Building a bracket with the Z-Ratings

The conference tournaments are complete, and the field of 65 has been selected for the 2003 NCAA Men's Basketball Tournament. The selection is basically done by ten guys (two conference commisioners and eight athletic directors) sealing themselves in a room and picking out whoever they like.

Well, I like to be a little more scientific. I haven't seen every candidate for selection, and can't speak about their relative strengths with any authority. Thus I rely on the Bradley-Terry system, a proven statistical method originally applied to college hockey several years ago, which I publish as the Z-Ratings. The final Z-Ratings will be our guide to selecting and placing the teams; I'll reference them extensively throughout this document.

Selecting the field

Step 1: the automatic bids

Each of the 31 Division I conferences gets one automatic bid to the tournament. Every conference except the Ivy League gives its automatic bid to its tournament champion. The Ivy League does not conduct a postseason tournament, and thus the regular season champion gets the bid. This year's champions are:

Rank - Team (Conference)
1 - Kentucky (Southeastern Conference)
3 - Pittsburgh (Big East Conference)
6 - Oklahoma (Big 12 Conference)
8 - Duke (Atlantic Coast Conference)
11 - Louisville (Conference USA)
14 - Illinois (Big Ten Conference)
16 - Dayton (Atlantic 10 Conference)
17 - Creighton (Missouri Valley Conference)
29 - Oregon (Pacific-10 Conference)
47 - Weber State (Big Sky Conference)
52 - Wisconsin-Milwaukee (Horizon League)
54 - Central Michigan (Mid-American Conference)
61 - Pennsylvania (Ivy League)
63 - Tulsa (Western Athletic Conference)
73 - Colorado State (Mountain West Conference)
76 - Manhattan (Metro Atlantic Athletic Conference)
77 - Western Kentucky (Sun Belt Conference)
82 - Troy State (Atlantic Sun Conference)
83 - UNC Wilmington (Colonial Athletic Association)
86 - Austin Peay (Ohio Valley Conference)
87 - Holy Cross (Patriot League)
89 - San Diego (West Coast Conference)
100 - Utah State (Big West Conference)
132 - East Tennessee State (Southern Conference)
142 - Sam Houston State (Southland Conference)
159 - IUPU Indianapolis (Mid-Continent Conference)
181 - Vermont (America East Conference)
182 - Wagner (Northeast Conference)
225 - UNC Asheville (Big South Conference)
236 - South Carolina State (Mid-Eastern Athletic Conference)
267 - Texas Southern (Southwestern Athletic Conference)

Step 2: the at-large teams

The committee has discretion to select the remaining thirty-four participants. If I were running this thing, I'd make it 33 so there would be 64 teams, but that's not my call. No complicated ballots or secret deals here. The top 34 teams without automatic bids get the at-large spots. The one exception to this is Georgia, which declared itself ineligible for postseason play after allegations of academic fraud. Here are the BT at-large selections - teams that were not actually selected for the field are in italics:

Rank - Team
2 - Arizona
4 - Wake Forest
5 - Syracuse
7 - Texas
9 - Florida
10 - Marquette
12 - Kansas
15 - Xavier
18 - Notre Dame
19 - Wisconsin
20 - Stanford
21 - Memphis
22 - Mississippi State
23 - Missouri
24 - Oklahoma State
25 - California
26 - Maryland
27 - Utah
28 - Louisiana St.
30 - St. Joseph's
31 - Connecticut
32 - Brigham Young
33 - Purdue
34 - Indiana
35 - Michigan State
36 - Alabama
37 - Arizona State
38 - Butler
39 - Tennessee
40 - Colorado
41 - Auburn
42 - Southern Illinois
43 - Texas Tech
44 - North Carolina State

32 of the 34 at-large teams above match the actual selections. The two that don't, Texas Tech and Tennessee, would have been the seventh selections from the Big 12 and SEC, respectively. The two teams that would have been left out under the BT system, but got into the actual bracket are Cincinnati and Gonzaga (who, interestingly enough, will meet each other in a first round game). I certainly don't mind Gonzaga, which was the regular-season champion of the 10th ranked conference, and lost the tournament championship on its opponent's home floor. Cincinnati, on the other hand, finished fifth in the 8th ranked conference, one from which three other teams got bids. They also lost in the first round of the Conference USA tournament to the twelfth seed, Southern Mississippi (ranked 120th). To me, their placement is not as justifiable. Though I'd prefer a strictly-by-the-numbers selection process, I think the committee did a pretty good job with the bubble this year.

Seeding the teams

Again, we go with straight logic - no S-Curve here. Order the teams by their ratings, and seed them in groups of four. In other words, the top four teams in the field get No. 1 seeds, the next four are No. 2's, and so on down to the No. 15 seeds. The next three are No. 16 seeds, and the final two teams (No. 64 and 65 in overall ranking) get sent to the opening round game. The table of seeds and rankings within the field appears below. These seeds may be modified later to satisfy certain bracketing principles.

Seed Rk FR Team Seed Rk FR Team Seed Rk FR Team Seed Rk FR Team
111Kentucky 51817Notre Dame 93433Indiana 137349Colorado State
22Arizona 1918Wisconsin 3534Michigan State 7650Manhattan
33Pittsburgh 2019Stanford 3635Alabama 7751Western Kentucky
44Wake Forest 2120Memphis 3736Arizona State 8252Troy State
255Syracuse 62221Mississippi State 103837Butler 148353UNC Wilmington
66Oklahoma 2322Missouri 3938Tennessee 8654Austin Peay
77Texas 2423Oklahoma State 4039Colorado 8755Holy Cross
88Duke 2524California 4140Auburn 8956San Diego
399Florida 72625Maryland 114241Southern Illinois 1510057Utah State
1010Marquette 2726Utah 4342Texas Tech 13258East Tennessee State
1111Louisville 2827Louisiana State 4443North Carolina State 14259Sam Houston State
1212Kansas 2928Oregon 4744Weber State 15960IUPU Indianapolis
41413Illinois 83029St. Joseph's 125245Wisconsin - Milwaukee 1618161Vermont
1514Xavier 3130Connecticut 5446Central Michigan 18262Wagner
1615Dayton 3231Brigham Young 6147Pennsylvania 22563UNC Asheville
1716Creighton 3332Purdue 6348Tulsa
OR23664South Carolina State
26765Texas Southern
Rk = overall ranking by the Z-Ratings.
FR = Field ranking - the ranking of the team among the field of 65.
OR = opening round.

Three comments on the seeds before we go on.
Wake Forest as a #1. People reading this must be going ballistic, thinking there's no way Wake is one of the top four in the nation. Let me remind readers, however, that the Z-Ratings do not place any additional emphasis on conference tournament games. The Demon Deacons did lose in their conference semifinal, but they also won the ACC regular season title, going 13-3 in the second best conference. That counts for enough to get them a #1.
Kansas as a #3. Many people had the Jayhawks as a potential #1. Keep in mind that the two wins over teams outside Division I don't count, reducing Kansas' win total to 23. They also lost seven games. And Andy Katz made a great point on Sportscenter: the Jayhawks played in the Big 12 North, so they had more games against the bottom of the conference (notably Iowa State, Nebraska, and Kansas State). These factors combine to put KU at the No. 12 overall ranking.
UNC Asheville not in the play-in game. Asheville may be under .500, but the Bulldogs' strength of schedule is ranked at 236; South Carolina State is at 307; Texas Southern is at 315. The guidelines are specific about the Nos. 64 and 65 teams meeting in the opening round game; for me, a losing record is no reason to override them.

Establishing the bracket

We adhere to the bracketing principles established by the committee. Some are absolute, while others are merely guidelines. In short, the hard-and-fast rules are:

We also will use similar procedures to those given in the basketball championship handbook. I'll list them as we go along.

Step 1: Place the No. 1 seeds.

No. 1 overall ranked Kentucky gets the South Region placement. Arizona goes to the West; Pittsburgh goes to the East; Wake Forest goes to the Midwest.

Step 2: Place the No. 2 seeds.

Syracuse has priority as the No. 5 overall, but cannot go to the East because Pittsburgh is there. They head to the Midwest. Oklahoma goes to the West, Texas to the South, and Duke to the East.

Step 3: Place the No. 3 seeds.

Florida can't go to the South because Kentucky is there; they get a placement in the East. Given that, we must therefore place Kansas in the Midwest (to avoid Texas and Oklahoma). Marquette gets shipped West, while Louisville gets placed in the South.

Step 4: Place the No. 4 seeds.

Illinois to the Midwest; Xavier to the East; Dayton to the South; Creighton to the West.

Step 5: Check for conflicts and balance in the Nos. 1 through 4 seeds.

No conflicts with the principles are present. We check for balance by adding the FR numbers of the teams in each region. As it currently stands, the bracket is perfectly balanced - the sum of the FR's is 34 in all four regions.

Step 6: Assign first and second round sites.

Proceeding in ranking order, determine which site each team's "pod" goes to. Here's what I come up with:
Boston - Pittsburgh, Syracuse
Tampa - Florida, Illinois
Birmingham - Duke, Xavier
Nashville - Wake Forest, Louisville
Oklahoma City - Texas, Oklahoma
Indianapolis - Kentucky, Marquette
Salt Lake City - Arizona, Dayton
Spokane - Kansas, Creighton

Now it's off to the bottom of the field.

Step 7: Place the No. 13 seeds.

Colorado State to the West; Manhattan to the East; Western Kentucky to the South; Troy State to the Midwest.

Step 8: Place the No. 14 seeds.

UNC Wilmington to the South; Austin Peay to the Midwest; Holy Cross to the East; San Diego to the West.

Step 9: Place the No. 15 seeds.

Utah State to the West; East Tennessee State to the South; Sam Houston State gets shipped to the East; IUPUI ends up in the Midwest.

Step 10: Place the No. 16 seeds.

Vermont goes to the East. Wagner is shipped to the West. UNC Asheville goes to the Midwest. I could have put either Wagner or UNCA in the South, but chose to leave it open for the winner of the opening round game between South Carolina State and Texas Southern. This was done so that Kentucky, at No. 1 overall, gets the lowest ranked team in the first round.

Step 11: Check for conflicts and balance in the Nos. 13 through 16 seeds.

There are no conflicts in this part of the bracket. In terms of balance, the lowest FR sum is 224 (West) and the highest is 229 (Midwest), so the balance is good.

It's time to take on the toughest part of the bracket - the muddled middle.

Step 12: Place the No. 5 seeds.

Notre Dame gets placed in the Midwest. Wisconsin gets shipped to the West. This enables Stanford to be placed in the South (thus avoiding Arizona) and Memphis in the East (in a different region than Louisville and Marquette). This is subject to change, however, as one of these teams will get bumped to a No. 6.

Step 13: Place the No. 6 seeds.

We place Mississippi State in the West to keep them separate from Kentucky and Florida. We then confront the most severe problem thus far - Missouri and Oklahoma State. They both cannot be placed as No. 6 seeds, because that would put five Big 12 teams in the bottom half of the regional draws (creating an intraconference matchup before the regional finals). Since Missouri has the higher overall ranking, we bump them up to a No. 5 - and into the other half of the bracket, placing them in the East. As for who to bring down, since Memphis has the lowest rating of the No. 5 seeds, they get dropped to a No. 6, and assigned to the Midwest. Oklahoma State goes to the East. This would leave California to the South, but that would put them in the same region as Stanford. California is therefore dropped to a No. 7, while Maryland gets brought up and placed in the South No. 6 slot.

Step 14: Place the No. 7 seeds.

California, as the highest No. 7, goes to the East (can't put them opposite Arizona in the West). Utah goes to the South (can't put them opposite Colorado State in the West). LSU goes to the Midwest, and Oregon goes to the West (since Oregon is the fourth team out of the Pac-10, they can be assigned to the same region as another Pac-10 team).

Step 15: Place the No. 8 seeds.

St. Joseph's can't to to either the East or the South, so they're in the Midwest. Connecticut can't go to the East, so they get sent to the South. We now come to Brigham Young, which is a special case because they must be bracketed to play Thursday/Saturday on the first two weekends, due to the school's religious requirements. (The selection committee got major egg on its face by forgetting this point when making the actual bracket.) The West regional, with a Salt Lake City pod, fits this criterion. To keep the Mountain West teams in different regions, we swap Colorado State and Troy State on the No. 13 line. And after all that, we put Purdue in the East.

Step 16: Check for conflicts and balance in the Nos. 5 through 8 seeds.

There is a conflict at this point - Notre Dame and Syracuse both being in the Midwest. We therefore shuffle the No. 5's around a bit: Wisconsin to the South, Stanford to the Midwest, and Notre Dame to the West. This shuffle also improves the balance of this set of seeds, with the lowest FR sum of 95 (Midwest) and the highest of 101 (East). There is a slight concern about having three true No. 6 seeds in the East (with Missouri bracketed as a No. 5 and California as a No. 7), but I can live with it.

49 down, 16 to go.

Step 17: Place the No. 9 seeds.

Another seed swap is inevitable here, as both Indiana and Michigan State are from the Big Ten, and only one can get placed in this half (in the West). Because Indiana has the higher overall ranking, they get the spot. We'll deal with the Spartans in the next step. Butler comes up, because they are the highest rated true No. 10 seed. Alabama can't go to the South, so we put them in the East. Arizona State must go to the South, because Butler cannot play in Indianapolis. The Bulldogs are placed in the Midwest. All this is pending, however - more seed swapping in the next step because of an abundance of SEC teams.

Step 18: Place the No. 10 seeds.

Michigan State gets placed in the Midwest. Now we deal with Tennessee and Auburn from the SEC. One of them has to be bumped to the other side of the bracket - i.e., up to a No. 9. Being the higher rated team, Tennessee gets the bump. This causes Butler to be bumped back down to a No. 10. The Volunteers take Butler's spot as Midwest No. 9. Butler goes to the East. The next team to be placed is Colorado. Guess what? They have to move too, because there are higher-seeded Big 12 teams in the bottom halves of all four regions. Other than Tennessee, the lowest No. 9 is Arizona State, but bumping the Sun Devils down would create an intraconference game (either between Arizona State and California or Oregon, or between Auburn and Florida or Mississippi State). We can't bump Alabama down, because that would give us the same problem that caused us to bump up Tennessee. That leaves Indiana, who goes to the West because Auburn must be placed in the South. Colorado takes the No. 9 West spot.

Step 19: Place the No. 11 seeds.

The first one in this step is simple - Southern Illinois goes to the Midwest. Texas Tech is next, but they have to be bumped down a seed because of overcrowding from the Big 12. Wisconsin-Milwaukee is the highest true No. 12, so they get bumped up. North Carolina State must go to the West. Weber State must go to the East, because Wisconsin-Milwaukee cannot be bracketed with Butler. Therefore, the Panthers end up in the South.

Step 20: Place the No. 12 seeds.

Texas Tech goes to the South - and with that, we're home free. Central Michigan to the Midwest; Ivy champion Pennsylvania in the East; and last, but not least, WAC champ Tulsa in the West.

Step 21: Check for conflicts and balance in the Nos. 9 through 12 seeds.

There are no conflicts here. The balance is very good here as well - the Midwest has an FR sum of 159, and the other three regions are all at 163.

Step 22: Re-check the entire bracket for conflicts.

No intraconference matchups are scheduled before the regional finals. The top four and bottom four seeds in each regional are from different conferences. No teams are playing on home courts or at sites where they are hosts. None of the top five seeds in any region are at a disadvantage. The first three teams from each conference are in different regionals. And perhaps most importantly, the top two seeds would be slated to meet in the national championship game if they both advanced that far. Go take a look at the bracket.

Potential improvements

Junk the play-in game

It's utterly stupid that there have to be 34 at-large teams. There should be 64 teams, so that each team in the field can have a seed. It works for the women; it worked for the men for over a decade; it's the right thing to do.

Allow some flexibility in the Final Four pairings

With the NCAA men's hockey tournament going to four regionals, that sport's committee elected not to specify which regional winners would meet in the national semifinals. The basketball committee should do the same, adopting something to the effect of:

"The regional winners shall be paired so that if all four No. 1 seeds advance to the Final Four, the No. 1 ranked team plays the No. 4 team in one national semifinal, and No. 2 plays No. 3 in the other."

This would enable the strongest teams to remain in their home regions, while attempting to prevent a situation like Arizona and Kentucky meeting in a national semifinal, as opposed to the championship game. The current static rotation serves no purpose in my mind.

Adopt some sort of open, objective system

Some people like a little mystery in their Selection Sunday. I don't. It's much more important to know that your team did or did not make the field because of what it did on the court, not because one member of the committee doesn't like a particular head coach (or something similar).

Having an objective system doesn't diminish the importance of the that Sunday in the middle of March. There is, after all, the fairly subjective question of how seeding is to be handled. In addition, the selection criteria can serve as a sort of "national standings," just as the Pairwise Rankings, or PWR, do in college hockey. I personally think the Bradley-Terry system is superior to the PWR (which is based heavily on the seriously flawed Ratings Percentage Index). But the important thing is to eliminate the smoke-filled room, as least as far as whether a particular team is in or out.

Having said my very lengthy piece in this document and the accompanying bracket, once the teams are selected and seeded, burn the rankings sheets and delete the computer files - settle the championship on the court. Let's jump it up.