## Bracket Breakdown, Bradley-Terry Style

[Bumped. Another tool for anyone who, like me, hasn't filled out their bracket yet.  Dan, can you provide a little more technical information on how you calculate these ratings?  Looks like a ton of work. -KJ]

(Alternate title: This Is Your Bracket on KRACH.)

Undoubtedly you have seen the log5 breakdowns of the bracket over at Basketball Prospectus, based on Ken Pomeroy's rankings. (If you haven't, what's stopping you? The link is to the Midwest preview, but the others are available as well.) You may also recall that back in December, I posted a short article on the Bradley-Terry ranking method (familiar to college hockey fans as KRACH) and applying it to college basketball as a potential RPI substitute. The main point in its favor is that exceptionally strong or weak opponents cannot have such a significant effect on strength of schedule that a loss raises your rating or a win lowers it; this is a significant flaw in the RPI (and some other record-only rating systems; it can happen in points-based rankings like Pomeroy's but that's because you won by less than his model predicted, so it's not really a flaw there). In fact, if you were to play the world's most awful team, a team that would never beat anyone, your rating would remain exactly unchanged (assuming you won). Likewise if you lost to a theoretically perfect team who had an infinite rating.

Another useful point about the Bradley-Terry method is that it, like Pomeroy's rankings, provides an easy method to calculate the odds of one team beating another: simply divide Team A's rating by the sum of (Team A's rating) and (Team B's rating), and you have Team A's probability of defeating Team B. (The formula for Pomeroy's rankings is only slightly more complex.) So I decided to do a similar bracket breakdown, giving odds of each and every team reaching each round.

Odds table after the jump.

Team Seed B-T Rating
2nd Round Sweet 16 Elite 8 Final 4 Title Game Champ
Kansas M1 95.3024 99.24% 87.98% 78.51% 67.24% 52.22% 38.24%
Kentucky E1 59.9940 98.36% 82.45% 63.84% 44.67% 32.60% 17.42%
Syracuse W1 43.7536 96.85% 77.40% 59.79% 42.18% 19.15% 10.84%
West Virginia E2 35.4880 95.98% 75.83% 50.67% 24.29% 15.01% 6.30%
Duke S1 29.6715 98.71% 76.44% 48.07% 29.81% 13.36% 5.11%
Villanova S2 24.3322 96.00% 66.38% 41.52% 21.70% 8.74% 2.99%
Kansas St W2 24.8326 93.90% 61.30% 40.01% 19.26% 6.59% 2.86%
New Mexico E3 23.2448 91.93% 64.65% 30.60% 11.80% 6.10% 2.03%
Purdue S4 23.8606 77.48% 50.70% 26.05% 14.83% 5.91% 2.00%
Temple E5 24.3970 71.62% 49.37% 17.60% 8.74% 4.62% 1.59%
Georgetown M3 20.1088 91.46% 54.82% 33.83% 9.15% 4.07% 1.57%
Baylor S3 19.3236 90.74% 62.27% 31.76% 14.84% 5.23% 1.56%
PIttsburgh W3 17.2214 82.78% 52.88% 24.71% 9.87% 2.74% 0.96%
Ohio St M2 16.2168 88.81% 57.34% 28.18% 6.68% 2.64% 0.89%
Butler W5 16.7066 65.70% 40.92% 14.37% 6.96% 1.89% 0.65%
BYU W7 16.4780 68.24% 29.35% 16.39% 6.39% 1.73% 0.59%
Tennessee M6 16.4854 64.64% 31.59% 18.03% 4.32% 1.72% 0.59%
Texas A&M S5 15.8677 70.84% 33.02% 13.84% 6.52% 2.03% 0.54%
Maryland M4 12.9482 86.71% 50.72% 8.46% 3.86% 1.34% 0.39%
Vanderbilt W4 12.8231 71.64% 35.07% 10.46% 4.40% 1.01% 0.29%
Wisconsin E4 13.6242 79.52% 33.61% 8.30% 3.01% 1.17% 0.28%
Xavier W6 12.1174 67.33% 31.97% 12.29% 4.01% 0.89% 0.25%
Michigan St M5 11.7461 76.19% 40.09% 6.22% 2.69% 0.88% 0.24%
N Iowa M9 14.0160 59.14% 7.95% 4.43% 2.10% 0.76% 0.24%
Texas E8 13.6891 62.03% 12.28% 5.56% 2.02% 0.79% 0.19%
Gonzaga W8 11.8179 54.00% 12.64% 6.14% 2.47% 0.54% 0.15%
St Mary's S10 11.2242 51.73% 17.58% 7.79% 2.67% 0.66% 0.14%
Marquette E6 11.5208 60.63% 22.63% 7.25% 1.81% 0.64% 0.14%
Notre Dame S6 10.3006 59.91% 23.56% 8.54% 2.77% 0.64% 0.13%
Richmond S7 10.4714 48.27% 15.70% 6.71% 2.20% 0.52% 0.10%
Oklahoma St M7 9.7051 54.87% 23.32% 8.61% 1.45% 0.42% 0.10%
Florida St W9 10.0687 46.00% 9.60% 4.29% 1.57% 0.31% 0.08%
Missouri E10 9.6902 51.40% 12.37% 4.57% 1.01% 0.32% 0.06%
Louisville S9 8.9170 51.61% 12.41% 4.34% 1.47% 0.31% 0.05%
San Diego St M11 9.0186 35.36% 12.51% 5.33% 0.85% 0.23% 0.05%
UNLV M8 9.6847 40.86% 4.03% 1.88% 0.73% 0.21% 0.05%
UTEP W12 8.7206 34.30% 16.09% 3.70% 1.23% 0.22% 0.05%
Clemson E7 9.1622 48.60% 11.25% 4.02% 0.85% 0.26% 0.05%
Cornell E12 9.6673 28.38% 13.63% 2.64% 0.77% 0.24% 0.05%
Georgia Tech M10 7.9837 45.13% 17.25% 5.63% 0.82% 0.21% 0.04%
California S8 8.3612 48.39% 11.10% 3.72% 1.21% 0.24% 0.04%
Florida W10 7.6704 31.76% 8.64% 3.27% 0.79% 0.13% 0.03%
Wake Forest E9 8.3804 37.97% 5.13% 1.75% 0.46% 0.13% 0.02%
Old Dominion S11 6.8942 40.09% 12.45% 3.47% 0.86% 0.15% 0.02%
Washington E11 7.4802 39.37% 11.31% 2.73% 0.50% 0.13% 0.02%
Siena S13 6.9356 22.52% 8.23% 2.06% 0.59% 0.10% 0.01%
Utah St S12 6.5318 29.16% 8.04% 1.93% 0.53% 0.09% 0.01%
Minnesota W11 5.8808 32.67% 10.38% 2.47% 0.49% 0.06% 0.01%
Murray St W13 5.0753 28.36% 7.92% 1.21% 0.28% 0.03% 0.005%
New Mexico St M12 3.6712 23.81% 6.62% 0.40% 0.08% 0.01% 0.001%
Oakland W14 3.5825 17.22% 4.78% 0.77% 0.10% 0.009% 0.001%
Wofford E13 3.5096 20.48% 3.39% 0.29% 0.04% 0.006% 0.0005%
Santa Barbara M15 2.0441 11.19% 2.10% 0.24% 0.01% 0.001% 0.00006%
Houston M13 1.9848 13.29% 2.57% 0.09% 0.01% 0.0009% 0.00005%
Sam Houston St S14 1.9723 9.26% 1.72% 0.18% 0.02% 0.0009% 0.00005%
Montana E14 2.0411 8.07% 1.42% 0.12% 0.008% 0.0007% 0.00004%
Ohio M14 1.8769 8.54% 1.08% 0.15% 0.007% 0.0005% 0.00003%
North Texas W15 1.6135 6.10% 0.71% 0.09% 0.006% 0.0003% 0.00001%
Morgan St E15 1.4848 4.02% 0.55% 0.05% 0.002% 0.0002% 0.000006%
Vermont W16 1.4211 3.15% 0.36% 0.04% 0.003% 0.0001% 0.000006%
Robert Morris S15 1.0140 4.00% 0.34% 0.03% 0.001% 0.00004% 0.000001%
East Tenn St E16 1.0015 1.64% 0.14% 0.009% 0.0004% 0.00002% 0.0000005%
Lehigh M16 0.7255 0.76% 0.04% 0.003% 0.0001% 0.000004% 0.0000001%
Ark Pine Bluf S16 0.3863 1.29% 0.06% 0.001% 0.00003% 0.0000004% 0.000000004%

Blue indicates a team that the Bradley-Terry method rates more highly than Pomeroy's (in terms of probability of beating an "average" team), red indicates the reverse. The darker the shade, the more significant the difference. The most extreme positive difference: New Mexico is expected to win 95.88% of the time against an average team by the Bradley-Terry method but just 88.21% by Pomeroy. Utah State, on the other extreme, is expected to win 86.72% by the Bradley-Terry method but 93.01% by Pomeroy. (Minnesota and Wisconsin, interestingly, are the two next most extreme in B-T pessimism. Oakland and Kentucky are second and third in B-T optimism.)

Some notable differences, bullet-style:

• Kansas is #1 by a mile in the Bradley-Terry rankings; only Kentucky and Syracuse even have a 30% chance of pulling the upset. This and the models' disagreement on Ohio State and Maryland (Pomeroy likes both considerably better) leads to a big jump in Kansas's title chances.
• Bradley-Terry gives Duke just over a 5% chance to win it all, compared to 24% by Pomeroy. Part of this is that Duke is more lightly regarded here, but the fact that all of the next six teams by seed in their region are rated at least as high here as by Pomeroy plays a role as well. West Virginia, despite having to get past Kentucky just to get to the Final Four, is considered to be more likely to win it all.
• Kentucky also sees a big jump in their chances of winning it all here, in large part due to difference of opinion on Wisconsin (who's actually the favorite, by a narrow margin, to escape the East according to Pomeroy's numbers).
• Both systems rate us about equally (24th Pomeroy, 26th Bradley-Terry). The main differences in our percentages are due to our opponents: New Mexico State is considered a much tougher opponent by this system (79th) than it is by Pomeroy's (115th), reporting a 23.8% chance of upset compared to 14.2%. But Maryland is not so highly regarded (10th Pomeroy, 22nd Bradley-Terry), so this model gives us a slightly better chance of reaching the Sweet 16. Then we meet the buzzsaw named Kansas, and Bradley-Terry considers them a much bigger busszaw, so our chances thereafter drop accordingly.
• Pomeroy's formula has Wisconsin tied for 3rd most likely to win it all (with Syracuse, behind Duke and Kansas). This one has them 21st.
• Just two 10 seeds (and no one lower) are favored in the first round by this method: St. Mary's and Missouri. Pomeroy favors Old Dominion, Utah State, Georgia Tech, and St. Mary's.
• Poor Arkansas-Pine Bluff. Pomeroy's model gives them a 1 in 2.5 billion chance of winning it all; this method isn't even that generous, at 1 in 25 billion. (That may be an artifact of their calculations simply being for the "play-in winner", which could have been Winthrop, who is rated slightly higher.)

I don't mean this to supplant Pomeroy's rankings and the analysis by the group over at Basketball Prospectus, of course; their data factors in scores and not just win-loss records. But by ignoring record entirely, their analysis also loses some information. Losing (or winning) close games isn't all luck, and a team that consistently failed in the clutch is more likely to do so again. Ideally some hybrid of the two could be used, perhaps by averaging the two rankings in some way before applying log5.

Finally, some major differences among the RPI and Bradley-Terry rankings:

• Teams the RPI overrates: San Diego State (19th vs. 38th), Cal (20th vs. 42nd), Siena (27th vs. 48th), Old Dominion (28th vs. 50th), Utah State (32nd vs. 54th), UAB (42nd vs. 58th), Wichita State (45th vs. 60th), Kent St. (47th vs. 74th), Oakland (51st vs. 80th), New Mexico State (52nd vs. 79th). Common thread, for most: Lots of games against the 100-200 range and/or low 200s, very few against the high 200s and 300s. Oakland seems to be the exception; theirs may be due to the opposite effect of playing teams way over their ranking and getting a huge SOS boost to offset the effect the loss has on RPI.
• Teams the RPI underrates: Gonzaga (37th vs. 25th), Missouri (46th vs. 33rd), Notre Dame (48th vs. 30th), Cornell (49th vs. 35th), Marquette (50th vs. 27th), Virginia Tech (58th vs. 36th), Ole Miss (62nd vs. 47th), Seton Hall (68th vs. 43rd). Again, there's a common theme: Tons of games against "RPI anchors" which aren't (for good teams) functionally that much different from playing teams in the 150-200 range but are punished far more severely in the RPI.

This seems to confirm what conventional wisdom has always said: RPI puts a big emphasis on choosing your cupcakes carefully. Choose poorly, and your RPI will suffer even if you win. Choose wisely, and you can get a bunch of near-guaranteed wins without making your strength of schedule look too embarrassing. This year, Virginia Tech chose ... poorly. And it is the reason they didn't make the tournament.

This is a FanPost, written by a member of the TOC community. It does not represent the official positions of The Only Colors, Inc.--largely because we have no official positions.