Friday, October 2, 2009

Virginia Tech, Really?

The midweek games have already come and gone, and I'm still a little puzzled about our RWFL rankings previously posted. Again, it's far too early in the season to expect good performance out of a computer ranking system that, like ours, ignores margin of victory, dates of games, and the previous season. But Virginia Tech (3-1) edging Iowa (4-0) for 2nd place?!? Now, sure, from a ranking violations standpoint, that's fine, since the loss came at the hands of #1 Alabama. But with so many undefeateds (including three potential BCS busters!), I found this result surprising.

The surprise disappears when we dig just a little further into the rankings. As we've done in this space for previous years, our tabulated results are for the specific bias value p=0.75 for the random walker algorithm. What this means is that each walker, considering a game between two teams, will decide that the winner is the better team 75% of the time. Why 75%? Seriously, essentially because it's halfway between 50% (ignoring the outcome altogether) and 100% (complete certainty that the outcome represents the better team). Okay, there's very slightly more to it than that: we tested the rankings across different p values and found that the middle of the range, around 75%, typically corresponds to the low values of rankings violations and the best values to predict bowl game outcomes in historical comparisons. And if you really press me for some other mathematical reasons, it turns out that RWFL rankings of round-robin tournaments appear (in numerical exploration) to agree perfectly with the resulting standings provided p is less than a value somewhere roughly around 0.75.

So, after all that mumbo jumbo, let's vary p and see what happens. You can see in the figure below that Virginia Tech and Miami both do well on the left (p closer to 0.5), but they fall quickly from these high perches as p increases moving to the right in the figure (VT and Miami are represented by the two curves moving quickly upwards towards worse rankings as p increases from left to right). Loosely speaking, this corresponds to the algorithm assigning an on-average stronger schedule to these teams on the left, while penalizing them for their losses on the right. At this point, the balance happens to be working out one way for them; but this high ranking is clearly tenuous at best.
We close today's post by briefly noting that the upcoming conference play might drastically change the above plot against Virginia Tech and Miami, even if they win, simply because the ACC is, on average, not ranked highly by this algorithm. In the plot below, we plot the average numbers of net RWFL votes (expressed as percentages) per team for each FBS conference (grouping the independents together). The way to read this plot is to look at vertical slices (fixed p values), wherein higher values correspond to greater numbers of net votes per team. So far, the ACC appears to be the weakest of the so-called major conferences at most p values, and indeed, it ranks weaker than some of the so-called mid-majors at higher values of p! No hate mail about this please; I'm just the messenger.

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