Self-Consistent CFB Ranker

Current Rankings (December 3, 2023):

Rank	Team	NAW	AAW	NCS	NRS	Record
1	Washington	15.325	1.179	15.325	0.000	13 - 0
2	Michigan	15.039	1.157	15.039	0.000	13 - 0
3	Texas	14.235	1.095	16.733	0.000	12 - 1
4	Alabama	13.828	1.064	16.755	0.000	12 - 1
5	Florida State	13.718	1.055	13.718	0.000	13 - 0
6	Georgia	12.286	0.945	15.157	0.000	12 - 1
7	Liberty	11.640	0.895	11.640	0.000	13 - 0
8	Ohio State	11.186	0.932	14.229	0.000	11 - 1
9	Oklahoma	10.600	0.883	14.894	0.000	10 - 2
10	James Madison	9.926	0.827	11.971	0.000	11 - 1
11	Oregon	9.578	0.737	15.750	0.000	11 - 2
12	Ole Miss	9.148	0.762	14.697	0.000	10 - 2
13	Penn State	9.050	0.754	14.650	0.000	10 - 2
14	Missouri	8.818	0.735	13.747	0.000	10 - 2
15	Troy	8.652	0.666	13.225	0.000	11 - 2
16	Tulane	8.068	0.621	12.696	0.000	11 - 2
17	Iowa	7.913	0.609	15.316	0.000	10 - 3
18	Toledo	7.895	0.607	12.144	0.000	11 - 2
19	Louisville	7.853	0.604	14.850	0.000	10 - 3
20	SMU	7.747	0.596	12.249	0.000	11 - 2
21	LSU	7.605	0.634	15.699	0.000	9 - 3
22	Miami (OH)	7.528	0.579	11.855	0.000	11 - 2
23	North Carolina State	6.565	0.547	13.049	0.000	9 - 3
24	Oklahoma State	6.492	0.499	15.476	0.000	9 - 4
25	Notre Dame	6.438	0.536	13.374	0.000	9 - 3
26	Arizona	5.726	0.477	12.862	0.000	9 - 3
27	Kansas State	5.675	0.473	15.180	0.000	8 - 4
28	Memphis	5.479	0.457	12.364	0.000	9 - 3
29	Kansas	5.106	0.425	14.357	0.000	8 - 4
30	Clemson	5.081	0.423	14.215	0.000	8 - 4
31	Utah	4.622	0.385	14.342	0.000	8 - 4
32	Oregon State	4.564	0.380	14.208	0.000	8 - 4
33	Tennessee	4.473	0.373	14.366	0.000	8 - 4
34	North Carolina	4.133	0.344	12.512	0.000	8 - 4
35	Nevada-Las Vegas	4.123	0.317	13.265	0.000	9 - 4
36	New Mexico State	4.077	0.291	13.510	0.000	10 - 4
37	Wyoming	3.913	0.326	13.003	0.000	8 - 4
38	Boise State	3.631	0.279	14.917	0.000	8 - 5
39	Iowa State	3.630	0.302	15.488	0.000	7 - 5
40	Miami (FL)	3.615	0.301	15.006	0.000	7 - 5
41	West Virginia	3.484	0.290	12.576	0.000	8 - 4
42	Ohio	3.387	0.282	9.778	0.000	9 - 3
43	Appalachian State	3.236	0.249	13.724	0.000	8 - 5
44	USC	3.204	0.267	14.973	0.000	7 - 5
45	Duke	2.923	0.244	14.375	0.000	7 - 5
46	Air Force	2.814	0.234	11.044	0.000	8 - 4
47	Kentucky	2.612	0.218	14.590	0.000	7 - 5
48	UTSA	2.592	0.216	11.031	0.000	8 - 4
49	Jacksonville State	2.502	0.208	11.203	0.000	8 - 4
50	Texas A&M	2.365	0.197	13.996	0.000	7 - 5
51	San Jose State	2.288	0.191	12.783	0.000	7 - 5
52	Coastal Carolina	2.277	0.190	12.739	0.000	7 - 5
53	Fresno State	2.176	0.181	10.454	0.000	8 - 4
54	Northwestern	2.061	0.172	12.748	0.000	7 - 5
55	Maryland	1.985	0.165	13.965	0.000	7 - 5
56	Wisconsin	1.836	0.153	12.813	0.000	7 - 5
57	Bowling Green	1.727	0.144	13.941	0.000	7 - 5
58	Western Kentucky	1.495	0.125	13.082	0.000	7 - 5
59	UCLA	1.438	0.120	11.848	0.000	7 - 5
60	Georgia Tech	1.276	0.106	14.715	0.000	6 - 6
61	Rutgers	1.010	0.084	15.273	0.000	6 - 6
62	Texas Tech	0.969	0.081	14.563	0.000	6 - 6
63	Auburn	0.851	0.071	15.114	0.000	6 - 6
64	Texas State	0.520	0.043	10.909	0.000	7 - 5
65	California	0.510	0.042	14.228	0.000	6 - 6
66	Virginia Tech	0.457	0.038	13.800	0.000	6 - 6
67	Georgia State	0.237	0.020	13.296	0.000	6 - 6
68	South Alabama	0.176	0.015	13.299	0.000	6 - 6
69	Old Dominion	0.173	0.014	13.285	0.000	6 - 6
70	Marshall	0.169	0.014	12.857	0.000	6 - 6
71	UCF	0.065	0.005	13.000	0.000	6 - 6
72	Arkansas State	-0.277	-0.023	12.700	0.000	6 - 6
73	Syracuse	-0.401	-0.033	12.649	0.000	6 - 6
74	Minnesota	-0.470	-0.039	15.266	0.000	5 - 7
75	Georgia Southern	-0.489	-0.041	12.006	0.000	6 - 6
76	Utah State	-0.584	-0.049	12.255	0.000	6 - 6
77	Boston College	-0.637	-0.053	12.706	0.000	6 - 6
78	Illinois	-0.744	-0.062	14.074	0.000	5 - 7
79	South Carolina	-0.820	-0.068	14.495	0.000	5 - 7
80	Texas Christian	-0.894	-0.074	14.838	0.000	5 - 7
81	Florida	-0.923	-0.077	15.380	0.000	5 - 7
82	Mississippi State	-1.144	-0.095	14.404	0.000	5 - 7
83	Rice	-1.220	-0.102	12.466	0.000	6 - 6
84	BYU	-1.335	-0.111	14.496	0.000	5 - 7
85	South Florida	-1.519	-0.127	11.734	0.000	6 - 6
86	Louisiana	-1.600	-0.133	10.859	0.000	6 - 6
87	Washington State	-1.776	-0.148	14.089	0.000	5 - 7
88	Nebraska	-1.838	-0.153	13.591	0.000	5 - 7
89	Army	-2.006	-0.182	10.995	0.000	5 - 6
90	Navy	-2.317	-0.211	10.518	0.000	5 - 6
91	Michigan State	-2.431	-0.203	16.909	0.000	4 - 8
92	Purdue	-2.571	-0.214	15.369	0.000	4 - 8
93	Eastern Michigan	-2.906	-0.242	9.580	0.000	6 - 6
94	North Texas	-3.232	-0.269	11.662	0.000	5 - 7
95	Northern Illinois	-3.403	-0.284	9.962	0.000	6 - 6
96	Wake Forest	-3.403	-0.284	13.974	0.000	4 - 8
97	Colorado State	-3.429	-0.286	10.899	0.000	5 - 7
98	Houston	-3.511	-0.293	13.888	0.000	4 - 8
99	Colorado	-3.621	-0.302	13.309	0.000	4 - 8
100	Arkansas	-3.631	-0.303	14.239	0.000	4 - 8
101	Central Michigan	-3.927	-0.327	10.986	0.000	5 - 7
102	Western Michigan	-4.091	-0.341	12.817	0.000	4 - 8
103	Virginia	-4.186	-0.349	14.871	0.000	3 - 9
104	Hawaii	-4.424	-0.340	12.684	0.000	5 - 8
105	Middle Tennessee State	-4.472	-0.373	13.741	0.000	4 - 8
106	UAB	-4.476	-0.373	12.723	0.000	4 - 8
107	Arizona State	-4.493	-0.374	15.533	0.000	3 - 9
108	Ball State	-4.506	-0.376	12.840	0.000	4 - 8
109	New Mexico	-4.696	-0.391	12.010	0.000	4 - 8
110	San Diego State	-4.713	-0.393	11.942	0.000	4 - 8
111	Tulsa	-4.995	-0.416	13.544	0.000	4 - 8
112	Pitt	-5.100	-0.425	14.286	0.000	3 - 9
113	Indiana	-5.186	-0.432	15.118	0.000	3 - 9
114	Florida Atlantic	-5.484	-0.457	11.386	0.000	4 - 8
115	Southern Mississippi	-5.508	-0.459	14.011	0.000	3 - 9
116	Baylor	-5.522	-0.460	13.804	0.000	3 - 9
117	Stanford	-5.652	-0.471	15.119	0.000	3 - 9
118	Cincinnati	-5.662	-0.472	13.624	0.000	3 - 9
119	Massachusetts	-5.682	-0.473	14.110	0.000	3 - 9
120	Florida International	-6.227	-0.519	11.037	0.000	4 - 8
121	Connecticut	-6.389	-0.532	12.534	0.000	3 - 9
122	UTEP	-6.555	-0.546	12.699	0.000	3 - 9
123	Vanderbilt	-7.062	-0.588	14.570	0.000	2 - 10
124	Temple	-7.161	-0.597	11.565	0.000	3 - 9
125	Louisiana Tech	-7.183	-0.599	12.174	0.000	3 - 9
126	Buffalo	-7.263	-0.605	12.801	0.000	3 - 9
127	Charlotte	-7.290	-0.607	11.290	0.000	3 - 9
128	Sam Houston	-7.418	-0.618	11.742	0.000	3 - 9
129	Louisiana-Monroe	-8.041	-0.670	12.866	0.000	2 - 10
130	East Carolina	-8.337	-0.695	13.691	0.000	2 - 10
131	Nevada	-9.456	-0.788	11.814	0.000	2 - 10
132	Akron	-10.644	-0.887	10.412	0.000	2 - 10
133	Kent State	-12.430	-1.036	10.858	0.000	1 - 11

Explanation:

I was looking at a couple of computer CFB rankings, and the following occurred to me:

• I can use a computer.
• I know about football.
• My opinions are better than other people's opinions.

So of course I had to do my own ranking system. Here are my principles:

• Games are won or lost. Point differentials are meaningless. Really, your team is better because your 8th-string QB rushed for a couple garbage time TDs?
• Wins and losses both matter. Equally.
• The order in which games are played doesn't matter. No late-season bias. No inertia.
• Beating a good team is better than beating a bad team, and similarly losing to a good team isn't as bad as losing to a bad team.
• No judgement should be involved. No adjustable parameters. In other words, there's no seed from a preseason poll, nor should there be sorting by P5/G5 etc. The rankings themselves have to determine what is a good win.
• The ratings used to generate rankings should be continuous rather than discrete and separate from the rankings themselves. The gap between #14 and #15, for example, may be small or large, and the value of beating these teams should reflect that.
• Rankings should be self-consistent. If the final set of ratings is used to generate rankings, there should be no change.

And thus the foundations of the Self-Consistent CFB Ranker (SCCR) were laid. The first step in executing this vision is choosing a function to generate the ratings. A new rating system, Net Adjusted Wins (NAW) was invented for this purpose.

Net Wins

Net wins is simply wins minus losses, like the team record with the dash interpreted as subtraction. So a 8-4 teams has 4 net wins. A 1-11 teams has -10 net wins. In the days of ties, a 4-5-1 teams has -1 net wins.

Adjusted Wins

As stated above, in CFB some wins are better than others, and some losses worse than others. To represent this, an Adjusted Wins function was selected. The adjusted win value ($\omega$) for an opponent strength of $\beta$ is:

$\omega_{win}$ = $\exp(\beta / \gamma)$

where $\gamma$ is a scaling factor. The adjusted win value for a loss against the same opponent would be:

$\omega_{loss}$ = $-\exp(-\beta/\gamma)$.

Net Adjusted Wins

Combining the two concepts above, Net Adjusted Wins for any team is simply:

NAW = $\sum_{games}$ $\omega_{game}$.

Now, what should the strength, $\beta$, be for each team? Simply that team's NAW!

Comparing the $\omega_{win}$ and $\omega_{loss}$ functions, it can be seen that the magnitude of a win against a team with a high NAW will be similar to that of a loss to a team with a low NAW. Note that an average NAW (i.e., for a team with an even record) will be around 0. This means for an opponent with an average NAW, $\omega_{win} \doteq 1$ and $\omega_{loss} \doteq -1$. In other words, the average adjustment factor is roughly unity.

The only remaining parameter to set is $\gamma$. The choice $\gamma$ = max(abs(NAW)) is arbitrary, but turns out to work pretty nicely. It ensures that the adjusted values of wins remain roughly constant over the course of a season, and defines the maximum possible ratio of values of FBS wins (or losses) to be exp(2) = 7.389. By this win metric, a team would do much better to split a series with the best team in FBS (+2.3504 net wins) than to beat the worst team in FBS twice (+0.7358 net wins). This feels reasonably fair, and also has the benefit of converging well.

A few more calculation details. The algorithm is initialized with the unadjusted net wins for each team. The adjusted wins are calculated and NAW summed iteratively until self-consistency of NAW for all teams is achieved. Multiple games against the same opponent are counted separately, rather than cancelling. Ties (though no longer part of CFB) are counted as 0.5 wins + 0.5 losses, so increase NAW if the opponent has NAW above 0, and decrease it if the opponent has NAW below 0.

Also calculated is Average Adjusted Wins (AAW), which is NAW per game played. The NAW of the Completed Schedule (NCS) and NAW of Remaining Schedule (NRS) for each team are also output as estimates of strength of schedule and strength of remaining schedule.

One caveat about the "no inputs" thing: only FBS teams are tracked, and all non-FBS teams have NAW set to (minimum FBS NAW - 1.0).

Results are downloaded from Sports Reference (i.e. https://www.sports-reference.com/cfb/years/2021-schedule.html click on "Share & Export" then "Get table as CSV (for Excel)").

Team Reports:

If team names are given as arguments, a report on the specified teams will be output instead of the full rankings. This includes the games played by that team, the outcomes of those games, and how they affected the computed team NAW. If games remain on the schedule, those are shown separately, along with estimated NAW changes for win/loss outcomes.

For example, California after week 13, 2021:

California (4 - 6)

	NAW	AAW	NCS	NRS
Value	-4.201	-0.420	8.933	2.047
Rank	100	102	125	9

Played	Outcome	Change
71 Texas Christian	Loss	- 0.997
44 Nevada	Loss	- 0.837
105 Stanford	Win	+ 0.692
64 Washington State	Loss	- 0.947
20 Oregon	Loss	- 0.606
52 Oregon State	Win	+ 1.132
108 Washington	Loss	- 1.523
96 Colorado	Win	+ 0.754
127 Arizona	Loss	- 2.210
131 Non-FBS	Win	+ 0.341

Remaining	If Win	If Loss
86 USC	+ 0.853	- 1.173
45 UCLA	+ 1.194	- 0.837

Examples:

2019

Let's take a look at the top 25 prior to bowl games for the 2019 season. I'll use the results as of December 8th to match the final CFP committee:

Rank	Team	NAW	CFP Committee Rank
1	Ohio State	16.622	2
2	LSU	15.179	1
3	Clemson	12.756	3
4	Memphis	12.101	17
5	Oklahoma	11.840	4
6	Boise State	11.224	19
7	Georgia	10.840	5
8	Appalachian State	10.437	20
9	Oregon	9.867	6
10	Notre Dame	9.632	15
11	Wisconsin	9.452	8
12	Penn State	9.320	10
13	Baylor	9.180	7
14	Utah	9.117	11
15	Cincinnati	8.558	21
16	Florida	8.429	9
17	Michigan	8.099	14
18	Auburn	7.988	12
19	SMU	7.937	NR
20	Minnesota	7.762	18
21	Alabama	7.713	13
22	Air Force	7.537	NR
23	Navy	7.487	23
24	Iowa	7.092	16
25	Florida Atlantic	6.777	NR

This ranking results in 3 of the 4 playoff selection, and Oklahoma is only edged out by Memphis by a razor-thin margin. Evidently this ranking system is not the hot-take generator I had feared. In total 4 of the top-25 teams differ from the CFP committee's list. Dropped are 8-4 USC (22), 9-4 Virginia (24), and 8-4 Oklahoma State (25). Added are 10-3 SMU, 11-2 Air Force, and 11-3 FAU. Overall this ranking prefers G5 teams with more wins over P5 teams with fewer, relative to the CFP. I agree.

So 2019 is a nice, uncontroversial year. LSU were champs, they deserved to be in the CFP, and after all bowls SCCR agrees that they were the best.

2017

But what about 2017? Starting with the same point (December 3rd), who should have been in the CFP?

Rank	Team	NAW	CFP Committee Rank
1	Clemson	14.609	1
2	Georgia	13.282	3
3	Oklahoma	13.282	2
4	UCF	12.599	12
5	Wisconsin	12.540	6
6	Ohio State	12.511	5
7	USC	11.728	8
8	Alabama	11.303	4
9	Auburn	10.581	7
10	Notre Dame	10.301	14

That's only a little different from the CFP as well.. According to SCCR, UCF should been given a chance to play for the title over Alabama.

Interestingly, the Colley Matrix still put UCF at 1 after bowl games, even though they didn't get to face off in the CFP. Since this ranking bears some similarity in philosophy to Colley, let's see if that's true for SCCR also.

Rank	Team	NAW
1	Alabama	16.426
2	Georgia	16.006
3	Ohio State	15.099
4	Wisconsin	14.802
5	UCF	14.489
6	Clemson	13.863
7	Oklahoma	13.681
8	Penn State	11.798
9	Notre Dame	11.595
10	Auburn	10.879

Sorry UCF, but SCCR agrees that Alabama were the 2017 national champions, with their playoff victories vaulting them to the top.

Comparison with Colley Matrix

It has come to my attention that there is already a self-consistent ranking method, the Colley Matrix (https://www.colleyrankings.com/index.html), which was part of the BCS system (prior to the CFP), and is still recognized by NCAA. No surprise that something similar has been done, as Daryl says: there's no such thing as an original sin. Anyway it seems some comparison is warranted. Please note that I am not an expert on the Colley system, but I will do my best to discuss it accurately. The similarities between the two are substantial:

• Only wins and losses are considered (no score differential).
• There is no sorting by conference or history (bias-free).
• The order in which games are played is irrelevant.
• The results are self-consistent.

However, there are also substantial differences between the two methods. First, to state the obvious, the two generate different rankings unless some miraculous coincidence has occurred. Here are some more details and examples:

Units.

SCCR outputs NAWs with units of wins, with wins against above-average teams being worth >1 win and wins against below-average teams worth <1 win, and with losses negative and opposite to wins. Thus as of week 5 2021, 4-0 Michigan is at #1 with 4.42142527 effective wins. Meanwhile the Colley matrix returns ratings which are more similar to win percentages. Colley also has Michigan at #1 for the same week, with a rating of 0.92392. In the Colley system, ratings are (almost) always between 0 and 1, so even an undefeated team with the highest strength-of-schedule will show a rating <1. In SCCR NAWs average around 0, and beating an average team yields exp(0) = 1 wins. In Colley, ratings average to 0.5, i.e. an even win-loss rate.

Strength-of-schedule adjustments for win/loss outcomes.

Both systems use the strengths of all opponents iteratively to determine the values of wins and losses. However, in Colley whether a particular game was won or lost does not affect the strength of schedule adjustment (to first order). In SCCR the win-loss state is directly accounted for in the NAW exponential function. This is best demonstrated by example. Again, let's take (the current at time of writing) week 5 2021. Rutgers is 3-1, having just lost to #1 Michigan. In SCCR Rutgers is ranked at #28 with NAW 1.98208119. In Colley, Rutgers is #16 with rating 0.7713. Now, let's imagine a hypothetical in which Rutgers had beaten Michigan, but lost a previous game against Temple. In Colley, Rutgers remains at #16 and has rating 0.7700. This is nearly identical to before. In SCCR, Rutgers rises to #25 (a jump of 3 positions) and has NAW 2.24361296 (+0.26153177 wins). This is because in SCCR the value of beating a very good opponent outweighs the penalty of losing to a middling opponent. On the other hand, in Colley, Rutger's rating only changes at all because of the changes in the strength-of-schedule of other teams (which is why it hardly budged).

Differing number of games completed.

As discussed in point 1, SCCR is effectively a cumulative resume rating system, while Colley is based on a rate. Therefore SCCR will prefer teams with larger numbers of games played (or more specifically, won) relative to Colley. I demonstrate this on the 2020 "regular" season (I had to use the Wayback Machine on the Colley website, and this only shows top 25 ranks and no ratings). Here are SCCR and Colley ranks for 2020 as of December 20th:

Rank	Team	NAW	Colley Team
1	Coastal Carolina	11.935	Alabama
2	Alabama	11.690	Cincinnati
3	Clemson	10.739	Coastal Carolina
4	Notre Dame	9.867	Clemson
5	Cincinnati	9.789	Ohio State
6	Louisiana	9.096	San Jose State
7	BYU	8.899	Louisiana
8	Miami (FL)	7.371	Notre Dame
9	San Jose State	7.199	BYU
10	Iowa State	6.949	Miami
11	Oklahoma	6.939	Ball State
12	Texas A&M	6.507	Texas A&M
13	Ohio State	6.122	Indiana
14	North Carolina	5.504	Oklahoma
15	Oklahoma State	5.369	Tulsa
16	Ball State	5.338	Iowa State
17	North Carolina State	5.306	Georgia
18	Army	5.261	USC
19	Liberty	5.109	Buffalo
20	Tulsa	4.851	Colorado
21	Georgia	4.687	North Carolina
22	Florida	4.681	North Carolina State
23	Marshall	4.633	Army
24	Appalachian State	4.610	Oklahoma State
25	Indiana	4.608	Northwestern

Notice that 6-0 Ohio State has dropped from playoff position at #3 in the CFP rankings to #13 here (#5 in Colley), due to not having earned enough wins in their shortened season. They would have been replaced by the Chanticleers! Now tell me there's a better rating system.

Comparing the Colley rankings to SCCR rankings, the big gainers under Colley were B1G and PAC-12 teams like Ohio State, Indiana, USC, and Colorado, who all played shortened seasons. So as expected, SCCR prefers teams with more wins under their belt relative to Colley. While neither is intended as a predictive method, it is probably fair to say that Colley has more traits of a predictive method and SCCR comes closer to a pure resume ranker.

Note that AAW (NAW per game played) is also available within SCCR, if you prefer an efficiency metric. The top four teams by AAW in 2020 were: Cincinnati, Coastal Carolina, Alabama, and San Jose State.

What's a win worth?

In SCCR, the answer is the same for everyone: exp({opponent NAW}/{maximum NAW}). In Colley, the answer depends on how your year has been going. Here's an example. As I write this, it is Monday morning following week 13 of 2021. Iowa sits at #13 in SCCR with NAW of 7.42832202, and #10 in Colley with rating 0.81598. Meanwhile Ohio is #121 (-8.07105492) by SCCR and #123 (0.25856) by Colley. Let's imagine these two took it upon themselves to play an unscheduled match here on Monday morning while the rest of the teams rest. Iowa wins of course. By SCCR, Iowa gains 0.54494081 to total 7.97326283 wins and moves up to #11. Per Colley, Iowa loses 0.00411 rating points and drops to #11 with 0.81187. Punished for winning!

Now suppose that instead of Iowa, it was the California Golden Bears who beat Ohio tonight. California is currently #100 (-4.20083795) by SCCR and #102 (0.36630) by Colley. After this hypothetical win, they gain 0.52891463 wins to move to #96 (-3.67192332) in SCCR, and gain 0.03103 points to go to #115 (0.39733) in Colley.

So what do we learn? In Colley, a win against a bad team can decrease your strength-of-schedule adjustment by enough to outweight the value of the win (especially if you are undefeated), while in SCCR beating a given opponent is worth (approximately) the same amount to every team. As we can see, the win value in SCCR is not identically the same because the outcome adjusts the NAWs of schedule for all teams (specifically, California beating Ohio decreases Ohio's NAW by much more than Iowa beating them would). This effect would be diminished if the example was after a complete season, but unfortunately the Colley website only allows for hypothetical games to be added to the current season-in-progress.