What Semivalues Cannot See: The Information Content of Anonymous Marginal Values

The semivalue family shares a common kernel: games invisible to every anonymous marginal value at once, nonzero from four players (Kleinberg and Weiss, 1985; Amer, Derks and Giménez, 2003). Crisman and Orrison (2015) ask what useful structure this kernel carries; this paper gives a concrete answer. In Harsanyi-dividend coordinates the joint information of all semivalues is exactly each player’s total synergy at each coalition size, so the kernel is synergy arranged in closed circuits. We prove: order-$\le d$ mixed-difference audits recover exactly the degree-$\le d$ dividend-slice harmonics, with closed-form dimension at every rung; nonzero blind games fail superadditivity, monotonicity, and core existence, yet distinct convex games with identical values under every semivalue exist from four players, with exact perturbation thresholds; the positive weighted Shapley family attains full information $2^n-1$, so anonymity is the binding axiom within the marginal framework; and a coalition of size $c$ defeats every audit of order $\le d$ precisely when $c\ge 2d+2$, within the convex class for small perturbations. An exhaustive census at $n=5$ exhibits non-isomorphic voting rules with identical values under every semivalue power index; no weighted game participates in any collision, prompting a swing-rigidity conjecture. Measured against the theory, classical cooperative games sit at $0.90$ to $1.00$ visibility to the family versus $0.089$ for a random game.