Solving the "Kagulo sa NCAA" mess

IMMEDIATELY after the NCAA games on Tuesday, I began thinking about the eight remaining scenarios and that possible five-way tie for second place.

So after going aboard a public transport from the FilOil Flying V Arena, I began doing the calculations while on the way to Robinson's Galleria. While at the Pasig mall attending a church service, I continued the math (I repent, Lord, for this).

Then from home, I spent the rest of the night at a nearby 24/7 resto with Wi-Fi to finish everything, and now, the solution to the huge problem known as "Kagulo sa NCAA".

First, the rules of destroying the ties. Unlike the UAAP where playoffs only happen for second and fourth places and after calculating the quotients, the team(s) with the worst quotient will either be relegated to the lowest seed or get eliminated at all, the NCAA uses a playoff to determine all the seeds and as well not eliminate everyone with the lousiest quotients.

Because of that, a stepladder format will be used in case of ties between three or more teams. As an example, here's how it looks like in a five-way logjam.
Now, the teams with playoff hopes hanging in the balance:

Letran (12-5), Arellano (12-6), JRU (11-6), Mapua (11-6), Perpetual Help (11-6)

Then, the aggregate scores (i.e., the combined scores of the two teams involved during both their meetings) of ten match-ups which will be the basis of computing the quotients:

Arellano 173 JRU 192, Arellano 149 Letran 164, Arellano 176 Mapua 177, Arellano 150 Perpetual Help 153
JRU 148 Letran 158, JRU 157 Mapua 155, JRU 145 Perpetual Help 146
Letran 157 Mapua 159, Letran 79 Perpetual Help 71 (they will face off again on Friday)
Mapua 149 Perpetual Help 147

Finally, the remaining games on Friday:

12:00 p.m.-JRU vs. San Sebastian
2:00 p.m.-EAC vs. Mapua
4:00 p.m.-Perpetual Help vs. Letran

The Heavy Bombers, the Cardinals, and the Altas will dictate their respective fates on Friday, so let's take a look at the possibility assuming n of them win, n=0, 1, 2, 3.

First, if n=1. If only either one of JRU, Mapua, and Perpetual Help wins, the semifinal cast will be complete and only the seeding will be determined on Tuesday.

1. If only JRU wins, Letran will have a chance to reclaim top spot from San Beda, while JRU will battle Arellano for third place.

2. If only Mapua wins, it will be Letran-San Beda for first and Arellano-Mapua for third.

3. If only Perpetual Help wins, a three-way tie for second happens between Arellano, Letran, and Perpetual Help. Since everything will still depend on the latter's winning gap, let's take a look at their respective head-to-head aggregate winning margins:

Arellano -18, Letran +23-x, Perpetual Help -5+x; x denotes Perpetual Help's winning margin.
Note: For both sub-scenarios, the loser of the first game will finish fourth and will take on San Beda, which will have a twice-to-beat advantage, in the Final Four. The other two teams will engage in an essential best-of-three Final Four series of their own (playoff for second included).

Now, what if either two of the three teams in focus win? For an early summary, if that happens, Arellano will not be out of the woods yet as it is the team with the worst quotient (in fact, Arellano has the worst quotient in all the ties where the team is involved).

4. If only JRU and Mapua win, Perpetual Help will be automatically eliminated, Letran will battle San Beda for first, and a three-way tie for third takes place. Here are their quotients:

JRU 1.064, Mapua .997, Arellano 946
Note: The loser of the Mapua-Arellano game will be out of the Final Four, and the match between the winner of the aforementioned tiff and JRU will determine who will finish third and fourth.

5. If only JRU and Perpetual Help win, a four-way tie for second happens. Again, since everything is dependent on Perpetual Help's lead, the total winning margin will be used here.

Letran +33-x, JRU +8, Perpetual Help -4+x, Arellano -37 (refer above for the denotation of x).
6. If only Mapua and Perpetual Help win, JRU is automatically out (take note that JRU plays first on Friday, so if the team loses, its Final Four is not on its hands anymore), and just like Scenario 5, we have a four-way deadlock at second. Here are the head-to-head winning margins of the four remaining teams.

Letran +21-x, Mapua +5, Perpetual Help -7+x, Arellano -19
Notes: For both Scenarios 5 and 6, in all sub-scenarios, the loser of the first stepladder phase will be eliminated, the loser of the second stepladder phase will finish fourth and will battle San Beda in the semis, while the third stepladder phase will become a virtual best-of-three series as both teams will also clash in the Final Four.

7. As I mentioned a while ago, a JRU loss and its playoff fate will now rely on EAC and Letran. If EAC also wins, JRU and Mapua can only hope Perpetual Help loses to Letran to stay in the race. What if Letran wins as well? Then Letran and San Beda clash for the top seed, Arellano will finish third with a twice-to-beat disadvantage, and a three-way tie for fourth occurs. The head-to-head quotients of JRU, Mapua, and Perpetual Help are as follows:

JRU 1.005, Mapua 1.000, Perpetual Help 0.995
Note: The losers in both games will be out of contention, and the remaining team, i.e., the winner of the second phase, will join San Beda, Letran, and Arellano in the Final Four.

Finally, the Super "Kagulo sa NCAA"! What if JRU, Mapua, and Perpetual Help win? I already mentioned the disadvantage of JRU playing first, but the advantage of it is that if JRU wins, this will put more pressure on Mapua and Perpetual Help. In fact, while Perpetual Help can get the easiest of comforts if both JRU and Mapua lose, it can also get the hardest of pressures should JRU and Mapua walk away victorious. Now, let's look at the potential five-way tie for second. Here are the five teams' head-to-head winning margins.

Letran +31-x, JRU +10, Mapua +3, Perpetual Help -6+x, Arellano -38
Notes: In all sub-scenarios, the losers of the first two phases will be eliminated, the loser of the third phase will battle San Beda in the Final Four, and the fourth phase will be the other Final Four series, which will be a virtual best-of-three affair.

Summary:

1. If Perpetual Help wins, a virtual best-of-three Final Four series between the second seed and the third seed will become imminent, but the third seed will be determined by another playoff(s).

2. If JRU, Mapua, and Perpetual Help have the same result (win or loss) on Friday, they are still in the Final Four race. Otherwise, the winner(s) will stay in the race and the loser(s) will be out.

3. Although Perpetual Help will play in the nightcap, the team must beat Letran by 16 or more to safely enter the Final Four regardless of the eventual remaining scenarios after the JRU and Mapua games and by 19 or more to have a longer rest come the stepladder playoffs.

4. Arellano can enter the Final Four outright if either two of San Sebastian, EAC, and Letran win.

Follow the writer on Twitter: @IvanSaldajeno