zeigen wir, daß die Einstimmigkeitsspiele und die Grundbesitzerspiele auch PMA-Spiele sind.
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Dragan, I., Driessen, T. & Funaki, Y. Collinearity between the Shapley value and the egalitarian division rules for cooperative games. OR Spektrum 18, 97–105 (1996). https://doi.org/10.1007/BF01539733
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DOI: https://doi.org/10.1007/BF01539733