Extremal Principle Solutions of Games in Characteristic Function Form: Core, Chebychev and Shapley Value Generalizations

By A. Charnes, Boaz Golany, M. Keane, J. Rousseau. 1988

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A Unified Approach to Interpreting Model Predictions (2017, background): “ns (Equation 11 or 12). 2During the preparation of this manuscript we discovered this parallels an equivalent constrained quadratic minimization formulation of Shapley values proposed in econometrics [2]. 6 f 3 f 2 f 1 f 3 f 2 f 1 hapley (A) (B) Figure 2: (A) The Shapley kernel weighting is symmetri”
Feature relevance quantification in explainable AI: A causality problem (2019, methodology): “By Charnes et al. (1988), the Shapley Values to the set function g are given as the solution (φ1, . . . , φn) of min φ1,...,φn ∑ T⊆U [ g(T )− (∑ j∈T φj )]2 k(U, T ) , (15) where k(U, T ) = (|U | − 1)/( (|U | |T | ) |T |(|U | − |T”
Explaining by Removing: A Unified Framework for Model Explanation (2020, methodology): “Since this connection has been noted in other model explanation works, we direct readers to existing proofs (Charnes et al., 1988; Lundberg and Lee, 2017).”
Explaining individual predictions when features are dependent: More accurate approximations to Shapley values (2019, unknown): “Charnes et al. (1988) and later Lundberg & Lee (2017a) define the Shapley values as the optimal solution of a certain weighted least squares (WLS) problem.”
Understanding Global Feature Contributions With Additive Importance Measures (2020, background): “6) can be understood as the solution to a weighted least squares problem [6, 25].”