not sure if it helps, but this is how I understand it:
you don't need to set the path. if you don't set the path, the cookie information is available in the directory where the script that set the cookie resides, and in all subdirectories of this, but not in the parent directory.
if you want to make it available to all files in the domain no matter where it was set, use "/" as path.
if on the other hand you want to restrict access to the cookie to a certain subdirectory, I think you need to give the full path from the web root, starting with a leading /.