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Values for Interior Operator Games J.M. BILBAO.pdf

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Values for Interior Operator Games J.M. BILBAO

Annals of Operations Research 137, 141–160, 2005 c? 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. Values for Interior Operator Games J.M. BILBAO mbilbao@us.es (http://www.esi2.us.es/?mbilbao/) A. JIME?NEZ-LOSADA E. LEBRO?N C. CHACO?N Matema?tica Aplicada II, Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain Abstract. The aim of this paper is to study a new class of cooperative games called interior operator games. These games are additive games restricted by antimatroids. We consider several types of cooperative games as peer group games, big boss games, clan games and information market games and show that all of them are interior operator games. Next, we analyze the properties of these games and compute the Shapley, Banzhaf and Tijs values. Keywords: cooperative games, antimatroids, Shapley value, Banzhaf value, Tijs value 1. Introduction A cooperative game describes a situation in which a finite set of n players can generate certain payoffs by cooperation. A value for cooperative games is a function which as- signs to every cooperative game a n-dimensional real vector which represents a payoff distribution over the players. Two well-known values are the Shapley value as proposed by Shapley (1953), and the Banzhaf value, initially introduced in the context of vot- ing games by Banzhaf (1965), and later on extended to arbitrary games by Dubey and Shapley (1979). Compromise values are a special type of values which assign to each game a payment based on two vectors. These vectors, so-called upper and lower, are the maximum and the minimum payment that the players can expect to get. The Tijs value (Tijs, 1981) was the first compromise value introduced for cooperative games. In a cooperative game the players are assumed to be identical in the sense that every player can cooperate with every other player. However, in practice there exist asymmetries among the players. For this reason, the game theore