On coalitional semivalues

• Autores: Miren Iosune Albizuri Irigoyen, José Manuel Zarzuelo Zarzosa
• Localización: Abstracts of the Fifth Spanish Meeting on Game Theory and Applications / coord. por Jesús Mario Bilbao Arrese, Francisco Ramón Fernández García, 2002, ISBN 84-472-0733-1, pág. 59
• Idioma: inglés
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• Resumen

  • In 1977 Owen defined and axiomatized the coalitional value for games with transferable utility, providing in this way a generalization of the Shapley value to the coalitional framework. Later, Hart and Kurz (1983) gave an alternative axiomatic characterization of the coalitional value by considering games with an infinite universe of players. In this paper we obtain two results. First, we propose another axiomatic characterization of the coalitional value by employing the same games as Hart and Kurz (1983). In fact, in our characterization we employ the same axioms in Hart and Kurz� characterization (1983) except one which is withdrawn. Moreover, we change one of their axioms, namely anonymity, becoming it stronger so that it turns also into a stability axiom.

    Second, we define the coalitional semivalues providing a generalization to the coalitional context of the semivalues, defined by Dubey et al. (1981). These authors obtained the semivalues by removing efficiency from the classical axiomatization of the Shapley value (1953) and by adding other axioms verified by the Shapley value. In our paper, to define the coalitional semivalues, we will employ our characterization of the coalitional value. We will remove also efficiency, we will add the translations to the coalitional framework of the axioms added by Dubey et al. (1981) and we will add a new axiom which is specific for the coalitional context and is verified by the coalitional value. The coalitional semivalues could also be got in a similar way as Owen obtained the coalitional value. Owen (1977) defined the coalitional value of a game by applying twice the Shapley value. First, the Shapley value is employed at the level of the coalitions of the coalitional structure formed by the players, and a new game for each coalition of the coalitional structure is obtained. After that, the Shapley value is applied to these new games. These results yield precisely the coalitional value of the original game. So, we can say that the coalitional value is obtained by means of a �composition� of the Shapley value with itself. In this paper we will show that the coalitional semivalues we have defined can be obtained also by means of a �composition� of semivalues. Furthermore, if one additional axiom is considered, the coalitional semivalues that result are �compositions� of a semivalue with itself. It can be pointed that if we consider the characterization provided by Hart and Kurz (1983) and we define the corresponding coalitional semivalues, we do not obtain all the �compositions� of semivalues, but only those in which a semivalue is �composed� with the Shapley value