The principle of competition does not generally apply to public organizations. This is because they typically do not have any competitors. It may apply within those organizations, but not for the organization as a whole.
Competition can occur within public organizations. Two Amtrak workers, for example, may compete with one another for a promotion. This presumably encourages them to do their best work. But Amtrak as a whole does not really have competition because there are no other passenger train lines to be its rivals. The same goes for something like a city parks department. It has no competition so does not benefit from the discipline imposed on private organizations by competition.