{"ID":6138087,"CreatedAt":"2026-07-09T01:07:32.349475501Z","UpdatedAt":"2026-07-11T05:01:04.438793932Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.07013","arxiv_id":"2607.07013","title":"What Semivalues Cannot See: The Information Content of Anonymous Marginal Values","abstract":"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 $c\\ge2d+2$, 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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2607.07013","url_pdf":"https://arxiv.org/pdf/2607.07013v1","authors":"[\"Matthew Fried\"]","published":"2026-07-08T05:14:06Z","proceeding":"cs.GT","tasks":"[\"cs.GT\"]","methods":"[]","has_code":false}