Why it works: nobody wins by arguing better. The only thing measured is whether the people you disagreed with recognised their own position in your description of it — which is the thing that makes a disagreement resolvable, and almost nobody practises it. Assigning sides at random matters too: a room that picks sides argues what it already thought, harder.
Do the draw with everyone watching. It is the thing that stops this becoming a debate about who believes what.
Each side now states the other side's strongest argument, out loud, in one or two sentences. The side being described marks it — not the facilitator.