I'm sorry for being so dense -- but I still don't have a clear idea of what "assignment of cases" means in the concrete. :-(
After the Supreme Court hears a case, they meet privately and take an initial vote. The Chief Justice, if he is in the majority (or the most senior justice in the majority, if the Chief is in the minority) assigns one justice in the majority to write the opinion for the Court.
There is an old saying that is applicable across government: people are policy.