  The Shapley value is a conventional and well-studied function for determining
the contribution of a player to the coalition in a cooperative game. Among its
applications in a plethora of domains, it has recently been proposed to use the
Shapley value for quantifying the contribution of a tuple to the result of a
database query. In particular, we have a thorough understanding of the
tractability frontier for the class of Conjunctive Queries (CQs) and aggregate
functions over CQs. It has also been established that a tractable (randomized)
multiplicative approximation exists for every union of CQs. Nevertheless, all
of these results are based on the monotonicity of CQs. In this work, we
investigate the implication of negation on the complexity of Shapley
computation, in both the exact and approximate senses. We generalize a known
dichotomy to account for negated atoms. We also show that negation
fundamentally changes the complexity of approximation. We do so by drawing a
connection to the problem of deciding whether a tuple is "relevant" to a query,
and by analyzing its complexity.
