Effect of Multiple Byes on Number of Games Played

Yesterday I investigated the impact having multiple byes has on winning conference tournaments, aiming to quantify the "benefit" of gaining/losing an additional bye. However, you could also argue there are other ways to measure a "benefit", such as the number of games you're expected to play (similar to the short-term effects of seeding). More games means more game experience, more practice time, more fan engagement, etc.

Recapping my findings: relative to chances of winning the whole thing, the current format:

1. Benefits the top seeds, who have to play 1 less game (except for Virginia)
2. Is about the same for the middle seeds, who have to play the same number of games, but have a slightly higher chance of playing a worse team in their first game
3. Damages the chances of the higher ranked worst seeds, who have to play an additional game

		Champion
		Current	No Byes	Dif
1	Virginia	49.35%	49.82%	-0.46%
2	Duke	26.62%	25.76%	0.86%
3	Miami FL	2.81%	2.02%	0.79%
4	Clemson	5.05%	4.35%	0.70%
5	North Carolina St.	1.67%	1.85%	-0.18%
6	North Carolina	7.03%	7.16%	-0.13%
7	Virginia Tech	1.58%	1.68%	-0.11%
8	Florida St.	1.31%	1.34%	-0.03%
9	Louisville	1.37%	1.40%	-0.03%
10	Notre Dame	1.21%	1.88%	-0.67%
11	Syracuse	0.57%	1.07%	-0.50%
12	Boston College	0.36%	0.76%	-0.40%
13	Georgia Tech	0.47%	0.34%	0.12%
14	Wake Forest	0.48%	0.51%	-0.04%
15	Pittsburgh	0.12%	0.05%	0.08%

But what about games played? It seems fairly straight forward that the teams with more byes will have less games under the current format, and the teams with no byes will have more games (since they play weaker teams at the start and have a greater chance to move on).

		E[Games Played]
		Current	No Byes	Dif
1	Virginia	2.04	2.05	-0.01
2	Duke	1.52	2.38	-0.86
3	Miami FL	0.49	0.97	-0.48
4	Clemson	0.69	1.29	-0.61
5	North Carolina St.	0.92	0.93	-0.01
6	North Carolina	1.41	1.40	0.01
7	Virginia Tech	0.76	0.71	0.05
8	Florida St.	0.63	0.63	0.00
9	Louisville	0.64	0.64	0.00
10	Notre Dame	1.32	0.74	0.58
11	Syracuse	0.88	0.55	0.33
12	Boston College	0.90	0.64	0.27
13	Georgia Tech	0.78	0.43	0.35
14	Wake Forest	0.67	0.54	0.13
15	Pittsburgh	0.34	0.11	0.24

This follows the above hypothesis as expected. As before, Virginia is the sole outlier, which is because they have a first round bye no matter how the bracket is constructed.

Effect of Multiple Byes on Win Probability + ACC Bracket Math

In the past I've always put out win probabilities for my MDS Model for the top 5 conference tournaments. Thanks to the Big 10 moving everything up a week to play in Madison Square Garden, I completely missed it this year. So here's the bracket math for the ACC and only the ACC:

Seed	Team	2nd Round	Quarters	Semis	Final	Champion
1	Virginia	100.00%	100.00%	84.04%	70.36%	49.35%
8	Florida St.	100.00%	49.58%	7.86%	3.93%	1.31%
9	Louisville	100.00%	50.42%	8.11%	4.09%	1.37%
4	Clemson	100.00%	100.00%	50.51%	12.96%	5.05%
5	North Carolina St.	100.00%	60.17%	25.27%	5.18%	1.67%
12	Boston College	56.39%	23.97%	8.36%	1.37%	0.36%
13	Georgia Tech	43.61%	15.86%	15.86%	2.12%	0.47%
2	Duke	100.00%	100.00%	72.32%	52.98%	26.62%
7	Virginia Tech	100.00%	55.27%	12.90%	5.98%	1.58%
10	Notre Dame	77.36%	39.45%	9.49%	4.49%	1.21%
15	Pittsburgh	22.64%	5.28%	5.28%	1.13%	0.12%
3	Miami FL	100.00%	100.00%	35.31%	11.00%	2.81%
6	North Carolina	100.00%	70.14%	44.93%	19.40%	7.03%
11	Syracuse	57.68%	18.78%	8.69%	2.46%	0.57%
14	Wake Forest	42.32%	11.07%	11.07%	2.55%	0.48%

Virginia is a cut above in the model, with Duke the only other serious threat. Nothing surprising there.

But this spawns a bigger question: which teams/seeds benefit from the double bye/single bye tournament format? All major conferences (including the ACC) use some iteration of this multiple bye system where the best teams get two byes (playing at most 3 games), the middle tier get one bye (playing up to 4 games), and the worst teams have to play 5 straight games (in 5 days) to win the conference title.

I've written on the short-term effect of seeding in the past, and so there needs to be a definition of what "benefits" a team. For my purposes here, I'm only looking at each team's chance of winning the tournament.

I reran the win probabilities with a standard 16-team bracket (where only the 1-seed gets a bye) and then compared this output to the current format:

Seed	Team	Quarters	Semis	Final	Champion
1	Virginia	100.00%	84.04%	70.70%	49.82%
8	Florida St.	49.58%	7.86%	3.98%	1.34%
9	Louisville	50.42%	8.11%	4.14%	1.40%
4	Clemson	70.72%	43.16%	11.07%	4.35%
13	Georgia Tech	29.28%	11.55%	1.54%	0.34%
5	North Carolina St.	57.50%	27.79%	5.69%	1.85%
12	Boston College	42.50%	17.50%	2.86%	0.76%
2	Duke	91.52%	69.82%	51.27%	25.76%
15	Pittsburgh	8.48%	1.95%	0.42%	0.05%
7	Virginia Tech	49.01%	13.66%	6.37%	1.68%
10	Notre Dame	50.99%	14.57%	6.94%	1.88%
3	Miami FL	61.30%	25.59%	7.90%	2.02%
14	Wake Forest	38.70%	12.16%	2.75%	0.51%
6	North Carolina	67.43%	45.74%	19.73%	7.16%
11	Syracuse	32.57%	16.50%	4.62%	1.07%

As expected, the current format:

1. Benefits the top seeds, who have to play 1 less game (except for Virginia, which we'll get to)
2. Is about the same for the middle seeds, who have to play the same number of games, but have a slightly higher chance of playing a worse team in their first game
3. Damages the chances of the higher ranked worst seeds, who have to play an additional game

		Champion
		Current	No Byes	Dif
1	Virginia	49.35%	49.82%	-0.46%
2	Duke	26.62%	25.76%	0.86%
3	Miami FL	2.81%	2.02%	0.79%
4	Clemson	5.05%	4.35%	0.70%
5	North Carolina St.	1.67%	1.85%	-0.18%
6	North Carolina	7.03%	7.16%	-0.13%
7	Virginia Tech	1.58%	1.68%	-0.11%
8	Florida St.	1.31%	1.34%	-0.03%
9	Louisville	1.37%	1.40%	-0.03%
10	Notre Dame	1.21%	1.88%	-0.67%
11	Syracuse	0.57%	1.07%	-0.50%
12	Boston College	0.36%	0.76%	-0.40%
13	Georgia Tech	0.47%	0.34%	0.12%
14	Wake Forest	0.48%	0.51%	-0.04%
15	Pittsburgh	0.12%	0.05%	0.08%

The outlier is Virginia, which has the same number of games in each setup, yet has a worse chance of winning it all. Their first (bye) and second (Florida St/Louisville) matchups remain the same, but Duke's chances of making the final are greatly increased (which specifically hurts Virginia).

Overall, playing a different number of games has the greatest effect on the odds of winning the tournament, but matchups do also matter. This is why the very worst seeds are helped slightly and the best (Virginia) is hurt, even though they have to play 1 more and 1 less game, respectively.

Points Per Minute Deficit (NCAAB)

As any diehard fan can tell you, when your team is down, you won't count them out until a comeback just isn't feasible anymore, models and simulations be damned. But there has to be some degree of reasonableness, right? 

Growing up a Carolina basketball fan, I always had certain heuristics in the back of my mind (instilled by my dad) on whether we could come back or not. The thinking went that if we could keep the score within the number of minutes left in the game, then we had a chance to chip away until we tied or took the lead at a rate of 1 point per minute. For example, if UNC trailed by 10 points with 9:50 left to play, then we needed to get to within 9 by 9:00 left to play. And then trail by 8 (or better) by 8:00 left, 7 by 7:00 left, etc. Note that this doesn't mean that the trailing team went on to win necessarily; all it means is that the trailing team either tied or took the lead at some point later in the game.

I aimed to determine if this heuristic had any basis in real-life results by looking at play-by-play NCAAB data for the entire 2016-17 season. I scraped KenPom's play-by-play win probability graphs to get scoring data on every possession, which ultimately gave me 393,719 individual possessions in 2nd half/overtime to analyze. Of those, 13,917 (3.5%) matched the criteria of being down x points with x minutes remaining.

There actually is a fairly linear relationship for this idea, with a largest deficit implying a lesser chance of a comeback (as you would expect):

Using linear regression, the relationship as shown above can be described as:

%_Comeback = -0.021*Min_Remaining + 0.409

This model has a fairly good fit too, with an r-squared value of 0.828.

For my purposes, I usually applied this thought process up until UNC trailed by 6 or less (a 2 possession game). At that point all you need is to make 2 threes and it's tied. However, based on the above graph it appears I should ONLY hold out hope when the deficit is within 2 possessions or less, since that is the only stretch in which the trailing team has better than a 1 in 4 chance of coming back. Good luck telling that to any fan of a team down by 7 or more though.