The Shapley Value of Classifiers in Ensemble Games

What is the value of an individual model in an ensemble of binary classifiers? We answer this question by introducing a class of transferable utility cooperative games called \textit{ensemble games}. In machine learning ensembles, pre-trained models cooperate to make classification decisions. To quantify the importance of models in these ensemble games, we define \textit{Troupe} -- an efficient algorithm which allocates payoffs based on approximate Shapley values of the classifiers. We argue that the Shapley value of models in these games is an effective decision metric for choosing a high performing subset of models from the ensemble. Our analytical findings prove that our Shapley value estimation scheme is precise and scalable; its performance increases with size of the dataset and ensemble. Empirical results on real world graph classification tasks demonstrate that our algorithm produces high quality estimates of the Shapley value. We find that Shapley values can be utilized for ensemble pruning, and that adversarial models receive a low valuation. Complex classifiers are frequently found to be responsible for both correct and incorrect classification decisions.