Second-order productivity, second-order payoffs, and the Shapley value (Q2231761)

A transferable utility coalitional game in a structure \(G = (N, v)\), where \(N\) is a finite se of players and \(v\) is a function that associates any subset of players \(S \subseteq N\) with its worth \(v(S)\), a real number. Solutions to a coalitional games are distributions of the available overall worth \(v(N)\) to players in \(N\) in a way that assures stability (players are not pushed to leave the grand-coalition \(N\)) or fairness (the solution complies with some general principles). Solutions, in their various forms, are defined to belong to solution concepts, some of which are denoted by points in the space of \(|N|\)-valued real vectors, while others as sets. Among the single-point solution concept, the so called Shapley value occupies a prominent place. As many other solution concept, the Shapley value can be proved to be the only solution that complies with sets of axioms as, for instance, efficiency, symmetry and strong monotonicity. In this paper, the authors introduce new properties of games encoding, so as to say, second order properties, that are, properties pertaining pairs of players with respect to others, other that the more usual properties pertaining one player with respect to the others. Basically, such second order properties are related to the influence of a player \(i\) on the productivity of another player \(j\). In particular, the authors introduce second-order versions of symmetry, marginality, and strong monotonicity properties and then prove several interesting results which refer to these newly introduced concepts, their main result being to show that the Shapley value is the unique solution that satisfies efficiency, second-order symmetry, and second-order marginality or second-order strong monotonicity.