More on "Simple Head-to-Head"

OK, you have two top-ranked teams - the Cartersville Chipmunks and the Portstown Possums. During the regular season, the two teams met head-to-head once in additional to eleven other common opponents. The Possums won the head-to-head matchup. Both finished 11-1. Which one deserves to play the undefeated Artersburg Aardvarks for the national championship?

Easy problem, right? The Possums won the head-to-head matchup. Therefore, they're the better team. Therefore, they should play in the championship. 'Nuff said.

Hmmm. You know, as long as we don't look any closer, the problem appears to be solved. Portstown and Artersburg play for the trophy. Everybody's happy.

But is "simple head-to-head" really so simple? Let's dig deeper...

Portstown won the head-to-head matchup but lost to someone else, in this case, the Forcester Fleas. Since Cartersville and Portstown played identical schedules and 11-1 Cartersville's only loss was to Portstown, then Cartersville must have beaten Forcester.

So, that means Portstown lost to Forcester who lost to Cartersville who lost to Portstown. Therefore, if "head-to-head" means that Portstown is better than Cartersville, then Cartersville must be better that Forcester and Forcester is better than Portstown. But if Cartersville is better than Forcester and Forcester is better than Portstown, then either Cartersville is better than Portstown or... since A better than B and B better than C doesn't mean A is better than C, we must have no clue what "better" means. So, what does all this complicated stuff mean?

It means many things?

#1 Ranking teams isn't as easy as ranking all winners ahead of losers. Sooner or later you will arrive at a contradition.

#2 If we had to define "head-to-head," it is simply that we hate to see teams ranked slightly behind teams they beat. If they're way behind in the rankings, we comfortably ignore it, but if they're close, we seem to come unglued. This is a very inconsistent position -- "big upsets" are tolerated while "little upsets" drive us crazy.

#3 In the above case, if we favor the winner of the head-to-head, then while we arguably have two teams whose "average" performance is identical, we're basically favoring the MORE INCONSISTENT of the two teams -- the team whose highs are higher but whose lows are lower.
But don't worry, it gets worse...

Let's assume that we somehow omniciently know that both teams are, on average, identical. Therefore, they played equally difficult schedules (since the only difference in their schedules is their head-to-head matchup) and they finished with identical records against those schedules. If we don't ask exactly which games were wins and which were losses, we would tend to say that they are equally good. Now, let's assume that, instead, we omniciently know that Portstown is the better team. Therefore, Cartersville played the tougher schedule since the only schedule difference is the head-to-head matchup and Cartersville's opponent (Portstown) is tougher than Portstown's opponent (Cartersville). Therefore, Cartersville had the same record against a tougher schedule. But a better record against a tougher schedule would imply that Cartersville is the better team. But we have this conclusion as a direct result of our assumption that Portstown is the better team...

Bottom line: There's no such thing as SIMPLE head-to-head.

Simple Head-to-Head

An irate fan calls into the local sports talk radio station Monday evening so angry he can barely get the words out. Between the coughing, gagging, and spitting, he manages to say: “This is ridiculous. It’s nonsense. How can they rank the Cartersville Skunks #5 ahead of my Waynesboro Lemmings. The Lemmings beat the skunks 17-14 the third week of the season. The rankings are stupid. It’s simple head-to-head. Simple head-to-head. I’ve got nothing more to say.” Click.

Amazingly, there are more than a few fans who think they’ve got it all figured out. The problem is, they’ve never taken a pencil and tried to do what they insist makes so much sense – just rank the teams so that the winners in each game are ranked higher than the losers.

Funny thing is, early in the season this is still possible, and yet fans fuss at the rankings because they don’t want to believe that the few games played are actually representative of how good (or bad) their team is. However, by midseason, this kind of ranking is no longer possible. Sooner or later, team A beats team B who beats team C who beat team A. Or some team beats a 10-1 team and loses to a 1-10 team.

As geeks who do computer rankings, we do understand the frustration. In fact, in the “ranking community,” we even have a lingo to describe all this stuff – “retrodiction,” “ranking violations,” “ranking by pairwise comparison,” and the like.

Before you’re tempted to call your local sports talk radio program, let me offer up a slightly different way to assess a set of rankings. Suppose a team is 12-0 at the end of the season. It would be reasonable for them to be ranked somewhere above all twelve of their opponents. Another team goes 0-12. It would seem reasonable for them to be ranked somewhere below all twelve of their opponents. Consider another team that goes 6-6. Would it not seem reasonable for them to be ranked above six of their opponents and below six others? Here’s the catch. Would this not seem reasonable even if this team actually beat one or two of their higher ranked opponents while losing to one or two of their lower ranked opponents? After all, teams have good days and bad days. Upsets are what makes football exciting, right?

For what it’s worth, any decent ranking method will approximately do just this? Why not exactly this, you might ask? Well, consider this one example of why it can’t always be done. Suppose no one goes undefeated. Someone must still be ranked #1. Since they have one loss, they are ranked above a team to whom they lost.

There is another problem with this scheme. If a team goes 12-0 against a schedule that includes no top 25 opponents, exactly how high should they be ranked? Somewhere between #1 and their best opponent, but where?

The bottom line is that ranking football teams is a really hard problem. Probably harder than any other sport because the teams play so few games.

Back to the irate caller. He’s got it all figured out. We just move the Lemmings up to #4. Of course, the Lemmings lost to the 4-6 Hedgehogs, so we’ll have to move the Hedgehogs up to #3. But the Hedgehogs lost to six other teams, and we don’t have enough slots for them, so we’ll have to move the Hedgehogs, Lemmings, and Skunks down to make room. Wait, one of those teams was the Skunks. I thought this was SIMPLE head-to-head.