Linear Transforms, Values and Least Square Approximation for Cooperation Systems

We study linear properties of TU-games, revisiting well-known issues like interaction transforms, the inverse Shapley value problem and the concept of semivalues and least square values. We embed TU-games into the model of cooperation systems and influence patterns, which allows us to introduce linear operators on games in a natural way. We focus on transforms, which are linear invertible maps, relate them to bases and investigate many examples (Möbius transform, interaction transform, Walsh transform, etc.). In particular, we present a simple solution to the inverse problem in its general form: Given a linear value Φ and a game v, find all games v′ such that Φ(v) = Φ(v′). Generalizing Hart and Mas-Colell’s concept of a potential, we introduce general potentials and show that every linear value is induced by an appropriate potential. We furthermore develop a general theory of allocations with a quadratic optimality criterion under linear constraints, obtaining results of Charnes et al., and Ruiz et al. and others as special cases. We prove that this class of allocations coincides exactly with the class of all linear values.