A new axiomatization of the Banzhaf semivalue

Abstract

A new characterization of the Banzhaf semivalue on the domain of monotonic simple games is given. We use the well-known valuation and dummy axioms plus two additional properties. The first one simply requires that the power-index be bigger for those players belonging to more winning coalitions. The second one is the proportionality axiom introduced by Owen in (1982) which is suitable for those simple games that represent an indirect voting process. JEL classification: C71

La escuela en la sociedad red: Un espacio de encuentro con los valores

“…Este progreso tecnológico en el que los medios de comunicación, Internet o las tecnologías están presentan en nuestra cotidianeidad genera el surgimiento de cuestiones éticas que pueden ser complejas de afrontar, pero necesarias (Albizuri, Samaniego & Torrientes, 2001). La multiculturalidad actual es a su vez, junto con estos elementos, un reto educativo fundamental, porque se presenta como la oportunidad de poder desarrollar una ciudadanía democrática y dialógica, que favorezca el intercambio cultural y la comprensión social.…”

Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value

“…A version of it was initially suggested by Penrose (1946), followed by two subsequent rediscoveries by Banzhaf (1965Banzhaf ( , 1966Banzhaf ( , 1968) and Coleman (1971). 1 The much-used probabilistic version described above 2 has its origin in the work of Dubey and Shapley (1979), who initiated the study of the Banzhaf index in the game-theoretic framework. Following the approach of Shapley and Shubik (1954), they model a voting situation as a simple cooperative game (or voting game); the Banzhaf index of a player (voter) is then the probability that he is a swinger for a random coalition of other players (which each player joins with probability 1 2 , independently of anyone else), meaning that he turns that coalition from losing to winning by joining it.…”

“…1 The much-used probabilistic version described above 2 has its origin in the work of Dubey and Shapley (1979), who initiated the study of the Banzhaf index in the game-theoretic framework. Following the approach of Shapley and Shubik (1954), they model a voting situation as a simple cooperative game (or voting game); the Banzhaf index of a player (voter) is then the probability that he is a swinger for a random coalition of other players (which each player joins with probability 1 2 , independently of anyone else), meaning that he turns that coalition from losing to winning by joining it. 3 The simple probabilistic model upon which the Banzhaf index is based has a fascinating implication for measuring voting power in compound voting.…”

“…3 The simple probabilistic model upon which the Banzhaf index is based has a fascinating implication for measuring voting power in compound voting. The latter 1 The name of John F. Banzhaf III has been the one most associated with that power index, due to the number of works he authored on the subject and the legal repercussions of his …ndings and recommendations. Thus, siding with most of the literature, we will use the term "Banzhaf power index" for brevity, although, and perhaps more appropriately, the index is sometimes referred to as the Penrose-Banzhaf-Coleman power index.…”

Shapley-Shubik and Banzhaf Indices Revisited

“…These indices had no axiomatic foundation till Dubey (1975) and Dubey and Shapley (1979) axiomatically characterized them on the class of simple games. Since then, several axiomatizations of both indices have been proposed (Owen 1978, Bolger 1982, Lehrer 1988, Haller 1994, Feltkamp 1995, Nowak 1997, Albizuri and Ruiz 1999, and Khmelnitskaya 1999. The main motivations of these papers are either the mathematical challenge of finding a self-contained characterization in subclasses of TU-games (like simple games or simple superadditive games) or the lack of intuitive appeal of some axioms.…”