Background

The inspiration for this post comes from this Chief Delphi thread asking what the maximum possible Offensive Power Rating (OPR) was in 2019. The easy answer is that there is no upper bound on 2019 OPRs because there was no upper bound for the score due to penalties. Oftentimes innocuous questions can lead down very intriguing paths though, and it got me interested enough to do a more detailed analysis of the question of what the maximum unpenalized OPR in 2019 would have been. Since the maximum unpenalized score for a 2019 match was 154, a first pass guess would say the max OPR would also be 154 (assuming a team scores 154 in all their matches and all other scores are 0). However, just like negative OPRs are theoretically possible, so too are OPRs greater than the max score.

OPR Theory

For a refresher on how OPR is calculated, check out Eugene’s great blog post from a few years ago, I’ll be skimming details here, so if you get lost try referencing back to that post. To summarize, the least squares solution to an overdetermined set of linear equations (or what is commonly called OPR around here) has the following closed form solution:

Where x is OPR, A is the schedule matrix, and b is the score matrix. From here on in, I’m going to refer to this:

As the multiplication matrix, since it multiplies with the score matrix to obtain the OPR matrix.

The schedule A is static, so the multiplication matrix is really just a static function of the schedule. Since all entries of b are bounded above (by the max score) and below (by 0), the logic to maximize an entry of x is simple: For each half-match (column), if the row corresponding to that entry of x in the multiplication matrix is positive, set that half match’s score equal to the max score. Otherwise, set that half-matches score equal to 0. I know that’s a lot to take in all at once, so here’s an example to illustrate.

Case Study

Let’s take a look at 5511 at the North Carolina State Championship. I’ve copied all of the entries in the row of the multiplication matrix corresponding to 5511 into the below table:

match	blue	red
1	0.085	-0.007
2	0.004	-0.010
3	-0.010	-0.002
4	-0.014	-0.005
5	0.004	-0.010
6	0.003	-0.010
7	-0.012	-0.012
8	-0.003	-0.010
9	-0.003	0.004
10	-0.009	-0.013
11	0.085	-0.006
12	0.004	-0.010
13	-0.013	0.083
14	-0.007	-0.011
15	-0.002	0.003
16	-0.011	-0.002
17	0.002	-0.005
18	-0.011	-0.003
19	-0.003	-0.005
20	-0.010	-0.012
21	-0.003	-0.009
22	0.083	-0.003
23	-0.002	-0.013
24	-0.014	0.010
25	-0.010	-0.013
26	0.000	-0.011
27	0.083	-0.003
28	-0.010	-0.002
29	-0.002	-0.002
30	-0.006	-0.007
31	-0.004	-0.001
32	0.081	-0.011
33	-0.003	-0.013
34	-0.010	-0.003
35	-0.011	0.004
36	-0.012	-0.009
37	0.083	0.012
38	-0.013	-0.003
39	-0.002	-0.009
40	0.081	-0.013
41	-0.003	-0.010
42	0.004	-0.005
43	-0.003	-0.014
44	-0.014	-0.003
45	-0.002	-0.010
46	-0.013	-0.003
47	0.003	0.086
48	-0.006	-0.003
49	-0.002	-0.004
50	-0.002	0.084
51	-0.010	-0.006
52	0.006	-0.012
53	-0.014	-0.004
54	-0.009	-0.014
55	-0.009	-0.011
56	0.005	-0.003
57	-0.005	0.003
58	-0.011	0.083
59	-0.005	0.004
60	-0.010	0.082
61	-0.003	-0.012
62	-0.005	-0.002
63	-0.011	-0.004
64	-0.002	-0.012

You can interpret these numbers as the “impact” of each half-match to 5511’s OPR. So you can see that by far the most important half-matches for 5511 are ones like blue 1, blue 11, and red 13, which not coincidentally are the half-matches 5511 is in. These are the obvious first-order effects of this methodology, if you only look at these, it is very similar to just looking at average scores for teams.

You’ll also see that the majority of the remaining half-matches have negative values. This indicates that, generally, it reflects better on 5511 if half-matches they aren’t in have low scores because that means their partners likely weren’t contributing as much when they were partnered with 5511. We’ll call these effects second-order.

What we are most interested in right now though, are the half-matches that don’t contain 5511, but still have slight positive importance to them such as blue 2, blue 5, blue 6, and red 9. These positive values are generally due to third-order effects. Just like it’s a good sign for you if your partners do poorly without you, it’s also a good sign if their partners do well without them.

This pattern continues forever, odd degrees of separation between teams have positive impact to your team, and even degrees of separation have negative impact. But the further out you go, the more watered down these effects are. Interestingly, 538 recently discussed this same effect for their NBA RAPTOR calculation methodology (jump to point 3 in the section titled RAPTOR On-Off) where they choose to stop at the third order effects or two degrees of separation.

Alright, so now that we have that out of the way, let’s use those half-match impacts to get 5511 a stupid-high OPR. Just go through and make the score 154 if it has positive impact for 5511, and 0 if it has negative impact for them. Here are those scores:

match	blue	red
1	154	0
2	154	0
3	0	0
4	0	0
5	154	0
6	154	0
7	0	0
8	0	0
9	0	154
10	0	0
11	154	0
12	154	0
13	0	154
14	0	0
15	0	154
16	0	0
17	154	0
18	0	0
19	0	0
20	0	0
21	0	0
22	154	0
23	0	0
24	0	154
25	0	0
26	154	0
27	154	0
28	0	0
29	0	0
30	0	0
31	0	0
32	154	0
33	0	0
34	0	0
35	0	154
36	0	0
37	154	154
38	0	0
39	0	0
40	154	0
41	0	0
42	154	0
43	0	0
44	0	0
45	0	0
46	0	0
47	154	154
48	0	0
49	0	0
50	0	154
51	0	0
52	154	0
53	0	0
54	0	0
55	0	0
56	154	0
57	0	154
58	0	154
59	0	154
60	0	154
61	0	0
62	0	0
63	0	0
64	0	0

Altogether, this gets 5511 up to an OPR of 165.8! That’s astoundingly high, but it isn’t even the highest possible at this event. Since each team’s schedule is slightly different, each team’s max OPR is also slightly different. So the highest possible OPR at NC Champs would actually have been 186.8 for team 7890. We can then find the highest possible OPR of the 2019 season by doing this same analysis at every event. Unfortunately, no sane person would do that much work.

Full 2019 Results

But at least I did :). Here is the max and min OPRs possible for every in season event along with some other details about those events.

event	matches per team	teams	matches	max OPR	min OPR
ausp	8	62	83	269.4	-218.0
flor	9	64	96	264.2	-212.9
flwp	9	64	96	260.1	-208.7
scmb	9	64	96	258.3	-207.0
mndu	9	63	95	258.0	-206.7
mnmi	9	63	95	257.6	-206.3
cada	9	59	89	257.1	-205.8
carv	10	68	114	256.4	-205.1
arli	9	58	87	254.8	-203.5
casj	9	59	89	254.6	-203.2
okok	9	62	93	253.9	-202.5
hop	10	68	114	253.7	-202.4
cars	10	68	114	253.5	-202.2
mndu2	9	60	90	252.6	-201.3
dal	10	68	114	251.5	-200.2
lake	9	59	89	250.5	-199.2
roe	10	67	112	250.2	-198.8
ohcl	9	60	90	250.2	-198.8
tes	10	68	114	250.1	-198.8
tur	10	66	110	249.9	-198.5
new	10	67	112	249.6	-198.3
cala	9	56	84	249.4	-198.1
ohmv	9	60	90	249.2	-197.9
mnmi2	9	60	90	248.3	-197.0
arc	10	68	114	248.3	-197.0
gal	10	67	112	248.0	-196.7
dar	10	68	114	247.9	-196.6
cur	10	68	114	247.7	-196.4
ilch	9	53	80	245.7	-194.3
nyny	9	51	77	239.4	-188.1
wila	10	54	90	232.9	-181.6
alhu	10	55	92	230.6	-179.3
azfl	10	53	89	227.0	-175.7
ndgf	10	52	87	226.5	-175.2
wimi	10	54	90	225.8	-174.5
code	10	53	89	225.7	-174.4
pncmp	12	64	128	225.4	-174.1
tnkn	10	51	85	225.3	-174.0
ftcmp	12	64	128	222.1	-170.8
necmp	12	64	128	221.0	-169.7
cafr	10	49	82	220.8	-169.5
mosl	10	48	80	220.4	-169.0
utwv	10	49	82	220.3	-168.9
nyli2	10	48	80	219.9	-168.6
caoc	10	49	82	219.0	-167.6
nysu	10	46	77	218.5	-167.1
nyro	10	48	80	217.8	-166.4
mrcmp	12	60	120	215.8	-164.5
paca	10	45	75	214.7	-163.4
iacf	10	48	80	214.5	-163.2
chcmp	12	58	116	211.6	-160.3
casd	10	47	79	211.3	-160.0
mxmo	10	43	72	210.1	-158.8
iscmp	11	45	83	209.4	-158.1
nvlv	11	44	81	198.9	-147.6
qcmo	11	42	77	197.9	-146.5
casf	11	43	79	196.6	-145.3
abca	10	37	62	196.4	-145.0
gacmp	12	45	90	196.0	-144.7
ilpe	11	40	74	194.5	-143.2
nhgrs	12	36	72	193.6	-142.2
ausc	11	40	74	192.5	-141.2
txpas	12	32	64	191.7	-140.3
txgre	12	42	84	191.5	-140.2
oncmp2	12	40	80	190.8	-139.4
cave	12	42	84	189.7	-138.4
wayak	12	31	62	189.7	-138.4
txdel	12	40	80	189.6	-138.2
misjo	12	40	80	188.9	-137.6
micmp2	12	40	80	188.8	-137.4
azpx	12	42	84	188.2	-136.9
qcqc	11	35	65	188.2	-136.9
milan	12	40	80	188.0	-136.7
mikng	12	40	80	188.0	-136.7
inmis	12	42	84	188.0	-136.7
mishe	12	40	80	187.8	-136.5
mitvc	12	40	80	187.7	-136.3
mifor	12	40	80	187.6	-136.3
mimil	12	40	80	187.4	-136.1
mitry	12	41	82	187.3	-136.0
ctwat	12	41	82	187.2	-135.9
mibel	12	40	80	187.0	-135.6
milsu	12	41	82	186.9	-135.6
nccmp	12	32	64	186.8	-135.4
onosh	12	40	80	186.8	-135.4
misou	12	40	80	186.6	-135.3
micmp4	12	40	80	186.4	-135.0
marea	12	38	76	186.2	-134.9
micen	12	40	80	186.2	-134.9
miken	12	41	82	186.2	-134.9
micmp3	12	40	80	186.2	-134.9
mimar	12	40	80	186.0	-134.7
inwla	12	39	78	186.0	-134.7
milin	12	41	82	186.0	-134.7
milak	12	39	78	185.8	-134.5
mxto	11	36	66	185.8	-134.4
midet	12	39	78	185.8	-134.4
miliv	12	40	80	185.7	-134.4
onlon	12	40	80	185.5	-134.2
onwin	12	40	80	185.4	-134.0
micmp1	12	40	80	185.2	-133.9
mxcm	11	27	50	185.0	-133.6
isde4	11	36	66	184.9	-133.6
onto3	12	32	64	184.9	-133.6
txdls	12	39	78	184.9	-133.5
caln	12	36	72	184.6	-133.3
nhdur	12	40	80	184.6	-133.3
macma	12	39	78	184.6	-133.2
mimus	12	39	78	184.5	-133.2
oncmp1	12	40	80	184.2	-132.9
miwmi	12	39	78	184.2	-132.8
mimid	12	39	78	184.2	-132.8
cthar	12	38	76	184.1	-132.8
bcvi	11	35	65	184.0	-132.7
gagai	12	38	76	184.0	-132.6
vagle	12	37	74	183.9	-132.5
migul	12	38	76	183.8	-132.5
melew	12	32	64	183.7	-132.3
wasno	12	38	76	183.6	-132.3
gafor	12	30	60	183.2	-131.9
mdoxo	12	39	78	182.8	-131.5
mike2	12	39	78	182.8	-131.5
mijac	12	40	80	182.8	-131.4
vahay	12	38	76	182.3	-131.0
mabri	12	38	76	182.3	-130.9
mialp	12	39	78	181.9	-130.6
txaus	12	36	72	181.9	-130.5
gadal	12	38	76	181.8	-130.4
vapor	12	37	74	181.6	-130.3
waamv	12	38	76	181.6	-130.3
mdowi	12	38	76	181.6	-130.2
njfla	12	37	74	181.5	-130.2
tuis	11	34	63	181.5	-130.1
txama	12	29	58	181.3	-129.9
nhsnh	12	37	74	181.2	-129.9
njski	12	37	74	181.0	-129.6
orwil	12	38	76	180.4	-129.1
mawne	12	30	60	180.3	-128.9
isde2	11	34	63	180.2	-128.8
onto1	12	30	60	179.9	-128.6
caav	12	37	74	179.7	-128.3
nyli	12	37	74	179.1	-127.8
onnyo	12	36	72	179.1	-127.8
mdbet	12	36	72	179.0	-127.7
mokc3	12	36	72	179.0	-127.6
mial2	12	36	72	178.5	-127.2
paphi	12	31	62	178.5	-127.2
ksla	12	36	72	178.5	-127.1
txsan	12	36	72	178.5	-127.1
ncwak	12	36	72	178.4	-127.1
mokc	12	36	72	178.3	-127.0
nytr	12	36	72	178.2	-126.9
ncgui	12	36	72	178.2	-126.8
camb	12	36	72	178.2	-126.8
orore	12	36	72	178.2	-126.8
waspo	12	29	58	177.9	-126.6
paben	12	31	62	177.3	-125.9
ncpem	12	36	72	177.2	-125.9
gaalb	12	30	60	177.0	-125.7
isde3	11	31	57	176.7	-125.4
ingre	12	35	70	176.3	-125.0
orlak	12	36	72	176.3	-125.0
tuis2	11	32	59	176.2	-124.9
cadm	12	35	70	175.9	-124.6
migib	12	35	70	175.9	-124.6
miket	12	35	70	175.9	-124.6
mabos	12	28	56	175.6	-124.3
gacol	12	35	70	175.6	-124.3
txpla	12	35	70	175.5	-124.1
njbri	12	35	70	174.9	-123.6
njtab	12	35	70	174.9	-123.5
nyut	13	31	68	174.8	-123.5
pahat	12	35	70	174.6	-123.3
waahs	12	34	68	174.5	-123.2
txcha	12	35	70	174.2	-122.9
wamou	12	34	68	174.0	-122.6
rismi	12	29	58	173.8	-122.5
onham	12	34	68	173.6	-122.3
vabla	12	34	68	173.3	-122.0
hiho	12	34	68	173.0	-121.7
isde1	11	31	57	172.8	-121.5
pawch	12	32	64	172.0	-120.7
incmp	12	32	64	171.4	-120.1
onnob	12	33	66	171.4	-120.0
ncash	12	32	64	171.3	-120.0
onbar	12	26	52	171.0	-119.6
idbo	12	27	54	170.9	-119.5
onwat	12	32	64	169.2	-117.8
txelp	12	25	50	167.0	-115.7

So, I believe the max possible OPR in 2019 was 269.4, which could only have been achieved by 6508 at South Pacific.

Here I’ve graphed Max OPR against all three of matches per team, total matches, and total teams:

Conclusions

There’s plenty of other interesting info in here. Firstly, all 2019 events had a max OPR greater than 154. This bodes well for my hypothesis that all but a select few schedules will have some team capable of achieving a max OPR greater than the max score (although a proof or counterexample would be needed to rule definitively on this).

Another cool tidbit is that the minimum possible normalized OPR is always 1/3-(normalized max OPR). This means that, since all events have a normalized max OPR greater than or equal to 1.0, all events also must have a normalized min OPR which is less than or equal to -2/3.

Finally, if you want to predict the max OPR without this excessive effort, then it seems the best linear predictor is team count (R^2 = 0.89). As you can see though there is no straightforward formula to determine precisely what it is as there is clear schedule dependence. Here is a rough exponential function that bounds the 2019 data points above:

It’s formula is (max OPR) = (max score) *(1+e^((teams at event – 63)/26)). I think for the future though, providing the scheduler stays the same and we don’t get more than 70 teams at an event, we can comfortably say that there will probably never be an opportunity for a team to achieve an unpenalized OPR greater than 2.5X the max score.