The Shapley NTU-Value via Surface Measures

We introduce the Maschler-Perles-Shapley value for NTU games composed by smooth bodies. This waywe extend the M-P-S value established for games composed by Cephoids (“sums of deGua Simplices”). The development is parallel to the one of the (generalized) Maschler-Perles bargaining solution. For Cephoidal bargaining problems this concept is treated in ([4], [11]). It is extended to smooth bargaining problems by the construction of surface measures. Such measures generalize the Maschler-Perles approach in two dimensions via a line integral -- what the authors call their “donkey cart” ([6], [11]). The Maschler-Perles-Shapley value for Cephoidal NTU Games extends the Cephoidal approach to Non Transferable Utility games with feasible sets consisting of Cephoids. The presentation is found in [10] and [11]. Using these results we formulate the Maschler-Perles-Shapley value for smooth NTU games. We emphasize the intuitive justification of our concepts. The original Maschler-Perles approach is based on the axiom of superadditivity which we rate much more appealing than competing axioms like IIA etc. As a consequence, the construction of a surface measure (Maschler-Perles' line integral) is instigated which renders concessions and gains of players during the bargaining process to be represented in a common space of "adjusted utility”. Within this utility space side payments -- transfer of utils -- are feasible interpersonally as well as intrapersonally. Therefore, the barycenter/midpoint of the adjusted utility space is the natural base for the solution concept. This corresponds precisely to the Maschler-Perles “donkey card” reaching the solution by calling for equal concessions in terms of their line integral. In addition, the adjusted utility space carries an obvious linear structure -- thus admitting expectations in the sense of the Shapley value or "von Neumann-Morgenstern utility". Consequently, we obtain a generally acceptable concept for bargaining problems as well as NTU games in the Cephoidal and in the smooth domain. We collect the details of this reasoning along the development of our theory in Remarks 1.7., 2.6., and 3.2.. These remarks constitute a comprehensive view on M-P-S concepts for Cephoidal and smooth NTU games.