Expected Shapley Value is Shapley Value for Expected Utility Game

Pratik Karmakar, Antoine Gauquier, Pierre Senellart

Abstract: The Shapley value provides a principled framework for attributing marginal contributions to players in coalitional games. While its axiomatic fairness guarantees have made it a cornerstone of value distribution in economics and multi-agent systems, recent computational advances have extended its applicability to data-driven domains. This paper bridges game-theoretic foundations with probabilistic reasoning by studying Shapley-like scores in stochastic environments. We prove that the expected Shapley value (\(\textsf{EShap}\)) – a player’s average impact in a game with an independent probabilistic setting – coincides with the Shapley value of the game whose utility is the expected utility of the original game (\(\textsf{ShapE}\)). This equality, however, fails for other power indices, such as the Banzhaf index, underscoring the Shapley value’s specificity of consistency in uncertain settings. We further identify that for a certain class of coefficients (including normalized Banzhaf indices) the equality persists, broadening the scope of reliable attribution mechanisms.

External IDs:doi:10.1007/978-3-032-05134-9_16