Characterization of Binomial Semivalues and their Modifications

• Autores: José Miguel Giménez Pradales, Rafael Amer Ramon
• 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. 93
• Idioma: inglés
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• Resumen

  • Each semivalue defined on a space of cooperative games with transferable utility comes determined by a vector of weights. These coefficients offer the payment to the different players weighting the marginal contributions from each one of them according to the cardinality of the coalitions to which they belong.

    Those semivalues whose weighting coefficients are arranged in geometric progression form the family that we designate as binomial semivalues. The Banzhaf value is the particular case in which the reason is the unit. For this solution different systems of axiomatic characterization exist, like the proposed by Owen in 1978, Feltkamp in 1988 or Lehrer in 1995.

    In this work we propose a characterization for each one of the members of the binomial family. We obtain these systems of characterization from the hand of a family of games that are introduced previously under the name of delegation games, which allow to establish an exclusive property for each one of the members of the family who is wanted to characterize. It deserves a special consideration the extreme cases corresponding to the marginal index and the dictatorial index.

    On the other hand, each binomial semivalue has a natural extension to a concept of solution modified for games with coalition structure, like the value of Shapley leads to the coalitional value of Owen (1977). For each member of this family of solutions also an axiomatic characterization sets out, so that this procedure includes to obtain an axiomatic system for the solution of Banzhaf modified for games with coalition structure.