Joint Shapley values: a measure of joint feature importance

The Shapley value is one of the most widely used measures of feature importance partly as it measures a feature's average effect on a model's prediction. We introduce joint Shapley values, which directly extend Shapley's axioms and intuitions: joint Shapley values measure a set of features' average contribution to a model's prediction. We prove the uniqueness of joint Shapley values, for any order of explanation. Results for games show that joint Shapley values present different insights from existing interaction indices, which assess the effect of a feature within a set of features. The joint Shapley values provide intuitive results in ML attribution problems. With binary features, we present a presence-adjusted global value that is more consistent with local intuitions than the usual approach. Published as a conference paper at ICLR 2022 SY symmetry : If two agents add equal worth to all coalitions that they can both join then they receive equal value: . This is strictly weaker than anonymity. Now extend each of these axioms in natural ways to conditions on sets rather than singletons. Below, φ S (v) denotes an index for coalition S on game v. JLI joint linearity : φ is a linear function on G N , i.e. φ(v + w) = φ(v) + φ(w) and φ(av) = aφ(v) for any v, w ∈ G N and a ∈ R. (This axiom has not been modified.) JNU joint null : A coalition that adds no worth to any coalition has no value, i.e. if v(S ∪T ) = v(S) for all S ⊆ N T , then φ T (v) = 0. JEF joint efficiency : The sum of the values of all coalitions up to cardinality k is equal to the worth of the entire set, i.e. for all v ∈ G N , JAN joint anonymity : For any σ on N and any v ∈ G N , φ T (v) = φ σ(T ) (σv), for all T ⊆ N . JSY joint symmetry : If two coalitions perform equally when joining coalitions that they can both join and for other coalitions they add no worth then they receive an equal value, i.e. if Axiom JSY only equates the joint Shapley values for coalitions T and T ′ if they contribute identically to coalitions that they may both join, and contribute nothing to the other coalitions. Axioms JLI, JEF and JAN are all also used in Dhamdhere et al. (2020). Our joint null and joint symmetry notions appear to be new: they reflect our interest in a set of features' contribution to a model's predictions, so that the set's cardinality should not play a role in determining its value. JOINT SHAPLEY VALUES Our main result is that there is a unique solution to axioms JLI, JNU, JEF, JAN and JSY, the joint Shapley value. The uniqueness is up to the k th order of explanation; we say nothing about |T | > k.