Reduced game properties of egalitarian division rules for cooperative games

Abstract

The paper focuses on a uniform treatment of a special type of one-point solutions for cooperative games, called the egalitarian non-individual contribution (ENIC-)value. The value in question is special in the sense that it is constructed in two stages: first and foremost, each player in the game is allocated some kind of a yet unspecified individual contribution and subsequently, the surplus of the total profits is equally divided among all the players. In addition to the general setting, the unspecified notion of individual contribution will be chosen to be one of the following four particular notions:

• the individual worth of the player to participate in the game as a solitary player. The corresponding ENIC-value is known as the center of the imputation set, called CIS-value.

• the so-called separable (or marginal) contribution of the player to participate in the game as a member of the player set. The corresponding ENIC-value goes by the name of egalitarian non-separable contribution (ENSC-)value. (cf. Driessen and Funaki (1991)).

• the so-called average contribution of the player in the game which is obtainable as some average of -marginal contributions of pairs of players including the player in question. The corresponding ENIC-valúe is called the egalitarian non-average contribution (ENAC-)value and has been studied in detail in Driessen and Funaki (1993).

• the so-called Banzhaf contribution of the player in the game which is interpretable as the average of all the player’s marginal contributions to participate in the game anyhow. The corresponding ENIC-value is called the egalitarian non-Banzhaf contribution (ENBC-)value. The notion of Banzhaf contribution itself is closely related to the well-known Shapley value as well as the Banzhaf index for simple games.