  Let $G$ and $H$ be infinite finitely generated amenable groups. This paper
studies two notions of equivalence between actions of such groups on standard
Borel probability spaces. They are defined as stable orbit equivalences in
which the associated cocycles satisfy certain tail bounds. In `integrable
stable orbit equivalence', the length in $H$ of the cocycle-image of an element
of $G$ must have finite integral over its domain (a subset of the $G$-system),
and similarly for the reverse cocycle. In `bounded stable orbit equivalence',
these functions must be essentially bounded in terms of the length in $G$.
`Integrable' stable orbit equivalence arises naturally in the study of
integrable measure equivalence of groups themselves, as introduced recently by
Bader, Furman and Sauer.
  The main result is a formula relating the Kolmogorov--Sinai entropies of two
actions which are equivalent in one of these ways. Under either of these tail
assumptions, the entropies stand in a proportion given by the compression
constant of the stable orbit equivalence. In particular, in the case of full
orbit equivalence subject to such a tail bound, entropy is an invariant. This
contrasts with the case of unrestricted orbit equivalence, under which all free
ergodic actions of countable amenable groups are equivalent. The proof uses an
entropy-bound based on graphings for orbit equivalence relations, and in
particular on a new notion of cost which is weighted by the word lengths of
group elements.
