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What is the value of an individual model in an ensemble of binaryclassifiers? We answer this question by introducing a class of transferableutility cooperative games called \textit{ensemble games}. In machine learningensembles, pre-trained models cooperate to make classification decisions. Toquantify the importance of models in these ensemble games, we define\textit{Troupe} -- an efficient algorithm which allocates payoffs based onapproximate Shapley values of the classifiers. We argue that the Shapley valueof models in these games is an effective decision metric for choosing a highperforming subset of models from the ensemble. Our analytical findings provethat our Shapley value estimation scheme is precise and scalable; itsperformance increases with size of the dataset and ensemble. Empirical resultson real world graph classification tasks demonstrate that our algorithmproduces high quality estimates of the Shapley value. We find that Shapleyvalues can be utilized for ensemble pruning, and that adversarial modelsreceive a low valuation. Complex classifiers are frequently found to beresponsible for both correct and incorrect classification decisions.

The Shapley Value of Classifiers in Ensemble Games