When it's a draw and there has been previous characters who has held prestige, the draw is settled like so:
In the case of a draw, the Hero who has held the leading Prestige for the longest is selected as the Prestige Leader. Source
This means, that if Rabbit held the prestige for one round and Wolf then drew prestige the next round, Rabbit would maintain prestige leader until they either lose prestige to a level beneath someone else, or someone overtakes them in prestige.
I can't remember what happens when there is a draw and there is no previous prestige, I need to double check that. I think no-one is given it and instead the king decides which decree to enact