  Valuation problems, such as feature interpretation, data valuation and model
valuation for ensembles, become increasingly more important in many machine
learning applications. Such problems are commonly solved by well-known
game-theoretic criteria, such as Shapley value or Banzhaf value. In this work,
we present a novel energy-based treatment for cooperative games, with a
theoretical justification by the maximum entropy framework. Surprisingly, by
conducting variational inference of the energy-based model, we recover various
game-theoretic valuation criteria through conducting one-step fixed point
iteration for maximizing the mean-field ELBO objective. This observation also
verifies the rationality of existing criteria, as they are all attempting to
decouple the correlations among the players through the mean-field approach. By
running fixed point iteration for multiple steps, we achieve a trajectory of
the valuations, among which we define the valuation with the best conceivable
decoupling error as the Variational Index. We prove that under uniform
initializations, these variational valuations all satisfy a set of
game-theoretic axioms. We experimentally demonstrate that the proposed
Variational Index enjoys lower decoupling error and better valuation
performance on certain synthetic and real-world valuation problems.
