I also figured out what I'm going to do for the tie-breaker. It might not be the coolest way to do it, but it's simple.
A paper bag (or other container) with slips of paper numbered 1-20 (or 30, 40, 50, etc.). To break the tie, each person takes out a piece of paper and the highest number wins. This can be reused several times without having to have any of the numbers put back into the bag, and when done, simply throw it out (no need to store it!).