I just received a new book today,

*Who's #1? The Science of Rating and Ranking*, by Langville and Meyer, two math professors. They state that they were frustrated that the methods they discuss in their book were not collected in any one other single book. I can say I share that frustration. At an initial look through it, the book is amazing. A lot of the methods they discuss I have looked at in one form or another in considering a tab program; many I have not. It is both validating and humbling. I have a lot of work to do! So, for quite a while, I am going silent on tab programs while I read this book. I will still post on neat mathematics problems and critical thinking course discussion ideas.

Before I go temporarily silent on the tab program topic, though, I thought it would be good to summarize what I have written so far. My positions have evolved in three and a half years considerably.

I started out looking at whether high-low and high-high pairings created fair schedules for teams, specifically looking at opponent wins. I looked at the Harvard tournament, a set of big national tournaments, and what was possible in theory. In general, I think the case is pretty clear that even at big tournaments, where constraints should not be an issue, the traditional methods fail to deliver fair schedules -- even within a bracket. Teams break with easy schedules; teams don't break with hard schedules.

I looked for a way to pair within brackets that I felt was more fair. The idea I hit on was strength-of-schedule pairings: a team would get an opponent that would balance out its schedule difficulty compared to other teams in the same bracket. This is a method only a computer can do, since it requires simultaneously evaluating whether team A is a good opponent for team B (as defined above) AND whether team B is a good opponent for team A. Keeping track of both ratchets up the complexity beyond a human's hands. It is still an understandable method, just too many calculations for a person to do. I tried it on a small tournament and a large tournament and found that it does indeed work to even out the schedules.

That problem "solved," I started thinking about evaluating a team's strength (and therefore also a team's schedule strength) in a more sophisticated ways than just wins and losses. I looked at graph theory, but for all its promise for some tiebreakers in round robins, it is not a good method for regular tournaments. I looked at one weighted wins scheme, where the points per win decline for each subsequent round, but this system is not good. I next looked at a weighted wins scheme where a team receives extra "points" for its defeated opponents' wins and loses points for its defeating opponents' losses. This does seem to work well for a simple method, and it lead me to think about more complex ways of getting, from the data, a team's strength on the affirmative and strength on the negative. I have also been thinking about how reliable even the best methods are. How many "upsets" are there in debate?

Along the way, I have also thought about side assignment here. I have written about how many teams will break here, here, and here. And I have sparred with A Numbers Game on whether there is topic side bias (sorry, it looks to me like novices muck up the average; varsity debaters get closer to parity) or judge side bias (again, sorry, it looks alright to me).

So where have I landed? I started out thinking all I wanted to do was propose a different within brackets pairing algorithm, which then expanded to thinking about measuring a team's strength and opponent strength. But a very early post on round robins planted the seed: preliminary rounds don't have to be elim rounds. They don't need winners and losers brackets. Teams need to meet all their opponents, or failing that, a good cross-section. So the fairest statement of what I believe now is that tournaments should ditch the brackets and start making sure that teams get mixed up by skill level and geography in prelims; let a sophisticated algorithm do the mixing, not random chance; let the best teams break, and use elims like they always have been: to pick the champion. And I think N.F.L. Nationals ought to start.

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