A weighted pseudo-potential approach to values for TU games

This paper provides a twofold generalization of the well-known characterization of the Shapley value for TU-games as the discrete gradient of a so-called potential function. On the one hand the potential approach is extended to the so-called weighted pseudo-potential approach in the sense that the extended representation may incorporate, besides a fraction of the discrete gradient, a fraction of the underlying pseudo-potential function itself, as well as a fraction of the average of all the components of the gradient. On the other hand the paper fully characterizes the class of values for TU-games that admit a weighted pseudo-potential representation. Besides two individual constraints, these values have to be efficient, symmetric, and linear. The theory developed is illustrated by several examples of such values and their weighted pseudo-potential representations are discussed.