I use the theories of duality and optimal branchings to find a necessary and sufficient characterization of stochastically stable limit sets (SSLS) that helps improve the radius-modified coradius test of Ellison (2000). The improved shortcut I offer may permit the identification of SSLS when Ellison's radius-modified coradius test fails to identify any, or may be able to pinpoint the true SSLS in cases where Ellison's test identifies only a superset. I also demonstrate precisely why the radius-modified coradius test is not universally applicable and illuminate the connection between the modified coradius and the Lagrange multipliers of the optimal branching problem.
