# Representations Of Positive Projections On Lipschitz Vector

Among the single-valued solution concepts studied in cooperative game theory and economics, those which are also positive projections play an important role. The value, semivalues, and quasivalues of a cooperative game are several examples of solution concepts which are positive projections. These solution concepts are known to have many important applications in economics. In many applications the specific positive projection discussed is represented as an expectation of

marginal contributions of agents to ``random" coalitions. Usually these representations are used to characterize positive projections obeying certain additional axioms. It is thus of interest to study the representation theory of positive projections and its relation with some common axioms. We study positive projections defined over certain spaces of nonatomic Lipschitz vector measure games. To this end, we develop a general notion of ``calculus" for such games, which in a manner extends the notion of the Radon-Nykodim derivative for measures. We prove several representation results for positive projections, which essentially state that the image of a game under the action of a positive projection can be represented as an averaging of its derivative w.r.t. some vector measure. We then introduce a specific calculus for the space $\mathcal{CON}$ generated by concave, monotonically nondecreasing, and Lipschitz continuous functions of finitely many nonatomic probability measures. We study in detail the properties of the resulting representations of positive projections on $\mathcal{CON}$ and especially those of values on $\mathcal{CON}$. The latter results are of great importance in various applications in economics.