The Geometry of Cooperative Game Solutions: Stratified Egalitarian Shapley Values
This work provides a unifying geometric perspective for comparing cooperative game solutions, benefiting game theorists and economists by offering a principled way to measure alignment with egalitarian principles.
The paper introduces a geometric framework for cooperative game solutions, showing that the space of efficient symmetric linear value maps is isomorphic to R^{n-1} with an orthogonal stratification by coalition size. The egalitarian Shapley family forms a diagonal slice, and the projection onto it yields an optimal parameter with a goodness-of-fit measure analogous to R^2. For n=4, the Banzhaf value has R^2 ~ 1%, the equal-surplus-division value R^2 ~ 38%, and the solidarity value R^2 ~ 99.6%; asymptotically, R^2 for Banzhaf tends to 1/2.
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.