A presentation on ranking methods

Here is the annotated PowerPoint from a recent presentation I gave on ranking methods.

Ranking Presentation

Here's a post on debate tabulation. And here's one on debate upsets. A post about weighted wins. Another post about network graphs.

Ski trail maps

I like snowboarding, but I always get confused by trail maps. The difficulty is that mountains have ridges that complicate the geography more than a front-on view can show.

This is Mt. Hood Meadows' perfectly normal trail map, but Heather Canyon requires a special, additional section of map because it is behind a ridge. Here is something completely different:

I downloaded topographical map 45121C6 from topoquest.com. Because it is a topographical map, there is no need to color-code runs by difficulty; the topo lines do that. For example, it is easy to see just how steep the Heather Canyon runs are (at the very top, up to the red-colored run). There are plenty of easy, rolling runs near the green, blue, and orange lifts (which are marked as up arrows, of course). And because the runs are not color-coded by difficulty, the runs can be color-coded to match runs and lifts. It is easy to see that once at the top of the yellow lift, the only way to another lift is to make an immediate right on the pink trail; every other run is yellow and ends back at the bottom of the yellow lift.

Now, this is not my ideal. First, I would remove everything except the topo lines; no green and white shading for below- and above-tree line areas. Instead, the whole catchment area of a lift would be very lightly shaded the appropriate color. (Thus ridge lines would usually be the boundaries, making the ridge lines pop out even more than in the version above.) Areas that could feed to two lifts would be shaded in striped colors. Specific trails would be marked with light-weight lines in the appropriate color, just enough to stand out, but not so much as to create visual clutter against the topo lines.

But even my rudimentary version above helps a skier or rider much more easily navigate and answer the question, "How do I get there?" than current trail maps. The few trails I did mark mostly show boundaries or unique connections. Furthermore, it is a quick step to turn this map into a network graph:

The lifts make the following connections:

Note: A to E and E to A are walkable, because they are opposite ends of the lodge. However, L to M and M to L are not walkable since they are on opposite sides of a big parking lot. And the runs:

Note carefully: There is no pathway from J to K; look carefully at the map. Also, only direct pathways are marked, e.g., H's pathways are only to B, C, and F directly, even though of course A, D, J, L, and M also possible locations a skier or rider can get to from H. (Oops, I just realized I missed H to K -- there is a run that does not go through C.) However, all these secondary pathways must go through B, C, or F first.

Here is another post on network graphs and maps.

Arlington National Cemetery

Every day for five years, I drove through the most confusing spaghetti set of interchanges in the world, around Arlington National Cemetery and the Pentagon in northern Virginia. It is a bold claim, I know, but in an approximately four square mile area, there are: two interstates, four major divided/limited access highways, four bridges over the Potomac, and about two dozen interchanges. Of these, only one is normal, eight-way interchange. The other 20 interchanges all have some restriction: for example, there is an exit for northbound but not southbound traffic. Here, check out Google Maps for yourself:

Google Maps will navigate you through correctly (it has the correct information on the interchanges programmed in), but you can not tell anything about the interchanges from a static map. To help myself, I made an interchange (node) map:

Here is how to read it: If an interchange dot is colored black, both exiting and entering are possible. For example, in the lower right-hand corner of the map between G.W. Parkway northbound and I-395 northbound, you can see that a driver can exit from the Parkway to I-395 and enter the Parkway from I-395 because the lower right node of the interchange is black.

On the other hand, a colored dot indicates that it is only possible to enter the road of the dot's color. For example, at the top of the map, you can see that it is only possible to enter Spout Run from G.W. Parkway northbound and to enter G.W. Parkway southbound from Spout Run.

Please let me know if you find any errors! Check out this beautiful visualization.

Here is another post on network graphs and trail maps.

Debate tournament math

Here's how a high school or college debate tournament works: for the first two rounds of debating, each team is randomly assigned an opponent; for the third round, the winners of the first two rounds are assigned other winners as opponents, while losers debate losers. This system continues for several preliminary rounds, "power matching" teams against opponents with the same record of wins and losses, until the top "brackets" with winning records (7-0s and 6-1s, for example) move on to elimination rounds. Thus, the preliminary rounds are a type of Swiss system tournament, a format that is used in chess competition, too. The number of teams in each final bracket follows a perfect binomial distribution (plus or minus one team or two for odd numbers that require a team to be "pulled up" from a lower bracket).

This approach is generally felt to be fair, although it is a recognized problem that a good team could lose the first or second round and would have easier opponents all the way through. How often does this happen? A visualization helps:


Click on image for more detail.

These are the preliminary varsity policy debate results at the 2009 Harvard invitational high school tournament. (I took out names because I don't want to seem like I'm ragging on any school; I'm really just interested in the math.) Each row represents a different bracket -- the 7-0 at the top, 6-1s one row down, etc., and the 0-7 at the bottom -- and each row is sorted best speaker points (left) to worst speaker points (right) in that bracket. Each arrow represents one actual debate between two teams, pointing to the winner but in the loser's row color. Every single round is there, but I bolded the rounds that the top nine teams won. You can see how differently the top teams (the 7-0 and 6-1s) got that record. Some 6-1s, circled, defeated at least three 5-2 or better teams. Other 6-1s, in squares, defeated only one or no 5-2s. The 6-1 on the far left defeated not one team in the top 20%. Perhaps they could have, but they never even faced off against one. They made it into elimination rounds on the basis of an easier schedule than any other 6-1.

Let me make it absolutely clear, I'm not criticizing the folks who run the Harvard tournament. They do a fine job. The problem is not with their execution. I'm sure that at every point, the 6-1s were given proper opponents for their records; the problem is that some of those opponents went on to lose many of their remaining rounds and revealed their weakness. The problem is the method, which is only as good as the current record of each team accurately reflects its true strength. Since this information can't be known in advance, the only solution so far has been to repeat the process many, many times to thoroughly test and properly rank each team in preliminary rounds. Potentially, what you're looking at above is a raw sort that still contains some errors, like ABCEDLFGJIMNP... it's getting better, but there's still a need for further sorting. Consider it this way: the first round is supposed to determine whether a letter is in the first half of the alphabet or not, by picking up two letters at the same and determining which comes first. Generally speaking, this works, and A, B, C, etc., are likely to end up in the first-half pile. But what happens if the letters you pick up to compare are T and W? T will be misleadingly placed in the first-half pile, and you hope that this doesn't happen two, or three, or seven times in a row, but clearly, it can and did happen, and a team made into the top 6% without ever facing an opponent in the top 20%.

Randomness isn't enough. There needs to be an element added to power-matching that controls for strength of schedule. If you need further convincing, here are the 5-2s highlighted:

Click on image for more detail.

The circled 5-2s defeated at least one other 5-2. (It's hard to see those blue arrows, so click on the image for expansion first.) The 5-2s in squares defeated only one or two 4-3s or better -- that is, they made it into the top 20% and elimination rounds on the basis of defeating only one or two teams in the top 40%. That's quite a disparate schedule: debating other 5-2s and several 4-3s, or debating a few 4-3s and then several teams that are weaker.