Sports commentators must recognize the need for tiebreakers in sports when, for example, two teams with identical records cannot both move on to some further round of play. Say, two teams with the same record, as happened recently in the FIFA Confederations Cup, cannot both move on to play in the next round. These tiebreakers are often based on a series of conditions such that eventually the two teams will differ with respect to one of them, so that one team can be judged superior so that it can move on. For example, in the aforementioned FIFA tournament, the US and Italy both had 1-2 records in their round, and the US moved on because they had a greater total margin of victory than Italy. Margin of victory can easily be a misleading statistic since it could result from a better team running up the score on a weaker team. Other tiebreaking conditions might be record vs. common opponents, difficulty of schedule, or head-to-head matchups. The last is the one most commentators prefer (at least, Tony Kornheiser and Michael Wilbon who yell about these kinds of issues on ESPN's Pardon the Interruption prefer it). Why? I believe they prefer it because it is simple, requires no sophisticated understanding of statistics and matches the presumption in sports that when two teams or individuals play, the better one wins. If your goal is to select the better team when two or more have identical records, then getting the simplest and easiest answer is no guarantee of getting the best answer. So, does head-to-head matchup more reliably pick the better team than other methods do? There is good reason to think it does not. Look at the number of upsets in the NCAA basketball tournament; while there is no guarantee that the seeding process reliably orders teams in order of their actual quality, the number of upsets in the tournament strongly suggests that the worse team often wins. All it takes is for one team to have a bad day and the other team to have a good day, and the worse team will win. Sports people know this; that's why professional basketball has longer games (48 vs. 40 minutes) and longer playoff series (best of seven instead of best of one). The more games the two teams play, the less likely it is that the worse team will have enough good games and the better team enough bad games for the worse team to win.

So, the head-t0-head matchup does not necessarily result in the better team being selected. A second problem is that the head-to-head matchup is more likely to give rise to paradox. Here's a famous voting paradox: Suppose you have three options A, B and C. Most people prefer A to B, B to C and C to A. If one had a choice among all three at once, they would each garner 1/3 of the vote. So they are, we stipulate, equally popular. Now, however, we get a paradox since, if we place the alternatives in pairs, and hold the votes in a different order we get a different result. So, if we have an election (say, a primary) between A and B, A will win. Then we have the vote of A against C, C wins. Suppose, instead, we start with B against C, which B wins, then hold the vote of A against B, which A wins. Finally, we could vote A against C, which C wins, then hold the vote of B against C, which C wins. So, each of the three ways of paired-votes gives a different result. So, a head-to-head matchup election elects a leader depends on an arbitrary fact besides relative popularity.

Sports has its equivalent in head-to-head matchups. If team A beats team B, team B beats team C, and team C beats team A, and all three have the same record, basing one's judgment on head-to-head matchups results in paradox. None of the three can be selected. Mathematically, this kind of stalemate is a lot less likely than a stalemate over margin-of-victory. The odds that three teams will all have exactly the same differential in total points scored over a season is much less likely than that the three teams might each beat one other of the three. If all three teams are evenly matched, then this paradox occurs whenever A beats B and loses to C (1/4 probability) or A loses to B and beats C (1/4). So, the head-to-head stalemate occurs, for equally matched teams, 1/16 of the time. That's a bit unusual, but common if tiebreakers come up a lot.

How often would margins of victory match? That depends on the sport's point system, but the only way the margin of victory stalemate could occur as often as the head-to-head stalemate is if the only margin of victory was 1-0 if there were no points but only wins and losses.

The conclusion to draw is that the head-to-head matchup is more prone to stalemate or paradox than at least some other methods, and there is no guarantee that it is reliable. I suspect that the best method is actually a linear regression model--essentially a weighting of various factors with the highest total being declared the winner--rather than a sequential list of conditions. That method, essentially the method used by the BCS, would really be the only way of accounting for every factor that might measure the quality of a team. And one can adjust the weights given to the various factors after every later match to strengthen the weights of the factors that gave the right predictions and weaken those that led to the wrong ones.

But ordinary humans do not like such methods; they prefer rough-and-ready shortcuts that give a quick result. We're satisficing machines; we mostly don't need the best answer, just one that is good enough. And so we prefer the methods we can easily keep track of. The problem with this is that the statistical measures (such as linear regression models) simply give better predictions than our rules of thumb. This lesson was a hard-learned one in science, so it's no wonder the humanities majors in sports reporting and commentary haven't yet twigged to it.

## Thursday, June 25, 2009

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