The space L of linear value maps on a finite-player cooperative game G^N is finite-dimensional, and admits a canonical inner product induced by the Harsanyi-dividend decomposition of G^N. We show that this inner product is intrinsic: the same value arises from any orthonormal basis of G^N with respect to the Harsanyi inner product. Within this geometry, the subspace L^{ESL} of efficient, symmetric, linear value maps admits a clean structure theorem. The induced orthogonal stratification of L by coalition size yields a canonical linear isomorphism L^{ESL} = R^{n-1}, under which every efficient symmetric linear value map decomposes uniquely into n-1 stratified epsilons, one per coalition size. The classical egalitarian Shapley family of Joosten (1996) is precisely the diagonal slice of this R^{n-1}. The orthogonal projection of any Psi in L^{ESL} onto this diagonal yields an optimal parameter eps*(Psi) equal to the weighted mean of the stratified epsilons under an explicit probability distribution {w_a} over coalition sizes, and the goodness-of-fit R^2(Psi) equals one minus the relative weighted variance of those epsilons. The framework is a literal regression-statistics analogue of the coefficient of determination. At n=4 it produces a clean three-way classification of the standard alternatives to the Shapley value: the Banzhaf value is nearly orthogonal to the egalitarian Shapley axis (R^2 ~ 1%); the equal-surplus-division value is moderately aligned (R^2 ~ 38%); the solidarity value is almost entirely aligned (R^2 ~ 99.6%). Asymptotically R^2(ESD) -> 1, R^2(So) -> 1, and R^2(Bz) -> 1/2, the last reflecting a structural identity between the efficiency defect and the egalitarian-Shapley deviation of the Banzhaf value at every coalition size.

翻译：有限玩家合作博弈G^N上的线性值映射空间L是有限维的，且由G^N的Harsanyi红利分解诱导出一个典范内积。我们证明该内积是本质的：对于G^N的任意关于Harsanyi内积的标准正交基，该内积产生相同的值。在此几何框架内，由有效、对称、线性值映射构成的子空间L^{ESL}具有简洁的结构定理。通过联盟规模对L进行正交分层，得到典范线性同构L^{ESL} = R^{n-1}，在此同构下，每个有效对称线性值映射唯一分解为n-1个分层epsilon（每个联盟规模对应一个）。Joosten (1996)的经典平等沙普利族正是此R^{n-1}的对角线切片。L^{ESL}中任意Psi在该对角线上的正交投影产生最优参数eps*(Psi)，它等于在联盟规模上的显式概率分布{w_a}下分层epsilon的加权均值，而拟合优度R^2(Psi)等于1减去这些epsilon的相对加权方差。该框架是决定系数的字面回归统计类比。当n=4时，它对沙普利值的标准替代方案产生清晰的三分类：Banzhaf值几乎正交于平等沙普利轴（R^2约1%）；等剩余分配值中等对齐（R^2约38%）；团结值几乎完全对齐（R^2约99.6%）。渐近地，R^2(ESD)→1，R^2(So)→1，而R^2(Bz)→1/2，后者反映了Banzhaf值在每个联盟规模上的效率缺陷与平等沙普利偏差之间的结构恒等式。