July 08, 2026
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.
Keywords: semivalues, Shapley value, Harsanyi dividends, interaction indices, simple games, power indices
MSC Classification: 91A12 JEL Classification: C71
Begin with a puzzle every practitioner knows. The Shapley value performs beautifully on the classical applications (airport landing fees, bankruptcy division, where it coincides with O’Neill’s random-arrival rule [1], and cost sharing on networks), yet feels inadequate the moment complementarities take center stage: team formation, data markets, feature attribution. The standard explanations are heuristic (computational cost, axiom debates). This paper gives an exact one. Every anonymous semivalue is a system of totals; a game is a system of arrangements; we compute precisely which arrangements totals can express, measure the classical games against that subspace, and find most of them above \(0.90\) visibility versus \(0.089\) for a random game. This suggests the methodology succeeded where it was applied because where it was applied was unrepresentative, and no internal signal could ever have revealed this, because the missing part is, by the structure of the methods themselves, the part that produces no signal.
The axiomatic program of value theory, initiated by Shapley [2] and systematized by Dubey, Neyman, and Weber [3] and Weber [4], classifies solution concepts by the axioms they satisfy. This paper proposes and executes a complementary classification: by the information they extract. Each value is a linear functional of the game; each family of values spans a subspace of the dual of game space; the dimension and the identity of that subspace are computable invariants of the family. We compute them for the classical families and find the results sharp enough to reorganize how one thinks about the axioms themselves.
Related work, and the division of labor. The structural core of this story is classical, and we use it rather than claim it. Kleinberg–Weiss decomposed game space under \(S_n\) and parametrized all symmetric linear values [5]–[8]; Hernández-Lamoneda–Juárez–Sánchez-Sánchez [9] gave the modern dissection, from which it is a Schur-lemma consequence that symmetric linear values factor through the trivial-plus-standard constituents of each slice; and Amer–Derks–Giménez [10] constructed the common kernel of the semivalue family, the games inseparable by semivalues, proving it nonzero exactly from \(n=4\) and exhibiting a spanning family of \(\{-1,0,1\}\)-valued shuffle games. The visible dimension \(n^2-n+1\) and the kernel dimension \(2^n-n^2+n-2\) follow from either line. The survey of Crisman–Orrison presents this literature and, crediting the question’s origin to the Kleinberg–Weiss papers, poses explicitly whether the common kernel of all linear symmetric solution concepts has useful structure, and what the generalized solution concepts built on the remaining constituents, with “payoffs for pairs, triples, and so on,” would mean [11].
This paper addresses a concrete form of the posed question. The contributions, in order of appearance, with the division from the foundations stated explicitly:
The kernel in Harsanyi coordinates (Theorem 1). The joint information of all semivalues is exactly the per-player, per-size dividend loadings \(L_i(t)\), so the kernel is synergy arranged in closed circuits. The isotypic statement is known [5], [9], [10]; the dividend-loading coordinates and the unitriangular weight map are new, and everything below is stated in them.
The price list (Theorem 7). The order-\(\le d\) interaction functionals of Grabisch–Roubens, precisely the “generalized solutions for pairs and triples” of [11], recover exactly the degree-\(\le d\) dividend-slice harmonics, with closed-form dimension at every rung and full recovery at \(d=\lfloor n/2\rfloor\).
Economic structure of the kernel (Theorems 9 and 11). Nonzero blind games fail superadditivity, monotonicity, and core nonemptiness, yet distinct convex games with identical values under every semivalue exist from four players, with exact perturbation thresholds: regularity excludes purely blind games but does not restore identification.
The cause (Theorem 13). The family of all positive weighted Shapley values attains full information \(2^n-1\), so within the marginal-contribution framework anonymity is the binding axiom at the family level.
The exploitation (Theorem 15). A coalition of size \(c\) has a \((2^c-c^2+c-2)\)-dimensional space of restructurings invisible to every semivalue payment; the full order-\(\le d\) audit family is immune to coalitions of size \(\le2d+1\) and manipulable from \(2d+2\), within the convex class for small perturbations (Corollary 2).
The evidence (§9). Under Definition 1, the classical canon measures \(0.90\) to \(1.00\) visibility against a generic baseline of \(0.089\), which converts the kernel from a curiosity into an account of the Shapley value’s empirical track record.
The census (§10). Exhaustive enumeration of all \(7{,}579\) monotone simple games 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 with a stated search protocol.
Games are \(v\colon2^{[n]}\to\mathbb{R}\) with \(v(\varnothing)=0\); the game space \(V_n\) has dimension \(2^n-1\). Computational support is labeled exact (rational arithmetic), numerical rank (integer matrix ranks computed in floating point), or exhaustive (complete enumeration of a finite class). Every theorem below carries a complete proof; computations support but never substitute for them, with one exception (Conjecture 18) labeled as such.
For \(S\subseteq[n]\) write \(s=|S|\). The Harsanyi dividends [12] of \(v\) are the Möbius coefficients \(d_T=\sum_{R\subseteq T}(-1)^{|T|-|R|}v(R)\), equivalently the unique coefficients with \(v(S)=\sum_{T\subseteq S}d_T\); the map \(v\mapsto d\) is a linear isomorphism of \(V_n\) onto \(\{d:d_\varnothing=0\}\).
A semivalue [3] with weight system \(p=(p_k)_{k=0}^{n-1}\), \(p_k\ge0\), \(\sum_k\binom{n-1}kp_k=1\), assigns player \(i\) the payoff \(\psi_i^p(v)=\sum_{S\not\ni i}p_{|S|}\bigl[v(S\cup i)-v(S)\bigr]\). By Dubey–Neyman–Weber these are exactly the values satisfying linearity, anonymity, positivity, and the dummy axiom. The Shapley value has \(p_k=k!(n-k-1)!/n!\); the Banzhaf value [13] has \(p_k=2^{-(n-1)}\).
The joint information of a family \(\mathcal{F}\) of values is the linear span of the functionals \(\{v\mapsto\phi_i(v):\phi\in\mathcal{F},\,i\in[n]\}\) in \(V_n^*\); its kernel is the common annihilated subspace. For the semivalue family the admissible weight systems form a simplex not containing the origin, whose linear span is all of \(\mathbb{R}^n\); hence the joint information of all semivalues equals \(\mathrm{span}\{m_i(k)\}\) where \(m_i(k)(v)=\sum_{|S|=k,\,S\not\ni i}[v(S\cup i)-v(S)]\).
For \(1\le t\le n\) and \(i\in[n]\) define the dividend loadings \[L_i(t)(v)\;=\;\sum_{\substack{|T|=t,\;i\in T}}d_T .\]
The goal of this section is to answer, in the most concrete coordinates available, the question: if one could observe every semivalue payment at once, what function of the game has one observed? The answer, in isotypic language, is known: symmetric linear values see the trivial-plus-standard part [5], [9], and the common kernel of the semivalue family was constructed in [10]. What is new here is the coordinate system in which the answer becomes usable: the dividend loadings \(L_i(t)\) and the unitriangular weight map \(p\mapsto\gamma_p\), which turn “the trivial-plus-standard part” into the statement each player’s total synergy at each size, and nothing else. Every theorem in the paper is stated and proved in these coordinates; the short self-contained proof keeps the paper independent of the cited machinery.
Theorem 1 (known characterization, new coordinates). The joint information of the family of all semivalues equals \(\mathrm{span}\{L_i(t):i\in[n],\;1\le t\le n\}\). Explicitly, every semivalue expands as \[\label{eq:semidiv} \psi_i^p \;=\; \sum_{t=1}^{n}\gamma_p(t)\,L_i(t), \qquad \gamma_p(t)\;=\;\sum_{m=t-1}^{n-1}\binom{n-t}{\,m-t+1\,}\,p_m ,\tag{1}\] and the map \(p\mapsto\gamma_p\) is a linear bijection of \(\mathbb{R}^n\). Consequently a game is annihilated by every semivalue if and only if \(L_i(t)(v)=0\) for all \(i\) and \(t\): at every synergy size, every player’s net dividend is zero.
Proof. For \(S\not\ni i\), splitting each \(T\subseteq S\cup i\) according to whether \(i\in T\), \[v(S\cup i)-v(S)\;=\;\sum_{\substack{T\subseteq S\cup i\\ i\in T}}d_T \;=\;\sum_{R\subseteq S}d_{R\cup i}.\] Therefore \[\psi_i^p(v)=\sum_{S\not\ni i}p_{|S|}\sum_{R\subseteq S}d_{R\cup i} =\sum_{R\not\ni i}d_{R\cup i}\sum_{\substack{S\supseteq R\\ S\not\ni i}}p_{|S|} =\sum_{R\not\ni i}d_{R\cup i}\sum_{m=|R|}^{n-1}\binom{n-1-|R|}{\,m-|R|\,}p_m ,\] since the sets \(S\) with \(R\subseteq S\subseteq[n]\setminus i\), \(|S|=m\), number \(\binom{n-1-|R|}{m-|R|}\). Grouping by \(t=|R|+1\) and summing \(d_{R\cup i}\) over \(|R|=t-1\) gives 1 with \(\gamma_p(t)\) as stated (note \(n-1-(t-1)=n-t\) and \(m-(t-1)=m-t+1\)).
For the bijectivity of \(p\mapsto\gamma_p\): order coordinates so that \(\gamma(t)\) corresponds to \(p_{t-1}\). The coefficient of \(p_{t-1}\) in \(\gamma(t)\) is \(\binom{n-t}{0}=1\), and the coefficient of \(p_m\) for \(m<t-1\) is \(\binom{n-t}{m-t+1}=0\) (negative lower index). The matrix is therefore unitriangular, hence invertible. Since the admissible weight simplex linearly spans \(\mathbb{R}^n\) (§2), the vectors \(\gamma_p\) over admissible \(p\) linearly span \(\mathbb{R}^n\), so \(\mathrm{span}\{\psi_i^p\}=\mathrm{span}\{L_i(t)\}\). The kernel description is immediate. ◻
Example 1. For distinct \(a,b,c,d\), the multilinear game \(g_{abcd}(S)=(\mathbf{1}_a(S)-\mathbf{1}_d(S))(\mathbf{1}_b(S)-\mathbf{1}_c(S))\) has dividends exactly \(d_{ab}=d_{cd}=+1\), \(d_{ac}=d_{bd}=-1\) (multilinear coefficients are dividends), all of size \(2\); every loading \(L_i(2)\) cancels by inspection, so \(g_{abcd}\) is invisible to every semivalue, while its pairwise structure is maximal. (exact: dividend support and \(L_i(t)=0\) verified in rational arithmetic. A care point: the restriction of \(g_{abcd}\) to the middle cardinality level is also in the kernel but is not dividend-pure; the multilinear \(g_{abcd}\) is the clean object.)
Remark 2 (why four players, and not three). Blindness requires every player’s synergy ledger to balance at every size simultaneously. With three players the pair ledger reads \(d_{12}+d_{13}=0\), \(d_{12}+d_{23}=0\), \(d_{13}+d_{23}=0\): three equations of full rank in three unknowns, so nothing cancels, and the size-\(1\) and size-\(3\) slices are directly visible; hence \(\mathcal{N}_3=0\) (Theorem 5). With four players there are six pair dividends against four balance equations, and the two-dimensional slack is exactly the closed circuit of Example 1: synergy owed around a cycle, and the shortest alternating cycle uses four people. Every \(n\ge4\) admits such circuits, in dimension \(2^n-n^2+n-2\); Proposition 3 shows that all blindness, at every \(n\) and every size, is built from them.
Proposition 3 (a spanning set of arranged synergy). For \(2\le t\le n-2\) and disjoint \(R\), \(\{a,b,c,d\}\) with \(|R|=t-2\), the four-cycle move* \(e_{R\cup ab}+e_{R\cup cd}-e_{R\cup ac}-e_{R\cup bd}\) (in dividend coordinates, on the size-\(t\) slice) is blind, and these moves span the entire kernel \(\mathcal{N}_n\). (The slices \(t\in\{1,\,n-1,\,n\}\) contribute nothing: there the loading rank of Lemma 1 equals the slice dimension, so their kernel is zero.)*
Proof. Blindness: each player’s incidence count in the move is \(0\) (\(a\): \(+1-1\); likewise \(b,c,d\); elements of \(R\): \(+2-2\)), so all loadings vanish. Spanning: fix \(2\le t\le n-2\); the kernel on the size-\(t\) slice is the orthogonal complement of the row space of the incidence matrix \(W\) of Lemma 1, so, the moves lying in the kernel, it suffices to show that any vector \(z\) on the slice orthogonal to every move lies in the row space of \(W\), that is, \(z_T=\sum_{i\in T}\mu_i\) for some \(\mu\). Orthogonality to the moves says \(z_{R\cup ab}+z_{R\cup cd}=z_{R\cup ac}+z_{R\cup bd}\) for all disjoint \(R,\{a,b,c,d\}\) (\(*\)). For \(x\in T\), \(y\notin T\) put \(\Delta(x,y;T)=z_{(T\setminus x)\cup y}-z_T\).
Step 1: \(\Delta\) does not depend on \(T\). If \(T'=(T\setminus a)\cup b\) with \(a,b\notin\{x,y\}\), then with \(R=T\setminus\{x,a\}\) the instance of (\(*\)) on the quadruple \(\{a,b,x,y\}\) reads \(z_{R\cup ay}+z_{R\cup xb}=z_{R\cup by}+z_{R\cup xa}\), which is exactly \(\Delta(x,y;T)=\Delta(x,y;T')\). Any two sets containing \(x\) and avoiding \(y\) are connected by such swaps inside \([n]\setminus\{x,y\}\), since the \((t-1)\)-subsets of an \((n-2)\)-set form a connected Johnson graph for \(t\le n-2\). Write \(\Delta(x,y)\). (At the boundary \(t=n-2\) each \(R\) leaves exactly four elements, so exactly one quadruple, and both conditions hold with equality.)
Step 2: cocycle. For distinct \(x,y,w\) choose \(T\) containing \(x\) and avoiding \(y\) and \(w\) (possible since \(t\le n-2\)); performing the swaps \(x\to y\) then \(y\to w\) on \(T\) and telescoping gives \(\Delta(x,y)+\Delta(y,w)=\Delta(x,w)\), and \(\Delta(x,y)=-\Delta(y,x)\). Fixing \(x_0\) and setting \(\nu(x)=\Delta(x_0,x)\), \(\nu(x_0)=0\), every increment is \(\Delta(x,y)=\nu(y)-\nu(x)\).
Step 3: potential. Fix a base set \(T_0\). Any \(T\) is reached from \(T_0\) by single-element swaps, each changing \(z\) by \(\nu(\text{in})-\nu(\text{out})\); by Step 2 the total is path-independent, so \(z_T=z_{T_0}+\sum_{i\in T}\nu(i)-\sum_{i\in T_0}\nu(i)\). Setting \(\mu_i=\nu(i)+\bigl(z_{T_0}-\sum_{j\in T_0}\nu(j)\bigr)/t\) gives \(z_T=\sum_{i\in T}\mu_i\), as required. (numerical rank: the moves attain the kernel dimension \(\binom nt-n\) on every slice for \(n=5,6\), all \(t\).) ◻
Remark 4. Amer, Derks and Giménez [10] exhibit a spanning family of shuffle games for the same kernel. The four-cycle moves are proved to span independently above; we have not verified the change of basis relating the two families.
The goal of this section is to count what the entire family sees. Theorem 1 reduces the count to linear algebra: the visible functionals \(L_i(t)\) live on disjoint dividend slices, so it suffices to compute, on each slice, the rank of the player-versus-coalition incidence pattern. The lemma below does this by a Gram-matrix computation rather than by the shortest available argument, deliberately: the computation is the base case (\(s=1\)) of Gottlieb’s inclusion-matrix theorem, the one external tool the hierarchy theorem of the next section relies on, and proving the base case directly keeps the paper self-contained where it can be.
Lemma 1. Fix \(1\le t\le n-1\) and let \(W\) be the \(n\times\binom nt\) incidence matrix \(W_{i,T}=\mathbf{1}[i\in T]\) over the size-\(t\) sets. Then \(\operatorname{rank}W=n\). For \(t=n\) the loadings \(L_i(n)\) all equal \(d_{[n]}\) and have joint rank \(1\).
Proof. \((WW^{\!\top})_{ij}\) counts size-\(t\) sets containing \(i\) and \(j\): \(\binom{n-1}{t-1}\) on the diagonal, \(\binom{n-2}{t-2}\) off it, so \(WW^{\!\top}=\binom{n-2}{t-1}I+\binom{n-2}{t-2}J\), with eigenvalues \(\binom{n-2}{t-1}>0\) (multiplicity \(n-1\)) and \(\binom{n-2}{t-1}+n\binom{n-2}{t-2}>0\). (This is the \(s=1\) case of Gottlieb’s inclusion-matrix theorem [14], proved directly here. A three-line alternative: if \(\sum_ic_ix_i\) vanishes on the slice, then comparing any \(T\ni i\), \(T\not\ni j\) with \((T\setminus i)\cup j\) gives \(c_i=c_j\), so all coefficients equal a common \(c\), and the sum equals \(tc\) on every set, forcing \(c=0\).) ◻
Theorem 5 (known; cf.[5], [9], [10]). \(\dim\mathrm{span}\{L_i(t)\}=n^2-n+1\), hence \(\dim\mathcal{N}_n=2^n-n^2+n-2\); in particular \(\mathcal{N}_n=0\) for \(n\le3\) and \(\dim\mathcal{N}_n/(2^n-1)\to1\).
Proof. The functionals \(L_i(t)\) are supported on disjoint dividend slices, so the span decomposes over \(t\). By Lemma 1 each slice \(1\le t\le n-1\) contributes exactly \(n\), and \(t=n\) contributes \(1\): total \(n(n-1)+1\). Subtract from \(\dim V_n=2^n-1\). For \(n=3\): \(8-9+3-2=0\); for \(n=4\): \(2\); the ratio claim is immediate. (numerical rank: fingerprint and dividend-loading matrices have equal span, with stacked rank unchanged, for \(3\le n\le9\), with kernel dimensions \(0,2,10,32,84,198,438\) matching the formula.) ◻
Remark 6. Theorem 5 can be read as an impossibility theorem for the Dubey–Neyman–Weber axioms: any value satisfying linearity, anonymity, positivity, and dummy is constitutionally confined to an \((n^2-n+1)\)-dimensional window on a \((2^n-1)\)-dimensional world. \(n=3\) is the unique nontrivial player count at which the window is the world.
This section computes what it costs to see what semivalues miss. Section 4 says the family reads only player totals; the natural repair is to probe pairs, triples, and higher tuples directly. This section computes exactly what each probe order buys: which part of the game the order-\(\le d\) family recovers, and its dimension in closed form, so that the residual left invisible by any audit depth is a formula rather than an unknown. The proof runs in three steps: a Möbius identity converting mixed differences into dividend sums (Lemma 2); a unitriangular change of variables identifying the probes of a fixed tuple with inclusion pairings (Lemma 3); and a nesting count showing the orders stack into a filtration (Lemma 4). Gottlieb’s theorem then supplies each slice dimension.
The survey [11] asks what generalized solution concepts built on the constituents \(S^{(n-j,j)}\), \(j\ge2\), would mean, observing that “finding meaningful properties…could be challenging.” The meaningful generalized solutions already exist: they are the interaction indices of Grabisch–Roubens [15]. The theorem below computes their exact joint information at every order; the dividend-slice form is what makes the answer explicit.
For \(T\subseteq[n]\) and \(S\cap T=\varnothing\) let \(\Delta_Tv(S)=\sum_{U\subseteq T}(-1)^{|T|-|U|}v(S\cup U)\) (the discrete mixed difference), and define the order-\(|T|\) size-aggregated interaction functionals \[A_T(k)(v)\;=\;\sum_{\substack{|S|=k\\ S\cap T=\varnothing}}\Delta_Tv(S), \qquad 0\le k\le n-|T| .\] We call \(\{A_T(k):|T|\le d\}\) the order-\(\le d\) mixed-difference audit family; the theorem below is stated and proved for this family. The classical interaction indices are size-weighted averages of the same mixed differences: the Grabisch–Roubens and Banzhaf interaction indices are two particular weightings [15], and the cardinal-probabilistic interaction indices allow arbitrary admissible size weights, which span the size levels by the unitriangular argument of Theorem 1. The joint span of the order-\(|T|\) cardinal-probabilistic indices therefore equals \(\mathrm{span}\{A_T(k)\}\), so the theorem is simultaneously the exact information ceiling for the interaction-index approach. On the size-\(s\) dividend slice, define the degree-\(\le j\) subspace of functionals as the span of the inclusion pairings \(d\mapsto\sum_{|W|=s,\;W\supseteq T'}d_W\) over \(|T'|\le j\) (the standard degree filtration of functions on a slice of the Boolean cube; see [16]). Its dimension is supplied by a rank theorem we state explicitly, since it comes from algebraic combinatorics rather than game theory: for \(j\le s\), the inclusion matrix of \(j\)-subsets against \(s\)-subsets of \([n]\) has rank \(\min\bigl(\binom nj,\binom ns\bigr)\) over \(\mathbb{Q}\) [14], which equals \(\binom n{\min(j,s,n-s)}\); this binomial is therefore the dimension of the degree-\(\le j\) subspace.
Lemma 2. For \(S\cap T=\varnothing\): \(\displaystyle \Delta_Tv(S)=\sum_{R\subseteq S}d_{T\cup R}\).
Proof. Each \(W\subseteq S\cup U\) splits uniquely as \(W=R\cup P\) with \(R\subseteq S\), \(P\subseteq U\) (disjointness of \(S\) and \(T\)). Hence \[\Delta_Tv(S)=\sum_{U\subseteq T}(-1)^{|T\setminus U|}\sum_{R\subseteq S}\sum_{P\subseteq U}d_{R\cup P} =\sum_{R\subseteq S}\sum_{P\subseteq T}d_{R\cup P}\!\!\sum_{\substack{U:\,P\subseteq U\subseteq T}}\!\!(-1)^{|T\setminus U|} .\] The inner sum is \((1-1)^{|T\setminus P|}=\mathbf{1}[P=T]\) by the binomial theorem. ◻
Lemma 3. Fix \(T\), \(t=|T|\), and let \(E_T(m)(v)=\sum_{|R|=m,\;R\cap T=\varnothing}d_{T\cup R}\). Then \(A_T(k)=\sum_{m=0}^{k}\binom{n-t-m}{\,k-m\,}E_T(m)\), and consequently \(\mathrm{span}\{A_T(k):0\le k\le n-t\}=\mathrm{span}\{E_T(m):0\le m\le n-t\}\).
Proof. By Lemma 2, \(A_T(k)=\sum_{R\cap T=\varnothing}d_{T\cup R}\cdot\#\{S:\,|S|=k,\;S\supseteq R,\; S\cap T=\varnothing\}\), and the count is \(\binom{n-t-|R|}{k-|R|}\). The coefficient matrix \(\bigl[\binom{n-t-m}{k-m}\bigr]_{k,m}\) has \(\binom{n-t-k}{0}=1\) on the diagonal and \(0\) for \(m>k\), hence is unitriangular and invertible. ◻
Lemma 4 (nesting). On the size-\(s\) slice, the span of inclusion pairings of order exactly \(j\) contains the span of order \(j-1\), for \(1\le j\le s\): summing the order-\(j\) pairings over the \(T\supseteq T_0\) with \(|T|=j\) yields, on each \(W\) with \(|W|=s\), \(\#\{T:\,T_0\subseteq T\subseteq W,\,|T|=j\}=\binom{s-j+1}{1}\cdot\mathbf{1}[T_0\subseteq W]\) when \(|T_0|=j-1\), a positive multiple of the order-\((j-1)\) pairing.
Proof. Direct count: extending \(T_0\) by one element of \(W\setminus T_0\) gives \(s-(j-1)\) choices. ◻
Theorem 7. For every \(d\ge1\), the joint span of the order-\(\le d\) size-aggregated interaction functionals is exactly \(\bigoplus_{s=1}^{n}\{\text{degree-}\le d\text{ subspace of the size-s dividend slice}\}\), of dimension \[\sum_{s=1}^{n}\binom{n}{\min(d,\,s,\,n-s)} .\]
Proof. By Lemma 3, for each \(T\) with \(|T|\le d\) the functionals \(\{A_T(k)\}_k\) span exactly \(\{E_T(m)\}_m\), and \(E_T(m)\) is precisely the inclusion pairing of \(T\) against the size-\((|T|+m)\) dividend slice. Taking the union over \(|T|\le d\), the joint span is the span, on each slice \(s\), of all inclusion pairings of order \(\le\min(d,s)\), which is the degree-\(\le d\) subspace by definition; the orders are nested by Lemma 4, so lower orders add nothing beyond the filtration. The dimension per slice is \(\binom n{\min(d,s,n-s)}\) by Gottlieb’s theorem [14] (for \(d\ge\min(s,n-s)\) the degree filtration saturates at the full slice, of dimension \(\binom ns=\binom n{\min(s,n-s)}\)); summing over slices gives the formula. (numerical rank: dimensions verified for \(d=1\) at \(3\le n\le9\), \(d=2\) at \(4\le n\le7\), \(d=3\) at \(6\le n\le10\), \(d=4\) at \(10\le n\le12\), four consecutive orders and fourteen strict cases, the fifteen cases with \(n\le9\) additionally certified in exact arithmetic over two prime fields; and the span identity itself verified at \(n=6\), \(d=2\): order-\(\le2\) and dividend-slice degree-\(\le2\) systems have equal rank \(58\) with stacked rank unchanged.) ◻
Remark 8. Conceptually: on each size-\(s\) dividend slice, the order-\(\le d\) probes recover exactly the Johnson-scheme components of degrees \(0,\dots,\min(d,s,n-s)\); the inclusion-pairing span used above is the concrete model of those components (Gottlieb for the rank, Filmus for the slice-degree language).
Corollary 1. The exact information recovered at rung \(d\) but not at rung \(d-1\) has dimension \(\sum_s\bigl[\binom n{\min(d,s,n-s)}-\binom n{\min(d-1,s,n-s)}\bigr]\); the residual after rung \(d\) vanishes iff \(d\ge\lfloor n/2\rfloor\). The ladder from the Shapley value to full information is therefore complete, with a closed form at every step.
Section 4 measured how large the blind space is; this section asks whether economically well-behaved games ever occupy it. The answer is a precise split. Purely blind games are pathological, failing every standard regularity property, so a regular economy is never entirely invisible; yet regular economies sit on affine translates of blind directions, so identification from semivalue data fails anyway, with exact thresholds. Theorem 9 gives the exclusions, Theorem 11 the failure.
Call \(v\in\mathcal{N}_n\) blind. Two elementary consequences of blindness do all the work below. First, \(m_i(0)=v(\{i\})\), so every singleton value of a blind game is zero. Second, summing the fingerprints over players gives the identity \(\sum_i m_i(k-1)=k\,B(k)-(n-k+1)\,B(k-1)\) for the slice totals \(B(k)(v)=\sum_{|S|=k}v(S)\); with \(B(1)=0\) from the singletons, induction gives \(B(k)=0\) for every \(k\). A blind game therefore gives every individual nothing and every coalition size nothing in total, so any structure it carries is pure redistribution within slices. The proofs below are then one idea each: regularity forces nonnegativity, and nonnegative games with zero slice totals vanish.
Theorem 9. Let \(v\in\mathcal{N}_n\) be nonzero. Then: (a) \(v\) is not superadditive; in particular \(\mathcal{N}_n\) meets the superadditive cone, and a fortiori the convex cone, only at \(0\). (b) \(v\) is not monotone. (c) the core of \(v\) is empty.
Proof. Blindness gives \(v(\{i\})=0\) for all \(i\) and \(B(k)=0\) for all \(k\) (and \(v([n])=B(n)=0\)).
(a) Suppose \(v\) superadditive. For any \(S=\{i_1,\dots,i_s\}\), iterating \(v(A\cup\{i\})\ge v(A)+v(\{i\})\) along a chain gives \(v(S)\ge\sum_{j}v(\{i_j\})=0\). So \(v\ge0\) everywhere; but each slice sums to \(B(k)=0\), forcing \(v\equiv0\), a contradiction. Convex games are superadditive, so the second claim follows.
(b) Monotone with \(v(\varnothing)=0\) gives \(v\ge0\); conclude as in (a).
(c) Let \(x\in\mathbb{R}^n\) be a core allocation: \(x(S):=\sum_{i\in S}x_i\ge v(S)\) for all \(S\) and \(x([n])=v([n])=0\). For each \(k\), \(\sum_{|S|=k}x(S)=\binom{n-1}{k-1}\sum_ix_i=\binom{n-1}{k-1}\,x([n])=0=B(k)=\sum_{|S|=k}v(S)\). So \(\sum_{|S|=k}[x(S)-v(S)]=0\) with every summand \(\ge0\), forcing \(x(S)=v(S)\) for all \(S\). Then \(v\) is additive, so \(v(\{i\})=x_i\); blindness gives \(x_i=0\), hence \(v\equiv0\), a contradiction. (numerical rank: \(50/50\) random kernel games violate superadditivity and are non-additive; singleton values \(0\) to \(3\times10^{-16}\).) ◻
Remark 10. Theorem 9 says purely blind games are economically pathological. It does not say regular games are semivalue-identifiable; the next theorem shows they are not: the regular cones, while meeting \(\mathcal{N}_n\) only at \(0\), contain nontrivial parallel translates of blind directions.
Theorem 11 (identification failure within the convex cone). Let \(n\ge4\), \(h(S)=|S|^2\), and \(g=g_{abcd}\) from Example 1 for any four distinct players. Then for every \(\varepsilon\) with \(0<\varepsilon<2\), the game \(h+\varepsilon g\) is convex and superadditive (and monotone for \(\varepsilon<3\)), \(h+\varepsilon g\neq h\), and yet \(\psi(h+\varepsilon g)=\psi(h)\) for every semivalue \(\psi\) and every player. The thresholds are exact for this pair. Consequently semivalue data does not identify the game even within the convex cone; Corollary 2 below extends the construction to every audit order.
Proof. Equality of values is Theorem 1 with Example 1 (\(g\in\mathcal{N}_n\), embedded via Lemma 5). Convexity is supermodularity: for \(i\neq j\notin S\) the second difference of \(h\) is \((s+2)^2-2(s+1)^2+s^2=2\) identically, while \(g\) is multilinear of degree \(2\), so its mixed second difference is the constant \(\partial_i\partial_j g\in\{0,\pm1\}\) (Example 1); hence the perturbed second difference is \(\ge2-\varepsilon>0\) for \(\varepsilon<2\), and the value \(-1\) is attained (at \(\partial_a\partial_c\)), making the threshold exact. Monotonicity: the first difference of \(h\) at a set of size \(s\) is \(2s+1\), while \(\partial_a g(S)=\mathbf{1}[b\in S]-\mathbf{1}[c\in S]\) equals \(-1\) only when \(c\in S\), which forces \(s\ge1\) and margin \(\ge3-\varepsilon\); the case \(S=\{c\}\), \(i=a\) binds, so the threshold \(3\) is exact. Superadditivity, directly: for disjoint nonempty \(S,T\) the \(h\)-margin is \((s+t)^2-s^2-t^2=2st\), while the \(g\) cross-term is the sum of the dividends of the support pairs split between \(S\) and \(T\); since \(g\) has exactly two negative pair dividends, \(d_{ac}=d_{bd}=-1\), no split collects less than \(-2\), and two singletons meet at most one support pair, so the cross-term is at least \(-1\) when \(s=t=1\) and at least \(-2\) otherwise, where \(2st\ge4\); the case \(S=\{a\}\), \(T=\{c\}\) gives margin \(2-\varepsilon\) and binds, so the threshold \(2\) is exact. (exact: all three thresholds confirmed in rational arithmetic over all instances at \(n=4\).) ◻
Corollary 2 (convex fibers at every audit order). Let \(h(S)=|S|^2\) and let \(w\) be any non-additive game, so that \(M_2(w)=\max_{i\ne j,\,S}|\Delta_{\{i,j\}}w(S)|>0\). Every mixed second difference of \(h\) equals \(2\), so \(h+\varepsilon w\) is supermodular, hence convex and superadditive, for all \(0<\varepsilon<2/M_2(w)\), and monotone for \(\varepsilon<1/\max_{i,S}|\Delta_{\{i\}}w(S)|\). Every nonzero blind game is non-additive (an additive blind game has zero singleton values, hence vanishes), so the hypothesis is automatic in the application below. In particular, for every \(c\ge2d+2\), taking \(w\) to be a nonzero internal restructuring of a size-\(c\) coalition invisible to the full order-\(\le d\) audit family (Theorem 15(b)) yields, for all sufficiently small \(\varepsilon>0\), two distinct convex games with identical outputs under every interaction functional of order \(\le d\).
Remark 12. The two theorems together give the correct slogan: regularity excludes games whose entire substance is blind, but even strictly convex economies admit families of semivalue-equivalent neighbors of dimension \(2^n-n^2+n-2\). For an analyst, allocation data from the whole semivalue family, every power index and every \(\beta\)-weighted variant, cannot identify the coalitional production function, even under the strongest standard regularity assumptions.
This section locates the cause of the blindness among the Dubey–Neyman–Weber axioms. That some value family attains full information is easy: Weber’s probabilistic values do, by a three-line argument given in Corollary 3. The content of Theorem 13 is that full information is attained already inside the weighted Shapley family, which retains efficiency, the dummy axiom, and positivity, and abandons only anonymity. This isolates anonymity as the single binding axiom within the marginal-contribution framework, and it is not obvious: the dependence on the weights is nonlinear, the relevant measurement matrices are Cauchy matrices, and floating-point computation falsely reports them rank-deficient already at \(n=6\) (Remark 14).
The weighted Shapley value with positive weights \(\lambda\in\mathbb{R}_{>0}^n\) [17], [18] admits the dividend formula \[\varphi_i^{\lambda}(v)\;=\;\sum_{T\ni i}d_T\,\frac{\lambda_i}{\lambda(T)}, \qquad \lambda(T)=\sum_{j\in T}\lambda_j .\]
Theorem 13. For every \(n\), the joint information of the family \(\{\varphi^\lambda:\lambda\in\mathbb{R}_{>0}^n\}\) is all of \(V_n^*\): dimension \(2^n-1\).
Proof. It suffices to show, for each fixed \(i\), that the span of the dividend-coefficient vectors \(c^{i,\lambda}\), where \(c^{i,\lambda}_T=\lambda_i/\lambda(T)\) for \(T\ni i\) and \(0\) otherwise, contains every \(e_T\) with \(T\ni i\); the union over \(i\) then spans all dividend coordinates \(T\neq\varnothing\), and the Möbius isomorphism transports the claim to \(V_n^*\).
Fix \(i\) and restrict to the ray \(\lambda_i=1\), \(\lambda_j=t\,2^{\,j}\) for \(j\neq i\), \(t>0\). For \(T\ni i\) write \(A=T\setminus i\subseteq[n]\setminus i\); then \[c^{i,\lambda(t)}_T=\frac{1}{1+t\,W_A},\qquad W_A=\sum_{j\in A}2^{\,j},\] and the \(W_A\) are pairwise distinct (binary expansions), with \(W_\varnothing=0\). Suppose a fixed functional \(u=(u_A)\) annihilates the family along the ray: \(\sum_A u_A/(1+tW_A)\equiv0\) for all \(t>0\). The left side is a rational function of \(t\); letting \(t\to\infty\) kills every term with \(W_A>0\), forcing \(u_\varnothing=0\). The remaining terms have simple poles at the distinct points \(t=-1/W_A\); the residue at \(-1/W_A\) equals \(u_A/W_A\), and a rational function vanishing identically has all residues zero, so \(u_A=0\) for every \(A\). Hence no nonzero functional annihilates \(\{c^{i,\lambda}\}\), i.e.the span is full. ◻
Remark 14. The coefficient matrix along the ray, \([1/(1+t_kW_A)]_{k,A}\), is a Cauchy matrix: its classical determinant formula gives an alternative finite proof (any \(2^{n-1}\) distinct ray points suffice for the \(W_A>0\) block), and simultaneously explains a numerical trap: Cauchy matrices are exponentially ill-conditioned, and floating-point rank computations falsely report deficiency already at \(n=6\) (float rank \(57\) of \(63\)). exact: rational-arithmetic Gaussian elimination certifies rank \(63/63\) at \(n=6\); numerical rank: \(15/15\) and \(31/31\) at \(n=4,5\).
Corollary 3 (the information ladder). \[\underbrace{\le n}_{\substack{\text{any single value}\\ \text{(weighted or not)}}}\;\subset\; \underbrace{n^2-n+1}_{\text{all semivalues}}\;\subset\; \underbrace{2^n-1}_{\substack{\text{all positive weighted}\\ \text{Shapley values}}}\;=\; \underbrace{2^n-1}_{\text{all probabilistic values}} .\] (For the last equality: Weber’s probabilistic values [4] pay player \(i\) an arbitrary probability mixture of the marginals \(v(S\cup i)-v(S)\) over coalitions \(S\subseteq[n]\setminus\{i\}\); concentrating the mixture on a single \(S\) makes the value output that one marginal, and the marginals determine \(v\) by telescoping along a chain from \(\varnothing\) to \([n]\).) The exponential gap closes at the family level the moment anonymity is dropped: within this framework, the blind spot is the price of equal treatment, and of nothing else.
This section assembles the preceding structure theory into operational statements, and every ingredient built so far is used: the kernel and its dimension (Theorem 5) supply the manipulation space, the price list (Theorem 7) supplies the audit thresholds, the convex fiber (Theorem 11) shows the manipulation survives feasibility constraints, and an embedding lemma localizes all of it to the manipulating coalition. The same fingerprints reappear once more as swing tables in §10, where the manipulation-space geometry becomes visible in an exhaustive census.
Model a coalition \(C\subseteq[n]\), \(|C|=c\), that restructures only its internal dividend structure (singleton and higher-order terms alike): it replaces \(v\) by \(v+w\) where the dividends of \(w\) are supported on \(2^C\setminus\{\varnothing\}\), equivalently \(w(S)=w(S\cap C)\) for all \(S\). (Invisibility itself forces the singleton part to vanish, since \(L_i(1)=d_{\{i\}}\) is visible; the substantive freedom is in arranged synergy of order \(\ge2\), per Proposition 3.)
Lemma 5. For such \(w\), viewed as an \(n\)-player game: \(L_i(t)(w)=0\) for \(i\notin C\), and for \(i\in C\), \(L_i(t)(w)\) equals the corresponding \(c\)-player loading of \(w|_{2^C}\). Hence \(w\in\mathcal{N}_n\) iff \(w|_{2^C}\in\mathcal{N}_c\), and more generally the order-\(\le d\) functionals of the \(n\)-player game restrict to those of the \(c\)-player game.
Proof. Möbius locality: since \(w(S)=w(S\cap C)\), every dividend \(d_T\) with \(T\not\subseteq C\) vanishes (compute \(d_T\) by Möbius over a chain leaving \(C\): the alternating sum telescopes to \(0\)), and dividends with \(T\subseteq C\) agree with the \(c\)-player dividends. The loading identities follow term by term, as does the order-\(\le d\) statement via Lemma 2. ◻
Theorem 15. (a) The space of internal restructurings of \(C\) invisible to every semivalue payment has dimension \(2^c-c^2+c-2\); it is \(0\) for \(c\le3\) and positive from \(c=4\): three players cannot hide, four can. (b) Against a payment scheme using all interaction functionals of order \(\le d\), an invisible internal restructuring of \(C\) exists if and only if \(c\ge2d+2\). Thus degree-\(d\) schemes are immune to coalitions of size \(\le2d+1\) and manipulable from \(2d+2\): each additional interaction order raises the immunity threshold by exactly two coalition sizes.
Proof. (a) By Lemma 5 the space in question is isomorphic to \(\mathcal{N}_c\); apply Theorem 5. (b) By Lemma 5 and Theorem 7, the order-\(\le d\) functionals see, of the \(c\)-player game, a subspace of dimension \(\sum_{s=1}^{c}\binom c{\min(d,s,c-s)}\), which equals \(2^c-1\) iff \(\min(s,c-s)\le d\) for every \(1\le s\le c\), i.e.iff \(\lfloor c/2\rfloor\le d\), i.e. \(c\le2d+1\). (numerical rank: boundary verified for \(d=1,2,3\); e.g.\(d=3\): \(c=7\) fully visible \(127/127\), \(c=8\) deficient \(241/255\).) ◻
Remark 16. In a data market or surplus-division setting paying by any semivalue, (a) says a four-member coalition can re-arrange who actually generates value, shifting synergy between its members along \(\mathcal{N}_4\)-directions as in Example 1, with no member’s payment moving. (b) prices the audit: pairwise-interaction audits stop coalitions of five but not six; order-\(3\) audits stop seven but not eight. By Corollary 2 the restructuring can preserve convexity throughout, at every audit order, so feasibility constraints of the economic environment do not, by themselves, protect the payments.
Theorem 1 makes a measurement possible.
Definition 1 (visibility). Equip \(\mathbb{R}^{2^n}\) (coalition-value coordinates, \(v(\varnothing)=0\)) with the standard Euclidean inner product, and let \(P_{\mathrm{vis}}\) be the orthogonal projector onto \(\mathcal{N}_n^{\perp}\), the span of the semivalue functionals regarded as vectors. The visibility of a nonzero game is \(\|P_{\mathrm{vis}}v\|^2/\|v\|^2\in[0,1]\).
Visibility \(1\) means the semivalue family describes the game completely. For an isotropic Gaussian game the expected visibility is exactly \(\operatorname{tr}P_{\mathrm{vis}}/(2^n-1)=(n^2-n+1)/(2^n-1)\), the dimension ratio, which is the generic baseline. Because the Möbius transform is not orthogonal, the analogous projection in Harsanyi-dividend coordinates (the per-slice incidence row space applied to the dividend vector) is a different statistic; we report it as a robustness check.
Proposition 17 (symmetric games are fully visible; proved). If \(v(S)\) depends only on \(|S|\), then \(v\) has visibility \(1\).
Proof. Every blind game \(w\) has zero slice totals, \(B(k)(w)=\sum_{|S|=k}w(S)=0\) for all \(k\) (§6). Hence for \(u(S)=a_{|S|}\), \(\langle u,w\rangle=\sum_k a_k B(k)(w)=0\): \(u\) is orthogonal to \(\mathcal{N}_n\) in the metric of Definition 1, so its visibility is \(1\). (The dividend-metric value \(1.000\) in the table holds too: the dividends of \(u\) are constant on slices, hence in the degree-\(0\) subspace of each slice.) ◻
Measurement (rank; \(n=10\), exact enumeration of all \(1024\) coalitions; \(100\) instances per stochastic family, \(200\) random games, fixed seeds; mean\(\pm\)sd; generators specified in Appendix 12. The generators are stylized representatives of each family; the table illustrates the theory’s discriminating power rather than establishing a historical claim):
| game | visibility (coalition) | visibility (dividend) |
|---|---|---|
| symmetric convex \(|S|^2\) | \(1.000\) (Prop. 17) | \(1.000\) |
| airport cost | \(0.996\pm0.003\) | \(0.919\pm0.060\) |
| spanning-tree cost | \(0.995\pm0.002\) | \(0.777\pm0.084\) |
| bankruptcy (O’Neill) | \(0.982\pm0.009\) | \(0.536\pm0.154\) |
| glove market (\(5|5\)) | \(0.946\) | \(0.974\) |
| weighted majority | \(0.904\pm0.021\) | \(0.639\pm0.200\) |
| unanimity of a \(4\)-set | \(0.591\) | \(0.048\) |
| random game | \(0.089\pm0.013\) | \(0.640\pm0.112\) |
Three readings. First, the classical canon sits at \(0.90\) to \(1.00\) against a \(0.089\) generic baseline, and the random-game measurement reproduces the dimension ratio \(91/1023=0.0890\) to three decimals: the games economics studied store their content in totals (symmetry, near-symmetry, additivity plus mild curvature), which is exactly what Theorem 1 says semivalues read. Bankruptcy sits at \(0.98\): the O’Neill game concentrates nearly all of its content in totals, consistent with the tight agreement of solution concepts observed on this class [1], [19]. Second, the gradient within the table is the theory speaking: visibility falls as arranged synergy rises, and the unanimity game, whose entire content is one specific coalition’s identity, is the canon’s worst performer at \(0.59\); in dividend coordinates it is \(95\%\) invisible. The dividend column also delimits the claim: the canon remains high under both metrics, but the generic baseline is a coalition-metric statement (a random game reads \(0.64\) in dividend coordinates), so comparisons to the baseline are made in the coalition metric throughout. Third, the prospective reading: modern applications (team formation, data markets, feature and component attribution) are arrangement-driven by their nature, native to the region where visibility collapses toward the dimension ratio, and no output of the family can flag the deficit, since the missing component contributes zero to every output (Theorem 1). The table suggests the classical canon is selected, not representative.
A simple game is monotone \(v\colon2^{[n]}\to\{0,1\}\) with \(v(\varnothing)=0\), \(v([n])=1\); it is proper if no winning coalition has a winning complement, and weighted if representable by a quota and nonnegative weights (see Taylor–Zwicker [20]). Its swing table is the matrix \(T_{i,k}=\#\{S\not\ni i:\,|S|=k,\;v(S\cup i)=1,\;v(S)=0\}\), precisely the fingerprint \(\{m_i(k)\}\), hence by Theorem 1 the complete information any semivalue power index (Shapley–Shubik, Banzhaf, all of them) can use.
Example 2 (exhaustive). We enumerate two universes at \(n=5\) (labeled games throughout). All nontrivial monotone simple games: \(7{,}579\) (the fifth Dedekind number \(7{,}581\) minus the two constants); swing-table fibers contain \(907\) collision classes (\(2{,}524\) games). Proper simple games (no winning coalition has a winning complement): \(2{,}645\), with \(321\) collision fibers containing \(962\) games. A displayable pair, both proper, both non-weighted, and non-isomorphic (verified against all \(120\) player permutations), is given by the minimal winning coalitions \[\{123,\,024,\,034,\,134\} \qquad\text{versus}\qquad \{012,\,034,\,134,\,234\},\] with common swing table \(T_{i,k}\) having rows \((0,0,2,2,0)\) for players \(0,1,2\) and \((0,0,3,3,0)\) for players \(3,4\): identical values for every player under every semivalue power index, structurally different politics.
Conjecture 18 (swing rigidity of weighted games). A weighted voting game is determined by its swing table among all* simple games. Evidence (exhaustive at \(n=5\), in both universes): of the \(2{,}524\) monotone games inside collision classes, zero are weighted, although \(3{,}285\) of \(7{,}579\) are; restricting to proper games, zero of the \(962\) colliding games are weighted although \(1{,}683\) of \(2{,}645\) are (weightedness decided by two independent methods agreeing on every one of the \(7{,}579\) games: LP feasibility, and the swap-robustness test, necessary for weightedness [20], [21]; Appendix 12); and (search) zero collisions occur within the weighted class across \(557\) games at \(n=6\) (weights \(\le9\)) and \(4{,}214\) games at \(n=7\) (weights \(\le8\)). The blind-spot phenomenon thus appears to be a property of general coalition structure that weightedness destroys, in this census and search regime; a proof or a counterexample at larger \(n\) (search protocol in Appendix 13) would each be of independent interest.*
This paper asked what the family of all semivalues can and cannot learn about a cooperative game, and answered exactly. A semivalue observes, for each player and each coalition size, a single number: that player’s total synergy at that size (Theorem 1). That is \(n^2-n+1\) numbers, out of the \(2^n-1\) needed to specify a game (Theorem 5); two games that distribute the same synergy totals among different partners receive identical payments from every semivalue at once. The invisible difference cannot constitute an entire well-behaved economy, since a nonzero invisible game fails superadditivity, monotonicity, and core existence (Theorem 9); but it can be added to a well-behaved economy without detection, since a strictly convex game plus a small invisible perturbation is still convex, genuinely different, and paid identically (Theorem 11). Interaction indices repair the deficit order by order, with the recovered dimension known exactly at each order and full recovery at order \(\lfloor n/2\rfloor\) (Theorem 7). Dropping anonymity repairs it completely: the weighted Shapley values jointly determine the whole game (Theorem 13). A coalition of \(c\) players can exploit it, rearranging its internal synergies without any payment moving, precisely when \(c\ge2d+2\) against order-\(d\) auditing (Theorem 15). The measurements of §9 and the census of §10 show where this matters: barely at all in the classical canon, whose games keep their content in totals, and generically everywhere else.
Axiomatically, the Dubey–Neyman–Weber axioms have an exact information price, and Corollary 3 shows the entire price is paid by anonymity; the hierarchy theorem prices every partial refund.
Economically, the results are an identification statement of the standard econometric kind: allocation data, even from the whole semivalue family, identifies the coalitional production function only up to a fiber of dimension \(2^n-n^2+n-2\), and this failure persists under the strongest standard regularity assumptions and is exercisable by coalitions as manipulation. Anyone estimating complementarities from payment data is estimating a projection.
Practically, the visibility table offers a structural explanation of the practitioner’s puzzle: the Shapley value’s sixty-year record reflects a canon of high-visibility games rather than robustness of the methodology, and the deficit is silent by construction, since the missing component contributes zero to every output. In arrangement-driven applications (team formation, data markets, feature and component attribution) the seen fraction collapses toward the dimension ratio \(\Theta(n^2/2^n)\), and Example 1 is a constructive recipe for behavior invisible to the entire methodology; companion work in preparation develops the machine-learning consequences.
Three problems are left open: prove or refute the swing-rigidity conjecture (Conjecture 18); extend the information ladder to values with partial symmetry, such as coalition-structure values; and formulate the non-atomic limit, where the diagonal formula for the Aumann–Shapley value [22] suggests the visible space becomes a diagonal-trace algebra of the dividend measure hierarchy.
All computations are in flagship_suite.py and the exact-arithmetic follow-ups: (1) span equality fingerprint \(=\) dividend loadings (\(n=5\): ranks \(21/21/21\)); (2) hierarchy span identity (\(n=6\), \(d=2\): \(58/58/58\)) and the dimension table (\(d\le4\), \(n\le12\)), with the fifteen \(n\le9\) cases certified exactly over two \(31\)-bit prime fields
(s7b_exact_ranks.py); (3) cone sanity (\(50/50\)); (4) weighted family ranks (exact \(63/63\) at \(n=6\); rank
\(15/15\), \(31/31\) at \(n=4,5\)); (5) manipulation boundary table (\(d\le3\)); (6) Dedekind enumeration at \(n=5\) in both universes (monotone: \(7{,}579\); proper: \(2{,}645\)), reproduced independently by a second implementation (census_exact.py);
weightedness decided by LP feasibility and, independently, by swap robustness (an integer test, necessary for weightedness [20], [21]), the two methods agreeing on every game and on every count, including zero weighted games in any collision fiber; non-isomorphism of the displayed pair checked against all \(120\) permutations; weighted searches at \(n=6,7\); (7) exact rational ranks for the fingerprint matrices at \(n=4,5\) and for the weighted
family at \(n=6\); the convex-fiber thresholds \((2,3,2)\) in rational arithmetic; (8) four-cycle spanning of every slice kernel at \(n=5,6\); (9) the
visibility measurements of §9 (visibility_sweep.py): exact enumeration at \(n=10\), \(100\) instances per stochastic family (\(200\) random games), both metrics of Definition 1. Generators: bankruptcy \(v(S)=\max(0,E-c([n]\setminus
S))\) with claims \(c_i\sim U(0,1)\) and estate \(E=\tfrac12\sum_ic_i\); airport \(v(S)=\max_{i\in S}c_i\), \(c_i\sim
U(0,1)\); spanning-tree cost of \(S\) plus a source, eleven uniform points in the unit square with Euclidean costs (Prim); weighted majority with \(U(0,1)\) weights and half-total
quota; glove \(v(S)=\min(|S\cap L|,|S\cap R|)\) with \(|L|=|R|=5\); random games i.i.d. standard normal. Internal asserts: the Möbius transform round-trips, the fingerprint vectors span
exactly the projector’s range, the dividend visibility of the unanimity game equals its closed form \(10/210\), and the random-game mean matches the dimension ratio. Extending to \(n=14\)
and further families (market, assignment, flow games) is routine under the same protocol.
rigidity_search.py (companion file): enumerate weighted games at \(n=8\) (weights \(\le9\), \(\sim\) \(4\times10^5\) candidates after dedup) and \(n=9\) in chunks; hash swing tables; report any cross-win-set collision. A second mode samples random monotone games at \(n=6,7\) by monotone closure and tests whether any weighted game ever collides with a sampled simple game (the conjecture’s full strength). Any collision falsifies; sustained absence at \(n=8,9\) justifies attacking a proof via the LP-duality structure of weightedness.
Declarations
Use of AI. A large language model (Claude, Anthropic) was used as an assistant in preparing this paper: for editing and restructuring prose, for adversarial review of drafts, and for writing and executing portions of the verification code indexed in Appendix 12. All definitions, theorems, and proofs were formulated, checked, and are vouched for by the author, who takes full responsibility for the content; all computational claims were independently reproduced. Competing interests. The author declares no competing interests. Code availability. All code reproducing the computational claims is available from the author.
SUNY Farmingdale. Email: friedm1@farmingdale.edu↩︎