k-Balanced games and capacities

In this paper we present a generalization of the concept of balanced game for finite games. Balanced games are those having a nonempty core, and this core is usually considered as the solution of the game. Based on the concept of k-additivity, we define the so-called k-balanced games and the corresponding generalization of core, the k-additive core, whose elements are not directly imputations but k-additive games. We show that any game is k-balanced for a suitable choice of k, so that the corresponding k-additive core is not empty. For the games in the kadditive core, we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion. Moreover, we look for necessary and sufficient conditions for a game to be k-balanced. For the general case, it is shown ∗Corresponding author: Pedro Miranda. Dept. of Statistics and O.R. Universidad Complutense de Madrid. Address: Plaza de Ciencias, 3, Ciudad Universitaria, 28040, Madrid (Spain). Tel: (+34) 91 394 44 19. Fax: (+34) 91 394 46 06. e-mail: [email protected]