## Wednesday, May 16, 2018

### Why debate tournaments have been doing side assignment wrong

Side assignment is easy, right? In odd rounds, assign teams to sides at random. In even rounds, assign each team to the opposite side as the previous round. What could be easier?

The problem is that this makes even rounds harder to pair. Any tournament director can tell you that even rounds often "lock up" and that one has to break brackets to make matches. I know I've sat at a screen, wishing the two 5-0s that are both due Aff could hit, instead of each getting a pull-up.

I stumbled on an alternative, what I call the constrained side equalization (C.S.E.) method. Instead of balancing Aff-Neg rounds at the end of even rounds, this method works its magic at the end of odd rounds. Here's the C.S.E. in action:

Rd 1 - paired at random
Rd 2 - paired at random, ignoring sides. If both teams were Aff in round 1, or both Neg in round 1, it's a computer flip-for-sides. If one team was Aff and the other was Neg, then the sides are equalized.

At the end of round 2, about 25% of teams will have two Affs, 25% two Negs, and 50% will be balanced. (It depends on the random pairings.)

Rd 3 - Teams with two Affs must go Neg; teams with Negs must go Aff. The balanced teams are not assigned to either side. If a balanced team is matched against a two-Aff team, then the two-Aff team goes Neg. Likewise, if a balanced team is matched against a two-Neg team, then the two-Neg team goes Aff. If a two-Aff team is matched against a two-Neg team, then the sides are equalized. And if a balanced team is matched against a balanced team, then it's a computer flip-for-sides.

At the end of round 3, every team will either have had two Affs and one Neg, or two Negs and one Aff. In other words, at the end of an odd round, the sides are "equalized."

The cycle repeats. Round 4 is paired at random, ignoring sides. Round 5 has the constraint that teams with three Affs must go Neg and teams with three Negs must go Aff; otherwise, any team can be paired against any other. If the tournament ends on an odd round, there's no special other consideration. If the tournament ends on an even round, you'd want to pair teams in the typical way for the final prelim.

Mathematically, it is as simple as this rule:
If the Aff rounds - Neg rounds is 2 or -2, then the team is assigned a side first, then paired with an opponent; otherwise, a team is assigned an opponent first, then assigned a side (to equalize if necessary).
This works in odd or even rounds.

But why go to all this bother? The reason is simple: constraints.

 Odd Even Avg. Trad. 100% 50% 75% Alt. 87.5% 100% 94%

In a traditional method, in odd rounds, 100% of possible matches-- 0.5 * (n (n - 1)) --could be considered. There are no side constraints in odd rounds, so anyone could be matched against anyone. But in an even round, a tournament is limited to a fourth of (n (n - 1)). A due-Aff team can only be matched against a due-Neg team. This is a huge constraint.

Using the C.S.E. method, in odd rounds, teams with more Affs must go Neg and vice versa. Aside from this small constraint (only about one-eighth of possible matches ruled out), nearly anyone can debate anyone. And in even rounds, it's 100% of possible matches that can be considered. The C.S.E. method has much lower overall constraints than the traditional method.

In other words, the odd C.S.E. round is considerably easier to pair than the even traditional round (21 times better odds of finding a good pairing, in fact). If a side assignment for C.S.E. happens to not turn up a suitable pairing, why, you can reshuffle the teams--switching some randomly selected teams' side, excepting the couple side-constrained teams--and try again. This works whether it's an odd or an even round. In the traditional method, you can only reshuffle with an odd round. You're stuck with the even round side assignments you get with the traditional method. This inability to reshuffle the teams means the tournament can lock up. In the C.S.E. method, because any round can be reshuffled, there's always another chance to find a good pairing.

I worked out an example here. At the end of five rounds of C.S.E., every team had either two or three Affs. The method yielded side "equivalence."

But, intriguingly, the teams took different paths to get there. Some went Aff two times in a row. Some alternated. Although all the paths end with one of two correct results--two or three Affs--there were more path types to get there and thus more options to pair the teams. More paths = more flexibility. We've been doing side assignment the hard way!