  The standard argument why gravity is not renormalisable relies on direct
powercounting of Feynman graphs to estimate the degree of UV divergence. This
analysis has in several (highly) supersymmetric examples be shown to
overestimate divergences considerably. In these examples the main improvements
arise from a conjectured duality between color and kinematics. In this paper we
initiate the systematic study of quite general powercounting under the
assumption that color-kinematic duality exists. The main technical tool is a
reformulation of the duality in terms of linear maps, modulo subtleties at loop
level mostly inherent to the duality. This tool may have wider applications in
both gauge and gravity theories, up to resolution of the subtleties. Here it is
first applied to the large Britto-Cachazo-Feng-Witten (BCFW) shift behavior of
gravity integrands constructed through the duality. Assuming color-kinematic
duality and reasonable technical requirements hold these shifts are shown to be
independent of loop order, which would imply massive cancellations with respect
to the Feynman graph expression. More speculatively, the same approach is then
applied to provide estimates of the overall degree of UV divergence in quite
general gravity theories, assuming the duality exists. The cancellations
obtained in these estimates depends on the exact implementation of the duality
at loop level, especially on graph topology. Finally, some evidence for the
duality to all loop orders is provided from an analysis of BCFW shifts of gauge
theory integrands through Feynman graphs.
