Subject: Re: Nationals Predictions (Scientific!)

From: price@carla.lbl.gov (Phillip N Price)

Date: 1995/11/02

Newsgroups: rec.sport.disc

OK, probably the last message on this subject. I've re-done the simulations based on my own odds matrix, which was influenced by Jim Parinella's but is not identical and in some ways is fairly different.

Using a standard method for this sort of thing, I gave each team an "ability" score, defined so that the probability of team A beating team B is given by

p(A beats B) = exp(A's ability)/(exp(A's ability)+exp(B's ability))

I adjusted the abilities so that the probabilities roughly matched my expectations, and seemed reasonable given recent tournament records. I used JP's information for some East Coast teams that I don't know very well.

Note that the "ability" scoring does NOT imply that if team A beats team B and team B beats team C, then team A will beat team C. It does imply that if team A is _likely_ to beat team B and team B is _likely_ to beat team C, then team A is likely to beat team C. Clear?

Here's the odds matrix I ended up using:

DoG Doub NY Ring PCS Z Sea Corn Mia Lem SD Chain

DoG 0.000 0.599 0.401 0.891 0.858 0.858 0.858 0.980 0.943 0.976 0.858 0.917

Doub 0.401 0.000 0.310 0.846 0.802 0.802 0.802 0.971 0.917 0.964 0.802 0.881

NY 0.599 0.690 0.000 0.924 0.900 0.900 0.900 0.987 0.961 0.984 0.900 0.943

Ring 0.109 0.154 0.076 0.000 0.426 0.426 0.426 0.858 0.668 0.832 0.426 0.574

PCS 0.142 0.198 0.100 0.574 0.000 0.500 0.500 0.891 0.731 0.870 0.500 0.646

Z 0.142 0.198 0.100 0.574 0.500 0.000 0.500 0.891 0.731 0.870 0.500 0.646

Sea 0.142 0.198 0.100 0.574 0.500 0.500 0.000 0.891 0.731 0.870 0.500 0.646

Corn 0.020 0.029 0.013 0.142 0.109 0.109 0.109 0.000 0.250 0.450 0.109 0.182

Mia 0.057 0.083 0.039 0.332 0.269 0.269 0.269 0.750 0.000 0.711 0.269 0.401

Lem 0.024 0.036 0.016 0.168 0.130 0.130 0.130 0.550 0.289 0.000 0.130 0.214

SD 0.142 0.198 0.100 0.574 0.500 0.500 0.500 0.891 0.731 0.870 0.000 0.646

Chain 0.083 0.119 0.057 0.426 0.354 0.354 0.354 0.818 0.599 0.786 0.354 0.000

So, for example, Seattle has a 10% chance of beating NY, a 57% chance of beating Ring of Fire, and a 50% chance of beating PCS or Z. Parinella's matrix (used for the previous simulations) rates NY stronger than mine, and rates Chain, Lemon, and Miami lower than mine. JP also rates PCS as being considerably stronger than Z (whereas I've put PCS, Z, and Seattle as equal in ability). It's not that I don't think one of those teams is likely to be stronger...I just don't know which one. As I've said, I really "should" do multiple simulations with different odds matrices. Feel free to do it yourself.

Despite the modifications, the overall probabilities for the championship teams barely change. The following table shows the results from 1000 simulations using the matrix above. (For the record, two-way ties in pool play were resolved with the head-to-head result, while three-way ties were resolved as follows: first, did the teams have the same "internal" record? In other words, considering just the group of three tied teams, did each team beat exactly one of the other teams? If one team had a better internal record, it advanced. If one team had a worse internal record, it was dropped and the head-to-head result was used to determine the winner from the other two. If all three teams had the same internal record, then a winner was selected at random. In real life they use point differential, but all I have is the win odds per team, not a model for points scored. )

The first column indicates the team, second shows the number of simulations (out of 1000) in which that team won Nationals, and the third column shows the fraction (sims/1000...tricky, eh?) which can be interpreted as my estimate of the probability of the team winning Nationals.

Team sims frac

"NY" 436 0.436

"DoG" 347 0.347

"Doub" 193 0.193

"PCS" 7 0.007

"Sea" 6 0.006

"Z" 6 0.006

Other 6 0.006 (Other = Ring, SD, or Chain)

I doubt anyone is interested in this stuff (besides me and Parinella) but there you have it anyway.