Beyond Midpoint Equivalence: A Full Axiomatic Characterization of the Moore–Shapley Value Under Interval Uncertainty

A cooperative interval game assigns a range of possible worths, rather than a single number, to each coalition. For the Shapley-like rule based on Moore subtraction, existing axioms determine the midpoint of each player’s allocated interval but need not determine its lower and upper endpoints. We close this identification gap on the full class of interval games. In midpoint–radius coordinates, the center is the ordinary Shapley value, whereas each Moore marginal adds the radii of its predecessor and successor coalitions. We introduce two requirements for this radius component. Total Moore exposure fixes the aggregate radius generated along random permutation chains, and boundary exposure balance divides each proper coalition’s contribution equally, in aggregate, between its members and nonmembers. The familiar center axioms together with these two conditions uniquely recover the Moore–Shapley rule, including both endpoints. Without boundary exposure balance, one insider-share parameter remains for each proper coalition size, yielding an (n−1)-dimensional family. We also derive an aggregate-envelope identity and the sharp Hausdorff–Lipschitz constant, establish logical independence of the six axioms for n≥3, and give an exact O(n2n−1) algorithm together with an unbiased permutation-sampling alternative. A stylized, non-empirical four-partner storage example separates total exposure from its allocation across players. All results concern the original Moore-based rule rather than other interval solution concepts.