Approximating the Shapley Value without Marginal Contributions

Whenever agents can federalize in groups (form coalitions) to accomplish a task and get rewarded with a collective benefit that is to be shared among the group members, the notion of cooperative game stemming from game theory is arguably the most favorable concept to model such situations. This is due to its simplicity, which nevertheless allows for covering a whole range of practical applications. The agents are called players and are contained in a player set N. Each possible subset of players S N is understood as a coalition and the coalition N containing all players is called the grand coalition. The collective benefit ν(S) that a coalition S receives upon formation is given by a value function ν assigning each coalition a real-valued worth. The connection of cooperative games to (supervised) machine learning is already well-established. The most prominent example is feature importance scores, both local and global, for a machine learning model: features of a dataset can be seen as players, allowing one to interpret a feature subset as a coalition, while the model's generalization performance using exactly that feature subset is its worth Cohen et al. [2007]. Other applications include evaluating the importance of parameters in a machine learning model, e.g.