New geometric framework classifies Shapley values with regression-style R²
Feys' paper shows Banzhaf value is nearly orthogonal to egalitarian Shapley axis (R² ~ 1%)
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Frank Feys' new paper on arXiv (2605.22847) presents a geometric reinterpretation of cooperative game theory's value maps. The space of linear values on finite-player games admits a canonical inner product derived from Harsanyi-dividend decomposition. Within this geometry, the subspace of efficient, symmetric, linear values becomes isomorphic to ℝⁿ⁻¹, with each component corresponding to a coalition size. The classical egalitarian Shapley family of values (Joosten, 1996) emerges as the diagonal slice of this space.
Using regression-style R² metrics, Feys quantifies how well alternative values align with this egalitarian axis. At n=4, the Banzhaf value is nearly orthogonal (R² ~ 1%), the equal-surplus-division value shows moderate alignment (R² ~ 38%), and the solidarity value is almost perfectly aligned (R² ~ 99.6%). Asymptotically, Banzhaf's R² converges to 1/2 due to a structural identity linking its efficiency defect and deviation from egalitarian Shapley. This framework provides a unified, interpretable geometry for comparing cooperative game solutions.
- Feys introduces intrinsic inner product on value maps using Harsanyi-dividends, enabling orthogonal decomposition by coalition size.
- For n=4 players, Banzhaf value has R² ~ 1%, equal-surplus-division ~ 38%, solidarity value ~ 99.6% alignment with egalitarian Shapley.
- As player count grows, Banzhaf R² → 1/2, while ESD and solidarity R² → 1, revealing structural differences.
Why It Matters
Provides a principled geometric tool for comparing cooperative game solutions, impacting AI fairness and explainability via Shapley values.