  In this paper, we propose an approximation method to study the regularity of
solutions to the Isaacs equation. This class of problems plays a paramount role
in the regularity theory for fully nonlinear elliptic equations. First, it is a
model-problem of a non-convex operator. In addition, the usual mechanisms to
access regularity of solutions fall short in addressing these equations. We
approximate an Isaacs equation by a Bellman one, and make assumptions on the
latter to recover information for the former. Our techniques produce results in
Sobolev and H\"older spaces; we also examine a few consequences of our main
findings.
