Values and Derivatives

The diagonal formula in the theory of nonatomic games expresses the idea that the Shapley value of each infinitesimal player is his marginal contribution to the worth of a “perfect sample” of the population of all players, when averaged over all possible sample sizes. The concept of marginal contribution is most easily expressed in terms of derivatives; as a result, the diagonal formula has heretofore only been established for spaces of games that are in an appropriate sense differentiable (such as pNA or pNAD). In this paper we use an averaging process to reinterpret and then prove the diagonal formula for much larger spaces of games, including spaces (like bv'NA) in which the games cannot be considered differentiable and may even have jumps (e.g., voting games). The new diagonal formula is then used to establish the existence of values on even larger spaces of games, on which it had not previously been known that there exists any operator satisfying the axioms for the Shapley value.