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.