Weighted Shapley levels values

This paper presents a collection of four different classes of weighted Shapley levels values. All classes contain generalisations of the weighted Shapley values to cooperative games with a level structure. The first class is an upgrade of the weighted Shapley levels value in Gómez-Rúa and Vidal-Puga (2011), who use the size of components as weights. The following classes contain payoff vectors from the Harsanyi set. Hence they satisfy the dummy axiom, in contrary to the values in the first class in general. The second class contains extensions of the McLean weighted coalition structure values (Dragan, 1992; Levy and McLean, 1989; McLean, 1991). The first two classes satisfy the level game property (the payoff to all players of a component sum up to the payoff to the component in a game where components are the players) and the last two classes meet a null player out property. As a special case, the first three classes include the Shapley levels value and the last class contains a new extension of the Shapley value.