Bracket season is upon us once again, which means that
millions of normally hardworking Americans will spend the next week carefully
examining an ordered list of 64 basketball teams and trying to select the
winners of the 63 games those teams will play. (Fine, it’s really 68 and 67,
but let’s not pretend that most people care overmuch about the opening-round
games, especially because most bracket pools ignore them.) Those picks will,
for the most part, be entered into online contests which will compare the predicted
brackets to the results of the games as they are played, and select the best
one as the winner. This should come as news to exactly nobody.
But for all the care put into the individual brackets by the
participants, the people running the contests generally put shockingly little
thought into the actual method used to evaluate them.
The most popular bracket challenge is run by ESPN.com, which
uses the following scoring system:
|
Round
|
Points
|
|
1
|
10
|
|
2
|
20
|
|
3
|
40
|
|
4
|
80
|
|
5
|
160
|
|
6
|
320
|
This is simple enough; the number of points double with each
round. The simplicity is appealing in and of itself, and a variation on this
method is also used by Yahoo and CBS Sports, among others. It also has a simple
mathematical basis – there are two teams competing for each second-round spot,
four competing for each Sweet Sixteen berth, eight for every slot in a regional
final, and so on. Your score for correctly selecting the team on a given line
of the bracket is directly proportional to the number of options you could put
there.
But a cursory examination of the actual brackets reveals the
unsatisfactory nature of this approach. Take the simplest example: If you pick a
#1 seed to beat their 16th-seeded opponent in the first round (no 1
seed has ever lost at this stage), you get 10 points. On the other hand, if you
successfully predict the biggest upset of the first day, say a #15 beating a #2
(which happens about once every four years), you get… 10 points. This, despite
the fact that the second prediction is vastly more unlikely than the first.
This problem becomes exacerbated if you examine the bracket
with a wider scope. Correctly selecting the national champion awards you with
320 points. Over the 29 years of the 64-team format, the tournament has been
won by a top-3 seed 26 times, with 18 of those champions being #1 seeds.
Barring an upset of rather monumental proportions (for instance, #8 seed
Villanova winning the title in 1985), someone in a pool of reasonable size is
probably going to pick the winner.
Meanwhile, the first round of the bracket is often famously
chaos-laden, making it highly difficult to nail down. ESPN’s bracket challenge
features millions of entrants, and in several years of participation, I have
never seen anyone come especially close to correctly picking the entire first
round; the highest number of correct predictions in the tournament’s first two
days is usually in the mid-to-high 20’s out of 32. If, however, someone were to
manage the titanic feat of exactly picking every 5-12 upset without picking any
that didn’t happen, nailing each of the coinflipesque 8-9 and 7-10 games, and
somehow anticipating the win or two by seeds 13 or higher that seem to slip in
every year, this astonishing achievement in prognostication would be rewarded
with… 320 points, the same number that would go to any of the thousands of
people who picked the right top seed as the last team standing. And that
understates things, because it doesn’t count the points those who correctly
pick the champ get for picking them to win their preceding five games, which
add up to a further 310. If every-first-round-game guy doesn’t nail the last
one as well, he’s still almost certain to lose the challenge despite his
once-in-a-thousand-years feat.
Surely there’s something that can be done about this, right?
This post wouldn’t be worth much if the answer were “no.”
What can be done is to take a look at the scoring system and what it should
ideally represent, which will hopefully result in reasonable rewards for the
various possible picks. I would define “reasonable rewards” based on the
proposition that any pick you make should have the same expected payout over an
infinitely large number of brackets. The way to manage this is to simply set
the following relation:
Points awarded for correct pick = 1/(Odds of the pick being
correct)
So, since #1 seeds have historically won all of their games
against 16s, the odds of that pick being correct are either 100% or very close
to it, and the reward for picking the 1 seed is 1 point, or very close to it.
Meanwhile, 15 seeds have a record of 7-109 against #2’s, and thus successfully
predicting such an upset would be expected to be worth 15.6 points (109/7),
give or take.
However, it’s not possible to run these numbers using simple
historical performance by the various seeds, because there are several possible
events that have never happened, and would therefore be expected to have
historical odds of 0 (thus resulting in infinite points for someone who
successfully picks the first one). For instance, before last year’s tournament,
no 15 seed had ever reached the Sweet Sixteen, and no 9 seed had made the Final
4 in the 64-team era. It famously remains the case that no 16 seed has ever
upset a #1; it is also true that no team seeded 13th or lower has
made the Elite Eight, and no 7 or 10 seed has made the Final Four (although
four 8’s, one 9, and three 11’s have done so). All of these things will happen
eventually, given time; it’s just that they haven’t happened in the 29 years of
64-team brackets to date.
So if not through direct historical performance, how can we
estimate the odds of a 16-seed beating a 1 seed – or, for that matter, a
16-seed making the Final Four or winning the title? The answer I’m using comes
in two parts. The first is Pythagorean winning percentage, a tool originally
created by Bill James for use in baseball. The idea is to estimate a team’s
winning percentage based on runs scored and allowed (or, in basketball,
points). The initial formula in baseball was:
Winning percentage = RS2 / (RS2 + RA2)
Later modifications have changed the value of the exponent
and made it dependent on the scoring level in the team’s games, neither of
which is relevant here. In basketball, the Pythagorean exponent is 13.9,
instead of 2.
I calculated the Pythagorean winning percentage for each
seed across every matchup that has occurred in the last 29 years of NCAA
tournaments. (I tried just lumping all the games together before calculating
Pythagorean winning percentage, but the early matchups skewed the numbers too
much – 1 seeds generally hammer 16s by such huge margins that their overall win
totals were vastly overstated. Going matchup-by-matchup resolved this issue.) Using
the historical points scored and allowed numbers, here are the expected winning
percentages for the higher seed across each first-round matchup:
|
Matchup
|
Actual W%
|
Pythag W%
|
|
1 vs. 16
|
1.000
|
.992
|
|
2 vs. 15
|
.940
|
.960
|
|
3 vs. 14
|
.853
|
.911
|
|
4 vs. 13
|
.784
|
.868
|
|
5 vs. 12
|
.647
|
.712
|
|
6 vs. 11
|
.664
|
.676
|
|
7 vs. 10
|
.603
|
.628
|
|
8 vs. 9
|
.483
|
.501
|
The correspondence is respectable, if imperfect; in
particular, the relatively common upsets are under-predicted. This is likely
caused by the fact that not all 14 seeds, for instance, are the same; the variation
in their quality should make upsets more likely than would otherwise be
expected.
Using Pythagorean wins provides a way to come up with
non-zero (if still minute) odds for upsets that have not yet occurred, but it
still leaves no option for projecting matchups that have never arisen. No 16
seed has ever faced a non-1-seeded foe. 15s have a few more data points, with
opponents including 2, 3, 7, and 10. There are gaps in every seed’s historical
opponent list, all the way to the top (no 1 seed has ever squared off with a 14
or 15). Since we want a way to predict the odds of a 16 seed making the Final
Four, we also need a way of calculating their chances against the 8 or 9 they
would face in the second round, and the teams that would be waiting beyond
that.
The second part of the answer comes in the form of a tool
I’ve used previously for tennis – the multiplicative Elo rating, which takes
each seed’s performance against its schedule of opposing seeds and generates a
strength rating, which is used to estimate winning percentage as follows:
Winning percentage = (Rating of team A) / (Rating of team A
+ Rating of team B)
The ratings are adjusted iteratively until they match the
historical Pythagorean win totals across seed matchups. In case anyone is
curious, they are:
|
Seed
|
Rating
|
|
1
|
36.60
|
|
2
|
21.56
|
|
3
|
16.26
|
|
4
|
11.91
|
|
5
|
10.63
|
|
6
|
9.27
|
|
7
|
8.20
|
|
8
|
5.81
|
|
9
|
5.33
|
|
10
|
5.48
|
|
11
|
4.65
|
|
12
|
5.01
|
|
13
|
1.83
|
|
14
|
1.64
|
|
15
|
0.85
|
|
16
|
0.29
|
The ratings decrease steadily, with a few slight exceptions
(10s score marginally higher than 9s, and 12s than 11s). More notable is the fact that the
sharp drop-offs are in sensible places; for instance, the 8-through-12 seeds
are bunched together (as the bottom tier of at-large teams), and there’s a big
dip down to the 13 seeds (the automatic qualifiers, who would not be in the field
had they not won their conference tournaments). The #1 seeds also tower
impressively over all of their competition, which does not come as an
Earth-shaking surprise.
Let’s repeat the earlier table of first-round matchups with
the Elo-projected winning percentages added in:
|
Matchup
|
Actual W%
|
Pythag W%
|
Elo W%
|
|
1 vs. 16
|
1.000
|
.992
|
.992
|
|
2 vs. 15
|
.940
|
.960
|
.962
|
|
3 vs. 14
|
.853
|
.911
|
.909
|
|
4 vs. 13
|
.784
|
.868
|
.867
|
|
5 vs. 12
|
.647
|
.712
|
.680
|
|
6 vs. 11
|
.664
|
.676
|
.666
|
|
7 vs. 10
|
.603
|
.628
|
.599
|
|
8 vs. 9
|
.483
|
.501
|
.522
|
Elo is still overly optimistic about 3 and 4 seeds in the
first round, but it has a more realistic picture of 5’s and 7’s. More to the
point, it also offers a way to extend projections beyond the first round. Here
are the projected odds of a team of each seed reaching each round (or further),
given the actual potential matchups they would face and the likelihoods
thereof:
|
Seed
|
Round of 32
|
Sweet 16
|
Elite 8
|
Final 4
|
Title game
|
Champ
|
|
1
|
0.992
|
0.861
|
0.672
|
0.466
|
0.283
|
0.164
|
|
2
|
0.962
|
0.725
|
0.460
|
0.218
|
0.106
|
0.048
|
|
3
|
0.909
|
0.621
|
0.313
|
0.129
|
0.054
|
0.021
|
|
4
|
0.867
|
0.507
|
0.155
|
0.067
|
0.024
|
0.008
|
|
5
|
0.680
|
0.355
|
0.101
|
0.041
|
0.013
|
0.004
|
|
6
|
0.666
|
0.272
|
0.102
|
0.030
|
0.009
|
0.002
|
|
7
|
0.599
|
0.179
|
0.073
|
0.020
|
0.006
|
0.001
|
|
8
|
0.522
|
0.075
|
0.028
|
0.008
|
0.002
|
0.000
|
|
9
|
0.478
|
0.064
|
0.023
|
0.006
|
0.001
|
0.000
|
|
10
|
0.401
|
0.091
|
0.029
|
0.006
|
0.001
|
0.000
|
|
11
|
0.334
|
0.090
|
0.022
|
0.004
|
0.001
|
0.000
|
|
12
|
0.320
|
0.113
|
0.019
|
0.005
|
0.001
|
0.000
|
|
13
|
0.133
|
0.025
|
0.002
|
0.000
|
0.000
|
0.000
|
|
14
|
0.091
|
0.017
|
0.002
|
0.000
|
0.000
|
0.000
|
|
15
|
0.038
|
0.004
|
0.000
|
0.000
|
0.000
|
0.000
|
|
16
|
0.008
|
0.000
|
0.000
|
0.000
|
0.000
|
0.000
|
The 0.000’s in the table are, of course, not actually 0;
they’re simply “less than 0.0005.” The smallest of them is naturally the chance
of a 16 seed winning the title, which is estimated at 4.58 * 10-11.
Not great odds for the 16s, which makes sense.
The next table will be the inverse of the odds, which also
makes it the number of points awarded for correctly picking a team of that seed
to reach the round in question:
|
Seed
|
Round of 32
|
Sweet 16
|
Elite 8
|
Final 4
|
Title game
|
Champ
|
|
1
|
1.01
|
1.16
|
1.49
|
2.15
|
3.53
|
6.09
|
|
2
|
1.04
|
1.38
|
2.18
|
4.59
|
9.46
|
20.78
|
|
3
|
1.10
|
1.61
|
3.20
|
7.78
|
18.54
|
47.56
|
|
4
|
1.15
|
1.97
|
6.44
|
14.86
|
42.40
|
131.55
|
|
5
|
1.47
|
2.82
|
9.88
|
24.27
|
74.30
|
248.42
|
|
6
|
1.50
|
3.68
|
9.83
|
33.17
|
110.88
|
406.80
|
|
7
|
1.67
|
5.57
|
13.67
|
49.95
|
181.29
|
725.39
|
|
8
|
1.92
|
13.36
|
35.97
|
129.33
|
599.61
|
3100.04
|
|
9
|
2.09
|
15.64
|
44.40
|
169.56
|
838.56
|
4637.28
|
|
10
|
2.50
|
10.95
|
34.29
|
165.42
|
801.35
|
4337.06
|
|
11
|
2.99
|
11.10
|
46.41
|
252.70
|
1387.61
|
8555.68
|
|
12
|
3.12
|
8.81
|
51.94
|
207.40
|
1075.42
|
6247.66
|
|
13
|
7.50
|
40.39
|
533.83
|
4751.24
|
5.69*104
|
7.84*104
|
|
14
|
10.94
|
58.53
|
558.72
|
7159.29
|
9.49*104
|
1.45*106
|
|
15
|
26.23
|
237.05
|
3211.98
|
7.41*104
|
1.79*106
|
5.01*107
|
|
16
|
126.69
|
2551.39
|
8.18*104
|
4.03*106
|
2.74*108
|
2.18*1010
|
Those are single-line totals, not cumulative (that is to say, accurately predicting a 7 seed in the final four would get you the sum of the first four columns in row seven, because you'd pick them in each of their three prior matchups as well). As a quick
example of how the system works, let’s use it on last year’s bracket.
Louisville, the top overall seed, won the national title.
ESPN’s scoring system would award 630 total points for that selection.
Michigan, a 4 seed, made the title game, resulting in 310 points for a correct
pick. Wichita State became the first #9 seed to make the Final Four in the
64-team era; 150 points for that one if you got it. And Florida Gulf Coast became
the first-ever 15 seed in the Sweet 16, which is worth all of 30 points for a
correct selection. So even if you had somehow picked Michigan, Wichita State,
and FGCU to reach the points in the bracket that they actually did, which would
be quite impressive, if you happened to have Louisville going out early, you
still didn’t necessarily fare well in your bracket pool.
By comparison, the method proposed here awards 15.42 total
points for correctly picking a 1 seed as the champ. It offers 66.83 for putting
the right 4 seed in the title game, 231.69 for a 9 seed in the Final 4, and
263.28 for a 15 in the third round. And if someone had completely aced the first round, they'd have started day 3 of the tournament with 92.43 points, more than would have been available from predicting Louisville over Michigan in the final. In this method, accurately selecting a
top-seeded titlist is beneficial, but if someone else in your pool nails this
year’s Cinderella team, they’re still likely to beat you. Which makes sense, because accurately picking Wichita State in the Final 4 last year
would have been vastly more impressive (or at least unusual) than listing
Louisville as the champion, and a good scoring system for a bracket should
reflect that fact.
Since the points awarded for each pick are intended as the inverse
of the odds of the pick, one would hope to have an expected return of N points
for picking any team to win N games in the tournament. Our final table will be
of the historic average return produced under this system by picking a team of
a given seed to win the national title (which ideally should be 6, since a
title requires 6 wins):
|
Seed
|
Avg points
|
|
1
|
5.68
|
|
2
|
5.57
|
|
3
|
6.62
|
|
4
|
6.67
|
|
5
|
5.76
|
|
6
|
9.65
|
|
7
|
2.79
|
|
8
|
45.77
|
|
9
|
3.98
|
|
10
|
4.95
|
|
11
|
10.88
|
|
12
|
3.07
|
|
13
|
3.71
|
|
14
|
2.61
|
|
15
|
3.63
|
|
16
|
0
|
With a couple of obvious exceptions (#8 seed Villanova’s
1985 title producing the most obvious among them), that comes very close to
achieving the system’s stated goal. In particular, seeds 1 through 5, which
have produced all but four of the last 58 title game participants, all give
expected returns between 5.5 and 6.75, which is… rather satisfyingly close to
6.
And there you have it - an alternate method of scoring brackets that properly rewards a correct surprise pick, and thereby provides adequate inducement for the prediction of upsets (or disincentive to picking favorites throughout).
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