MathNFL-PowerRank

I have never done something like this. I would like to apply various quantitative approaches to better understand NFL results and create power rankings. My first ideas are

• Applying Elo to measure skill level of teams.
• A Bayesian approach to latent skill variable.

Below I explain each of these approaches, the results will be below in a table.

Eventually I would like to evaluate these results against betting lines and my own thoughts. I will try and record what I think the win probability for each game is every week.

All data for matchups comes from Pro Football Reference.

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Elo

The information on Elo rating system comes from the respective Wikipedia page. Here each team $i$ will have a rating $R_i$. I follow the Elo suggested scaling, namely that a difference of 200 points in 2 teams ratings should yield an expected score of 0.75, this means that the team has a 75% chance of winning (if we ignore draws). In reality we will account for draws with a draw being worth 0.5 wins. Hence the expected score for team $j$ when playing against team $j$ will be $$E_i = \frac{1}{1 + 10^{(R_j - R_i)/400}}.$$ The rank update will occur after each week and works as follows. A team $i$ shall either achieve 1, 0.5, or 0 points based on realized results each week (by winning, drawing, or losing respectively, denoted by $S_i$. The difference between this score and their expected score will be multiplied by a factor $K$, finally this adjustment will be added to their original score. $$R_i^{\textrm{new}} = R_i^{\textrm{old}} + K (S_i - E_i).$$ $K$ represents the maximal possible adjustment per game, we will experiment with different values here, chess has used 16 for masters and 32 for noobs. Each team will start out with a ranking of 1000, and each week their ranking will be updated to tell us where they stand. Furthermore a predicted winner of each match will be calculated.

Note that we do not take much into account here, things such as home field advantage are not looked after at all. At week 1, each team will have a equal probability to win or lose, so it may take a few weeks for the ranks to establish themselves. To address this, I will start the simulation from the 2017/18 season. We will reset the ranks using the rank reset factor chosen by optimization.

Another factor I wanted to address was to account for was how much a team won by. In week 3 the Giants 19-16 win over the Panthers should not be valued the same as the Bills 41-7 absolute decimation of the Titans. To do this a win will not count as a score of 1, instead we will use a logistic function based of the point differential. I will fit this logistic function such that the 3 point win is worth around 0.65 points, a 7 point win around 0.8, and a 10 point win around 0.9 points. The 41-7 beatdown will now be worth 0.999 points for the winning team. A logistic factor $\beta=-0.1$ fulfills these conditions, and yields a score curve which looks like:

In implementing the model a range a values were tested across the 2020/21 and 2021/22 season to minimize the mean square error between the expected and realized scores. The values which were minimized over were the logistic rate, the maximal rank change $K$, and a 'reset factor' $r$ which shrinks the ranks after each season. So a reset factor of $0.1$ means that after each season, a team only keep 10% of their previous rank. Testing a range of values for these 3 variables yielded a minimal mean square error for a logistic rate of 100, $K=11$, and $r=0.68$. This has yielded the following rankings over the 2022/23 season:

	Week ConfChamp	Week Division	Week WildCard	Week 18	Week 17	Week 16	Week 15	Week 14	Week 13	Week 12	Week 11	Week 10	Week 9	Week 8	Week 7	Week 6	Week 5	Week 4	Week 3	Week 2	Week 1	Week 0
0	Kansas City Chiefs	San Francisco 49ers	San Francisco 49ers	Buffalo Bills	Buffalo Bills	Buffalo Bills	Buffalo Bills	Buffalo Bills	Buffalo Bills	Kansas City Chiefs	Kansas City Chiefs	Kansas City Chiefs	Buffalo Bills	Buffalo Bills	Buffalo Bills	Buffalo Bills	Buffalo Bills	Buffalo Bills	Buffalo Bills	Buffalo Bills	Kansas City Chiefs	Kansas City Chiefs
1	Cincinnati Bengals	Cincinnati Bengals	Buffalo Bills	San Francisco 49ers	San Francisco 49ers	San Francisco 49ers	San Francisco 49ers	Kansas City Chiefs	Kansas City Chiefs	Buffalo Bills	Buffalo Bills	Buffalo Bills	Kansas City Chiefs	Kansas City Chiefs	Kansas City Chiefs	Kansas City Chiefs	Kansas City Chiefs	Kansas City Chiefs	Tampa Bay Buccaneers	Tampa Bay Buccaneers	Buffalo Bills	Buffalo Bills
2	San Francisco 49ers	Kansas City Chiefs	Kansas City Chiefs	Kansas City Chiefs	Kansas City Chiefs	Kansas City Chiefs	Kansas City Chiefs	San Francisco 49ers	Dallas Cowboys	Dallas Cowboys	Dallas Cowboys	Minnesota Vikings	Philadelphia Eagles	Dallas Cowboys	Cincinnati Bengals	Tampa Bay Buccaneers	Tampa Bay Buccaneers	Tampa Bay Buccaneers	Kansas City Chiefs	Kansas City Chiefs	Tampa Bay Buccaneers	Tampa Bay Buccaneers
3	Buffalo Bills	Buffalo Bills	Cincinnati Bengals	Cincinnati Bengals	Cincinnati Bengals	Cincinnati Bengals	Cincinnati Bengals	Dallas Cowboys	San Francisco 49ers	San Francisco 49ers	San Francisco 49ers	Dallas Cowboys	Dallas Cowboys	Philadelphia Eagles	Dallas Cowboys	Philadelphia Eagles	San Francisco 49ers	Green Bay Packers	Los Angeles Rams	Los Angeles Rams	Los Angeles Rams	Los Angeles Rams
4	Philadelphia Eagles	Philadelphia Eagles	Dallas Cowboys	Dallas Cowboys	Dallas Cowboys	Dallas Cowboys	Philadelphia Eagles	Philadelphia Eagles	Philadelphia Eagles	Cincinnati Bengals	Tennessee Titans	San Francisco 49ers	Cincinnati Bengals	Minnesota Vikings	Philadelphia Eagles	Dallas Cowboys	Dallas Cowboys	San Francisco 49ers	Green Bay Packers	Green Bay Packers	Miami Dolphins	Green Bay Packers
5	Dallas Cowboys	Dallas Cowboys	Philadelphia Eagles	Philadelphia Eagles	Philadelphia Eagles	Philadelphia Eagles	Dallas Cowboys	Cincinnati Bengals	Cincinnati Bengals	Philadelphia Eagles	Baltimore Ravens	Philadelphia Eagles	Baltimore Ravens	San Francisco 49ers	Tampa Bay Buccaneers	Cincinnati Bengals	Green Bay Packers	Dallas Cowboys	Miami Dolphins	Miami Dolphins	Green Bay Packers	San Francisco 49ers
6	Pittsburgh Steelers	Pittsburgh Steelers	Pittsburgh Steelers	Minnesota Vikings	Green Bay Packers	Minnesota Vikings	Minnesota Vikings	Baltimore Ravens	Minnesota Vikings	Minnesota Vikings	Cincinnati Bengals	Cincinnati Bengals	Minnesota Vikings	Tennessee Titans	Los Angeles Rams	San Francisco 49ers	Cincinnati Bengals	Cincinnati Bengals	Dallas Cowboys	San Francisco 49ers	Minnesota Vikings	Dallas Cowboys
7	Green Bay Packers	Green Bay Packers	Green Bay Packers	Pittsburgh Steelers	Los Angeles Chargers	Baltimore Ravens	Baltimore Ravens	Minnesota Vikings	Baltimore Ravens	Tennessee Titans	Philadelphia Eagles	Baltimore Ravens	San Francisco 49ers	Cincinnati Bengals	Tennessee Titans	Los Angeles Rams	Philadelphia Eagles	Los Angeles Rams	San Francisco 49ers	Dallas Cowboys	New Orleans Saints	Cincinnati Bengals
8	Los Angeles Chargers	Los Angeles Chargers	Los Angeles Chargers	Green Bay Packers	Baltimore Ravens	Green Bay Packers	New England Patriots	New England Patriots	Tennessee Titans	Baltimore Ravens	Tampa Bay Buccaneers	Tennessee Titans	Tennessee Titans	Baltimore Ravens	Minnesota Vikings	Minnesota Vikings	Baltimore Ravens	Miami Dolphins	Philadelphia Eagles	Cincinnati Bengals	Cincinnati Bengals	Tennessee Titans
9	Minnesota Vikings	Minnesota Vikings	Minnesota Vikings	Los Angeles Chargers	Minnesota Vikings	Los Angeles Chargers	Miami Dolphins	Tampa Bay Buccaneers	Tampa Bay Buccaneers	Miami Dolphins	Minnesota Vikings	Tampa Bay Buccaneers	Seattle Seahawks	Tampa Bay Buccaneers	San Francisco 49ers	New England Patriots	Los Angeles Rams	Philadelphia Eagles	Cincinnati Bengals	Philadelphia Eagles	Dallas Cowboys	New Orleans Saints
10	Detroit Lions	Detroit Lions	Detroit Lions	Baltimore Ravens	New Orleans Saints	New England Patriots	Cleveland Browns	Miami Dolphins	Miami Dolphins	Tampa Bay Buccaneers	New England Patriots	Seattle Seahawks	Tampa Bay Buccaneers	Seattle Seahawks	Baltimore Ravens	Tennessee Titans	Minnesota Vikings	Baltimore Ravens	Baltimore Ravens	New Orleans Saints	San Francisco 49ers	Indianapolis Colts
11	Baltimore Ravens	Baltimore Ravens	Baltimore Ravens	Detroit Lions	New England Patriots	Tampa Bay Buccaneers	Las Vegas Raiders	Tennessee Titans	Seattle Seahawks	New England Patriots	Seattle Seahawks	Miami Dolphins	New England Patriots	Los Angeles Rams	Seattle Seahawks	Baltimore Ravens	Tennessee Titans	Tennessee Titans	Minnesota Vikings	Baltimore Ravens	Indianapolis Colts	Miami Dolphins
12	New England Patriots	New England Patriots	New England Patriots	Seattle Seahawks	Tampa Bay Buccaneers	Pittsburgh Steelers	Detroit Lions	Seattle Seahawks	New England Patriots	Seattle Seahawks	Miami Dolphins	New England Patriots	Los Angeles Rams	Miami Dolphins	Miami Dolphins	Green Bay Packers	Miami Dolphins	Minnesota Vikings	Tennessee Titans	Los Angeles Chargers	Tennessee Titans	New England Patriots
13	New Orleans Saints	New Orleans Saints	New Orleans Saints	New England Patriots	Cleveland Browns	Las Vegas Raiders	Green Bay Packers	Detroit Lions	Cleveland Browns	Washington Commanders	Washington Commanders	Los Angeles Rams	Miami Dolphins	New England Patriots	Green Bay Packers	Indianapolis Colts	New England Patriots	Arizona Cardinals	Indianapolis Colts	New England Patriots	Baltimore Ravens	Seattle Seahawks
14	Miami Dolphins	Miami Dolphins	Miami Dolphins	New Orleans Saints	Pittsburgh Steelers	Miami Dolphins	Tampa Bay Buccaneers	Washington Commanders	Las Vegas Raiders	New York Jets	Los Angeles Chargers	Green Bay Packers	Cleveland Browns	New Orleans Saints	Las Vegas Raiders	Los Angeles Chargers	New Orleans Saints	Los Angeles Chargers	Cleveland Browns	Tennessee Titans	Seattle Seahawks	Baltimore Ravens
15	Seattle Seahawks	Seattle Seahawks	Jacksonville Jaguars	Miami Dolphins	Seattle Seahawks	New Orleans Saints	Tennessee Titans	Las Vegas Raiders	Washington Commanders	Los Angeles Chargers	Indianapolis Colts	Arizona Cardinals	Los Angeles Chargers	Green Bay Packers	New England Patriots	Miami Dolphins	Los Angeles Chargers	Indianapolis Colts	New Orleans Saints	Minnesota Vikings	Los Angeles Chargers	Minnesota Vikings
16	Jacksonville Jaguars	Jacksonville Jaguars	Seattle Seahawks	Jacksonville Jaguars	Las Vegas Raiders	Cleveland Browns	Los Angeles Chargers	Los Angeles Chargers	Green Bay Packers	Cleveland Browns	New Orleans Saints	Los Angeles Chargers	New Orleans Saints	Cleveland Browns	Arizona Cardinals	New Orleans Saints	Indianapolis Colts	Cleveland Browns	New England Patriots	Arizona Cardinals	Pittsburgh Steelers	Los Angeles Chargers
17	Cleveland Browns	Cleveland Browns	Cleveland Browns	Tampa Bay Buccaneers	Detroit Lions	Seattle Seahawks	Seattle Seahawks	Cleveland Browns	New York Jets	Las Vegas Raiders	Green Bay Packers	New York Jets	New York Jets	Indianapolis Colts	Indianapolis Colts	Seattle Seahawks	Arizona Cardinals	New England Patriots	Arizona Cardinals	Pittsburgh Steelers	New England Patriots	Arizona Cardinals
18	Las Vegas Raiders	Las Vegas Raiders	Las Vegas Raiders	Cleveland Browns	Miami Dolphins	Detroit Lions	Pittsburgh Steelers	Green Bay Packers	Pittsburgh Steelers	Green Bay Packers	Los Angeles Rams	Indianapolis Colts	Green Bay Packers	Los Angeles Chargers	Los Angeles Chargers	Atlanta Falcons	Cleveland Browns	Seattle Seahawks	Denver Broncos	Seattle Seahawks	Philadelphia Eagles	Cleveland Browns
19	Washington Commanders	Washington Commanders	New York Giants	Las Vegas Raiders	Jacksonville Jaguars	Tennessee Titans	Washington Commanders	New York Jets	Detroit Lions	Arizona Cardinals	Arizona Cardinals	Cleveland Browns	Atlanta Falcons	Arizona Cardinals	New Orleans Saints	Las Vegas Raiders	Las Vegas Raiders	New Orleans Saints	Seattle Seahawks	Indianapolis Colts	Cleveland Browns	Pittsburgh Steelers
20	Tampa Bay Buccaneers	Tampa Bay Buccaneers	Washington Commanders	Washington Commanders	New York Giants	Carolina Panthers	New Orleans Saints	Pittsburgh Steelers	Los Angeles Chargers	Pittsburgh Steelers	New York Jets	Washington Commanders	Arizona Cardinals	Las Vegas Raiders	New York Giants	Arizona Cardinals	Seattle Seahawks	Las Vegas Raiders	Los Angeles Chargers	Cleveland Browns	Arizona Cardinals	Philadelphia Eagles
21	Carolina Panthers	Carolina Panthers	Tampa Bay Buccaneers	Carolina Panthers	Tennessee Titans	Jacksonville Jaguars	New York Jets	New Orleans Saints	Arizona Cardinals	New Orleans Saints	Cleveland Browns	New York Giants	Indianapolis Colts	Atlanta Falcons	Atlanta Falcons	New York Giants	Denver Broncos	Denver Broncos	Pittsburgh Steelers	Denver Broncos	Las Vegas Raiders	Las Vegas Raiders
22	New York Giants	New York Giants	Carolina Panthers	New York Giants	Carolina Panthers	Los Angeles Rams	New York Giants	Los Angeles Rams	New Orleans Saints	Los Angeles Rams	Las Vegas Raiders	New Orleans Saints	New York Giants	New York Jets	New York Jets	Cleveland Browns	Atlanta Falcons	Atlanta Falcons	Las Vegas Raiders	Las Vegas Raiders	Chicago Bears	Denver Broncos
23	Tennessee Titans	Tennessee Titans	Tennessee Titans	Tennessee Titans	Los Angeles Rams	Washington Commanders	Indianapolis Colts	Carolina Panthers	Los Angeles Rams	Indianapolis Colts	Atlanta Falcons	Pittsburgh Steelers	Las Vegas Raiders	New York Giants	Cleveland Browns	Pittsburgh Steelers	New York Giants	Pittsburgh Steelers	Chicago Bears	Chicago Bears	Washington Commanders	Washington Commanders
24	Atlanta Falcons	Atlanta Falcons	Atlanta Falcons	Atlanta Falcons	Washington Commanders	New York Giants	Jacksonville Jaguars	Arizona Cardinals	Indianapolis Colts	Detroit Lions	Pittsburgh Steelers	Las Vegas Raiders	Denver Broncos	Denver Broncos	Pittsburgh Steelers	Denver Broncos	Pittsburgh Steelers	Chicago Bears	Atlanta Falcons	Detroit Lions	Denver Broncos	Chicago Bears
25	Los Angeles Rams	Los Angeles Rams	Los Angeles Rams	Los Angeles Rams	New York Jets	New York Jets	Los Angeles Rams	Indianapolis Colts	New York Giants	Atlanta Falcons	Detroit Lions	Atlanta Falcons	Washington Commanders	Washington Commanders	Chicago Bears	New York Jets	Chicago Bears	Detroit Lions	Detroit Lions	Washington Commanders	Atlanta Falcons	Atlanta Falcons
26	Denver Broncos	Denver Broncos	Denver Broncos	Denver Broncos	Arizona Cardinals	Indianapolis Colts	Carolina Panthers	Atlanta Falcons	Carolina Panthers	Carolina Panthers	New York Giants	Denver Broncos	Pittsburgh Steelers	Pittsburgh Steelers	Denver Broncos	Washington Commanders	New York Jets	New York Giants	Carolina Panthers	Atlanta Falcons	Houston Texans	Detroit Lions
27	New York Jets	New York Jets	New York Jets	New York Jets	Atlanta Falcons	Arizona Cardinals	Denver Broncos	New York Giants	Atlanta Falcons	New York Giants	Denver Broncos	Carolina Panthers	Chicago Bears	Chicago Bears	Washington Commanders	Chicago Bears	Houston Texans	Jacksonville Jaguars	Washington Commanders	Houston Texans	Detroit Lions	Houston Texans
28	Arizona Cardinals	Arizona Cardinals	Arizona Cardinals	Arizona Cardinals	Denver Broncos	Atlanta Falcons	Arizona Cardinals	Jacksonville Jaguars	Denver Broncos	Jacksonville Jaguars	Chicago Bears	Chicago Bears	Detroit Lions	Carolina Panthers	Carolina Panthers	Houston Texans	Detroit Lions	Carolina Panthers	Jacksonville Jaguars	New York Giants	Carolina Panthers	Carolina Panthers
29	Indianapolis Colts	Indianapolis Colts	Indianapolis Colts	Indianapolis Colts	Indianapolis Colts	Denver Broncos	Atlanta Falcons	Denver Broncos	Jacksonville Jaguars	Denver Broncos	Carolina Panthers	Detroit Lions	Carolina Panthers	Detroit Lions	Detroit Lions	Detroit Lions	Washington Commanders	Washington Commanders	Houston Texans	Carolina Panthers	New York Giants	New York Giants
30	Houston Texans	Houston Texans	Houston Texans	Houston Texans	Chicago Bears	Chicago Bears	Chicago Bears	Chicago Bears	Chicago Bears	Chicago Bears	Jacksonville Jaguars	Jacksonville Jaguars	Jacksonville Jaguars	Houston Texans	Houston Texans	Jacksonville Jaguars	Carolina Panthers	Houston Texans	New York Giants	Jacksonville Jaguars	New York Jets	New York Jets
31	Chicago Bears	Chicago Bears	Chicago Bears	Chicago Bears	Houston Texans	Houston Texans	Houston Texans	Houston Texans	Houston Texans	Houston Texans	Houston Texans	Houston Texans	Houston Texans	Jacksonville Jaguars	Jacksonville Jaguars	Carolina Panthers	Jacksonville Jaguars	New York Jets	New York Jets	New York Jets	Jacksonville Jaguars	Jacksonville Jaguars

And gave the Rams a 52% chance to win in the season opener against the Bills (not a great start, but much better than the experts on the pregame show who went around 70% for the Rams). I will continue to update the Elo power rankings each week, hopefully get that automated here soon, as well as some graphics to display the evolution over time.

Matchup Predictions

The following table lists the matchups, and the predicted chance for team 1 to win (0.46 corresponds to 46%). This is for the Superbowl of 2022/23.

Team 1	Team 2	Prediction
Kansas City Chiefs	Philadelphia Eagles	0.568233

Bayes

While Elo is an easy to implement and understand model, I am an appreciator of Bayesian statistics. So it only makes sense that I try to apply it here. I will preface everything that follows, that I really do not know what I am doing here with applying mathematical models to sports, and am just trying to learn and play with some ideas. Also all my homies hate frequentist statistics, so Bayes all the way baby.

Intro to Bayes

If you know about Bayesian statistics, this will probably not be helpful to you, or if you just don't care. Also I tried to use very little formulas, and without a chalkboard, that is not easy.

First, a quick lesson on Bayesian approaches (here is a proper one). Essentially we start out any situation with our prior beliefs (for instance a certain aboreal YouTuber may believe that the Steelers are going to the Superbowl). We quantify these beliefs in terms of a probability distribution. Then upon observing an event, we update our beliefs according to how likely we quantified the event to be.

The following is for demonstrative purposes only, and not actually how the model works. I will take my Giants as an example, say we want to know on average how many of their first 4 games they will win, call this $\mu$. Now I have my beliefs about $\mu$, these are represented by the prior distribution.

Now the actual amount of games they win is something random around the average, for simplicity say they they have a probability $p$ of winning each game, so the probability that they win all 4 would be given by $p^4$ or the probability that they lost all 4 would be given by $(1-p)^4$.

The Giants play the Titans, Panthers, Cowboys, and Bears. So I would say there is about a 5% chance they win 4 games, a 25% chance they win 3, 50% they win 2, 20% they win 1, and 5% they win 0. Now I have my 'prior' distribution. Note that before we said that each game had a probability $p$ for the Giants to win, and didn't take into account who the opponents were, so we shouldn't have either, but this is an example and shouldn't be taken that seriously.

Now for a given $\mu$ we can actually find the probability $p$, just take average amount of wins $\mu$ and divide it by the number of games (4), to find the probability $p$ for this given $\mu$.

After week 4, I will have a realization of how many times the Giants won (hopefully it isn't 0) that was partially due to the value of $\mu$. Remember, we don't know the value of $\mu$ and only have some belief. Now for every possible value of $\mu$ we evaluate what the likelihood of our observation was, and call this $f(x | \mu)$. Say the Giants won all 4 games, if $\mu$ was equal to 4, then $p = 4/4 =1$, so $p^4 = 1^4 =1$, and this had a 100% chance of happening (in math terms $f(x=4| \mu=4) = 1$). The story isn't so simple for another value of $\mu$, say 3. We get that $p = \frac{3}{4}=0.75$, then the likelihood to have won 4 games is $\binom{4}{3}0.75^3(1-0.75) \approx 0.42$. The wierd term at the beginning is the binomial coefficient and just deals with how many different ways you can win 3 out of 4 games, and all that is relevant here is that we have that the likelihood $f(x=4 | \mu=3) = 0.42$ of what we observed (4 wins) given the wins on average (3), is about 42%. The model will continue to calculate the likelihoods for $\mu = 0,1,2$, but that would be a pain (this is why we make computers do it).

In the end we would have a likelihood of the observation for each value of $\mu$, we then would weight this value by our prior belief, so in the case of our prior guess being that $\mu$ equals 3 with probability 25% and the likelihoood of 42% , we get that the probability that $\mu$ being equal to 3, given the fact that we observed 4 wins is $25% * 42% * C$ where $C$ is a constant we have to calculate as well, but is not needed to be understood here (it is about 0.25). So we have that the probability that $\mu$ is equal to 3, given the observed 4 wins is 25%. The probability of $\mu=4$ is now 20%, so we can see that now we believe that our prior beliefs have changed quite a lot (we used to think that $\mu=4$ has probability 5%).

This was probably very confusing and my apologies are in order

The actual model

I am writing this half to explain it, and half to figure out exactly how exactly the model should work. Hence, the lack of clarity.

Although probably not as powerful as the almighty algorithm by Jackson Krueger, I hope to be able to compare his ideas to mine. From now on we will use skill level and rating interchangebly. We will use a latent variable to measure the skill level of a given team $i$. We will call this $S_i$. This is the variable we are interested in measuring, but we will never see it directly. We will say that in a given game, each team realizes a skill level based on some distribution. The team with the higher realized skill level then wins the match. Hence, we have that $$\mathbb{P} [ i \textrm{ wins against } j ] = \mathbb{P} [ S_i > S_j ] .$$ We have the problem of draws here (if the distributions are continuous), I have not figured out exactly how I am going to deal with that yet.

For simplicity, we will work with normal distributions for now. We will say each team's skill is distributed around a mean $mu_i$ with standard deviation $\sigma_i$ (this may end up being the same for all teams). Our prior is then on the $mu_i$ (may have to have one on the $\sigma_i$ as well if we do not fix those at the beginning), let this just be a standard normal for now. Then with a match between $i$ and $j$, we are observing the probability that $i>j$. We then get a posterior after the match (say $i$ wins) $$\pi(\mu_i = x, \mu_j = y | i \textrm{ won}] = \frac{f(i>j | x,y) \pi(x,y)}{\int f(i>j | x,y) \pi(x,y) dxdy }.$$ Where $f(i>j | x,y)$ is the probability that the difference of independent normals centered around $x$ and $y$ is positive, i.e. $$\mathbb{P}\left[Z > - \frac{\mu_i - \mu_j}{\sigma_i + \sigma_j} \right].$$ This is not looking like it is going to have a nice conjugate prior. But that is an effort for later. So for now this will have to be done numerically.

To do this numerically each team will start out with a standard normally distributed random variable, this is our original prior. After observing a result, we shall be able to update this, with the observation (say team $i$ wins) conditional on the skill difference being normally distributed about the difference. The probability to observe $i$ winning will the be the area under the distribution to the right of 0 (if the difference is $S_i - S_j$, otherwise to the left). The parameters used for the numerical simulation are that each distribution will be represented by 1000 points between $[-5,5]$ (step size 0.01).

Historical performance

Elo Predictions

Here is the performance of the Elo Algorithm. 1 Thumbs up means the correct winner was predicted, 2 if the winner was correct and the point differential was within 7, and 3 if the winner was correct and the differential was within 3. 1 Thumbs down means the incorrect winner was predicted and the point differential was more than 7, 2 if the winner was incorrect and the point differential was within 7, and 3 if the winner was incorrect and the differential was within 3. Note that the parameter optimization was done for the mean squared error of logistic scores, so another choice of parameters may perform better here. Trying to optimize the parameters to minimize the difference between predicted and realized score differential is something I still want to try. So far the prediction record is: 131-77 (63.0%).

Date	Winner/tie	Loser/tie	Game Score	Accuracy	Normalized Score	Prediction
2023-01-29	Kansas City Chiefs	Cincinnati Bengals	23-20	👍👍👍	0.666	0.541
2023-01-29	Philadelphia Eagles	San Francisco 49ers	31-7	👎👎👎	0.996	0.462
2023-01-22	San Francisco 49ers	Dallas Cowboys	19-12	👎👎👎	0.834	0.499
2023-01-22	Cincinnati Bengals	Buffalo Bills	27-10	👎👎👎	0.980	0.424
2023-01-21	Philadelphia Eagles	New York Giants	38-7	👍	0.999	0.711
2023-01-21	Kansas City Chiefs	Jacksonville Jaguars	27-20	👍👍👍	0.834	0.811
2023-01-16	Dallas Cowboys	Tampa Bay Buccaneers	31-14	👍	0.980	0.599
2023-01-15	Cincinnati Bengals	Baltimore Ravens	24-17	👍👍	0.834	0.533
2023-01-15	New York Giants	Minnesota Vikings	31-24	👎👎👎	0.834	0.322
2023-01-15	Buffalo Bills	Miami Dolphins	34-31	👍👍👍	0.666	0.620
2023-01-14	Jacksonville Jaguars	Los Angeles Chargers	31-30	👎	0.557	0.387
2023-01-14	San Francisco 49ers	Seattle Seahawks	41-23	👍	0.984	0.616
2023-01-08	Detroit Lions	Green Bay Packers	20-16	👎👎	0.715	0.453
2023-01-08	Seattle Seahawks	Los Angeles Rams	19-16	👍👍👍	0.666	0.589
2023-01-08	Philadelphia Eagles	New York Giants	22-16	👍👍	0.799	0.666
2023-01-08	Denver Broncos	Los Angeles Chargers	31-28	👎👎	0.666	0.412
2023-01-08	Washington Commanders	Dallas Cowboys	26-6	👎👎👎	0.990	0.345
2023-01-08	San Francisco 49ers	Arizona Cardinals	38-13	👍	0.997	0.661
2023-01-08	Miami Dolphins	New York Jets	11-6	👍👍	0.760	0.601
2023-01-08	Houston Texans	Indianapolis Colts	32-31	👎👎	0.557	0.326
2023-01-08	Pittsburgh Steelers	Cleveland Browns	28-14	👎👎👎	0.962	0.475
2023-01-08	Minnesota Vikings	Chicago Bears	29-13	👍	0.975	0.683
2023-01-08	Carolina Panthers	New Orleans Saints	10-7	👎👎	0.666	0.411
2023-01-08	Atlanta Falcons	Tampa Bay Buccaneers	30-17	👎👎👎	0.952	0.352
2023-01-08	Cincinnati Bengals	Baltimore Ravens	27-16	👎👎👎	0.926	0.498
2023-01-08	Buffalo Bills	New England Patriots	35-23	👍	0.941	0.630
2023-01-07	Jacksonville Jaguars	Tennessee Titans	20-16	👎👎👎	0.715	0.257
2023-01-07	Kansas City Chiefs	Las Vegas Raiders	31-13	👍	0.984	0.739
2023-01-01	Los Angeles Chargers	Los Angeles Rams	31-10	👎👎👎	0.992	0.482
2023-01-01	Green Bay Packers	Minnesota Vikings	41-17	👎👎👎	0.996	0.386
2023-01-01	San Francisco 49ers	Las Vegas Raiders	37-34	👍👍👍	0.666	0.670
2023-01-01	Seattle Seahawks	New York Jets	23-6	👍	0.980	0.585
2023-01-01	Pittsburgh Steelers	Baltimore Ravens	16-13	👎👎	0.666	0.351
2023-01-01	New Orleans Saints	Philadelphia Eagles	20-10	👎👎👎	0.909	0.351
2023-01-01	New England Patriots	Miami Dolphins	23-21	👎	0.613	0.492
2023-01-01	Jacksonville Jaguars	Houston Texans	31-3	👍	0.998	0.533
2023-01-01	Kansas City Chiefs	Denver Broncos	27-24	👍👍👍	0.666	0.765
2023-01-01	New York Giants	Indianapolis Colts	38-10	👎👎👎	0.998	0.490
2023-01-01	Cleveland Browns	Washington Commanders	24-10	👍	0.962	0.500
2023-01-01	Detroit Lions	Chicago Bears	41-10	👎👎👎	0.999	0.496
2023-01-01	Tampa Bay Buccaneers	Carolina Panthers	30-24	👍👍👍	0.799	0.678
2023-01-01	Atlanta Falcons	Arizona Cardinals	20-19	👎	0.557	0.437
2022-12-29	Dallas Cowboys	Tennessee Titans	27-13	👍	0.962	0.520
2022-12-26	Los Angeles Chargers	Indianapolis Colts	20-3	👍	0.980	0.539
2022-12-25	Tampa Bay Buccaneers	Arizona Cardinals	19-16	👍👍👍	0.666	0.611
2022-12-25	Los Angeles Rams	Denver Broncos	51-14	👍	1.000	0.603
2022-12-25	Green Bay Packers	Miami Dolphins	26-20	👎👎	0.799	0.447
2022-12-24	Pittsburgh Steelers	Las Vegas Raiders	13-10	👎👎	0.666	0.467
2022-12-24	Dallas Cowboys	Philadelphia Eagles	40-34	👎👎	0.799	0.488
2022-12-24	San Francisco 49ers	Washington Commanders	37-20	👍	0.980	0.661
2022-12-24	Minnesota Vikings	New York Giants	27-24	👍👍👍	0.666	0.646
2022-12-24	Kansas City Chiefs	Seattle Seahawks	24-10	👍	0.962	0.619
2022-12-24	Houston Texans	Tennessee Titans	19-14	👎👎👎	0.760	0.272
2022-12-24	New Orleans Saints	Cleveland Browns	17-10	👎👎👎	0.834	0.485
2022-12-24	Cincinnati Bengals	New England Patriots	22-18	👍👍	0.715	0.544
2022-12-24	Carolina Panthers	Detroit Lions	37-23	👎👎👎	0.962	0.479
2022-12-24	Buffalo Bills	Chicago Bears	35-13	👍	0.994	0.776
2022-12-24	Baltimore Ravens	Atlanta Falcons	17-9	👍👍	0.863	0.633
2022-12-22	Jacksonville Jaguars	New York Jets	19-3	👎👎👎	0.975	0.384
2022-12-19	Green Bay Packers	Los Angeles Rams	24-12	👎👎👎	0.941	0.465
2022-12-18	Las Vegas Raiders	New England Patriots	30-24	👎👎	0.799	0.456
2022-12-18	New York Giants	Washington Commanders	20-12	👍	0.863	0.523
2022-12-18	Los Angeles Chargers	Tennessee Titans	17-14	👎👎	0.666	0.400
2022-12-18	Cincinnati Bengals	Tampa Bay Buccaneers	34-23	👍	0.926	0.516
2022-12-18	Denver Broncos	Arizona Cardinals	24-15	👎👎👎	0.888	0.447
2022-12-18	Kansas City Chiefs	Houston Texans	30-24	👍👍👍	0.799	0.803
2022-12-18	Detroit Lions	New York Jets	20-17	👎👎	0.666	0.412
2022-12-18	Jacksonville Jaguars	Dallas Cowboys	40-34	👎👎👎	0.799	0.223
2022-12-18	Philadelphia Eagles	Chicago Bears	25-20	👍👍👍	0.760	0.715
2022-12-18	Pittsburgh Steelers	Carolina Panthers	24-16	👍	0.863	0.522
2022-12-18	New Orleans Saints	Atlanta Falcons	21-18	👍👍👍	0.666	0.540
2022-12-17	Buffalo Bills	Miami Dolphins	32-29	👍👍👍	0.666	0.705
2022-12-17	Cleveland Browns	Baltimore Ravens	13-3	👎👎👎	0.909	0.430
2022-12-17	Minnesota Vikings	Indianapolis Colts	39-36	👍👍👍	0.666	0.606
2022-12-15	San Francisco 49ers	Seattle Seahawks	21-13	👍	0.863	0.532
2022-12-12	New England Patriots	Arizona Cardinals	27-13	👍	0.962	0.507
2022-12-11	Los Angeles Chargers	Miami Dolphins	23-17	👎👎	0.799	0.462
2022-12-11	San Francisco 49ers	Tampa Bay Buccaneers	35-7	👎👎👎	0.998	0.475
2022-12-11	Carolina Panthers	Seattle Seahawks	30-24	👎👎👎	0.799	0.369
2022-12-11	Kansas City Chiefs	Denver Broncos	34-28	👍👍👍	0.799	0.754
2022-12-11	Baltimore Ravens	Pittsburgh Steelers	16-14	👍👍👍	0.613	0.614
2022-12-11	Philadelphia Eagles	New York Giants	48-22	👍	0.997	0.636
2022-12-11	Jacksonville Jaguars	Tennessee Titans	36-22	👎👎👎	0.962	0.275
2022-12-11	Detroit Lions	Minnesota Vikings	34-23	👎👎👎	0.926	0.299
2022-12-11	Dallas Cowboys	Houston Texans	27-23	👍👍👍	0.715	0.738
2022-12-11	Cincinnati Bengals	Cleveland Browns	23-10	👍	0.952	0.661
2022-12-11	Buffalo Bills	New York Jets	20-12	👍👍	0.863	0.745
2022-12-08	Los Angeles Rams	Las Vegas Raiders	17-16	👍👍👍	0.557	0.562
2022-12-05	Tampa Bay Buccaneers	New Orleans Saints	17-16	👍👍👍	0.557	0.627
2022-12-04	Dallas Cowboys	Indianapolis Colts	54-19	👍	1.000	0.573
2022-12-04	Las Vegas Raiders	Los Angeles Chargers	27-20	👎👎👎	0.834	0.471
2022-12-04	Cincinnati Bengals	Kansas City Chiefs	27-24	👎👎	0.666	0.425
2022-12-04	Seattle Seahawks	Los Angeles Rams	27-23	👎👎	0.715	0.421
2022-12-04	San Francisco 49ers	Miami Dolphins	33-17	👍	0.975	0.578
2022-12-04	Green Bay Packers	Chicago Bears	28-19	👍👍	0.888	0.652
2022-12-04	Pittsburgh Steelers	Atlanta Falcons	19-16	👎👎	0.666	0.458
2022-12-04	Philadelphia Eagles	Tennessee Titans	35-10	👍	0.997	0.562
2022-12-04	Washington Commanders	New York Giants	20-20	👎	0.500	0.435
2022-12-04	Minnesota Vikings	New York Jets	27-22	👍👍👍	0.760	0.634
2022-12-04	Detroit Lions	Jacksonville Jaguars	40-14	👍	0.997	0.535
2022-12-04	Baltimore Ravens	Denver Broncos	10-9	👍👍👍	0.557	0.601
2022-12-04	Cleveland Browns	Houston Texans	27-14	👍	0.952	0.577
2022-12-01	Buffalo Bills	New England Patriots	24-10	👍	0.962	0.670
2022-11-28	Pittsburgh Steelers	Indianapolis Colts	24-17	👎👎👎	0.834	0.420
2022-11-27	Philadelphia Eagles	Green Bay Packers	40-33	👎👎👎	0.834	0.499
2022-11-27	San Francisco 49ers	New Orleans Saints	13-0	👍	0.952	0.613
2022-11-27	Kansas City Chiefs	Los Angeles Rams	26-10	👍	0.975	0.623
2022-11-27	Los Angeles Chargers	Arizona Cardinals	25-24	👍👍👍	0.557	0.506
2022-11-27	Las Vegas Raiders	Seattle Seahawks	40-34	👍👍	0.799	0.522
2022-11-27	New York Jets	Chicago Bears	31-10	👎👎👎	0.992	0.490
2022-11-27	Carolina Panthers	Denver Broncos	23-10	👎👎👎	0.952	0.416
2022-11-27	Washington Commanders	Atlanta Falcons	19-13	👎👎👎	0.799	0.428
2022-11-27	Jacksonville Jaguars	Baltimore Ravens	28-27	👎👎	0.557	0.304
2022-11-27	Miami Dolphins	Houston Texans	30-15	👍	0.969	0.642
2022-11-27	Cleveland Browns	Tampa Bay Buccaneers	23-17	👎👎👎	0.799	0.344
2022-11-27	Cincinnati Bengals	Tennessee Titans	20-16	👍👍	0.715	0.533
2022-11-24	Minnesota Vikings	New England Patriots	33-26	👍👍	0.834	0.532
2022-11-24	Dallas Cowboys	New York Giants	28-20	👍👍	0.863	0.680
2022-11-24	Buffalo Bills	Detroit Lions	28-25	👍👍👍	0.666	0.792
2022-11-21	San Francisco 49ers	Arizona Cardinals	38-10	👍	0.998	0.596
2022-11-20	Kansas City Chiefs	Los Angeles Chargers	30-27	👍👍👍	0.666	0.682
2022-11-20	Dallas Cowboys	Minnesota Vikings	40-3	👍	1.000	0.558
2022-11-20	Cincinnati Bengals	Pittsburgh Steelers	37-30	👍👍	0.834	0.645
2022-11-20	Las Vegas Raiders	Denver Broncos	22-16	👍👍	0.799	0.521
2022-11-20	New Orleans Saints	Los Angeles Rams	27-20	👎👎👎	0.834	0.402
2022-11-20	Philadelphia Eagles	Indianapolis Colts	17-16	👍👍👍	0.557	0.576
2022-11-20	Baltimore Ravens	Carolina Panthers	13-3	👍👍	0.909	0.667
2022-11-20	Atlanta Falcons	Chicago Bears	27-24	👍👍👍	0.666	0.539
2022-11-20	New England Patriots	New York Jets	10-3	👍👍	0.834	0.643
2022-11-20	Washington Commanders	Houston Texans	23-10	👍	0.952	0.522
2022-11-20	Detroit Lions	New York Giants	31-18	👍	0.952	0.505
2022-11-20	Buffalo Bills	Cleveland Browns	31-23	👍👍	0.863	0.688
2022-11-17	Tennessee Titans	Green Bay Packers	27-17	👎👎👎	0.909	0.432
2022-11-14	Washington Commanders	Philadelphia Eagles	32-21	👎👎👎	0.926	0.343
2022-11-13	San Francisco 49ers	Los Angeles Chargers	22-16	👍👍	0.799	0.590
2022-11-13	Arizona Cardinals	Los Angeles Rams	27-17	👎👎👎	0.909	0.367
2022-11-13	Green Bay Packers	Dallas Cowboys	31-28	👍👍👍	0.666	0.539
2022-11-13	Indianapolis Colts	Las Vegas Raiders	25-20	👍👍	0.760	0.574
2022-11-13	Pittsburgh Steelers	New Orleans Saints	20-10	👎👎👎	0.909	0.473
2022-11-13	Kansas City Chiefs	Jacksonville Jaguars	27-17	👍👍	0.909	0.743
2022-11-13	Detroit Lions	Chicago Bears	31-30	👎	0.557	0.470
2022-11-13	New York Giants	Houston Texans	24-16	👎👎👎	0.863	0.496
2022-11-13	Tennessee Titans	Denver Broncos	17-10	👍👍	0.834	0.539
2022-11-13	Miami Dolphins	Cleveland Browns	39-17	👍	0.994	0.594
2022-11-13	Minnesota Vikings	Buffalo Bills	33-30	👎👎	0.666	0.338
2022-11-13	Tampa Bay Buccaneers	Seattle Seahawks	21-16	👍👍👍	0.760	0.684
2022-11-10	Carolina Panthers	Atlanta Falcons	25-15	👎👎👎	0.909	0.467
2022-11-07	Baltimore Ravens	New Orleans Saints	27-13	👎👎👎	0.962	0.496
2022-11-06	Kansas City Chiefs	Tennessee Titans	20-17	👍👍👍	0.666	0.660
2022-11-06	Tampa Bay Buccaneers	Los Angeles Rams	16-13	👍👍👍	0.666	0.583
2022-11-06	Seattle Seahawks	Arizona Cardinals	31-21	👎👎👎	0.909	0.497
2022-11-06	Jacksonville Jaguars	Las Vegas Raiders	27-20	👎👎👎	0.834	0.403
2022-11-06	New England Patriots	Indianapolis Colts	26-3	👍	0.995	0.524
2022-11-06	Miami Dolphins	Chicago Bears	35-32	👍👍👍	0.666	0.651
2022-11-06	Cincinnati Bengals	Carolina Panthers	42-21	👍	0.992	0.666
2022-11-06	Los Angeles Chargers	Atlanta Falcons	20-17	👍👍👍	0.666	0.606
2022-11-06	Minnesota Vikings	Washington Commanders	20-17	👍👍👍	0.666	0.593
2022-11-06	Detroit Lions	Green Bay Packers	15-9	👎👎👎	0.799	0.333
2022-11-06	New York Jets	Buffalo Bills	20-17	👎👎👎	0.666	0.183
2022-11-03	Philadelphia Eagles	Houston Texans	29-17	👍	0.941	0.647
2022-10-31	Cleveland Browns	Cincinnati Bengals	32-13	👎👎👎	0.988	0.447
2022-10-30	Buffalo Bills	Green Bay Packers	27-17	👍	0.909	0.611
2022-10-30	Seattle Seahawks	New York Giants	27-13	👍	0.962	0.656
2022-10-30	Washington Commanders	Indianapolis Colts	17-16	👎	0.557	0.408
2022-10-30	San Francisco 49ers	Los Angeles Rams	31-14	👎👎👎	0.980	0.448
2022-10-30	Tennessee Titans	Houston Texans	17-10	👍👍	0.834	0.638
2022-10-30	New England Patriots	New York Jets	22-17	👍👍👍	0.760	0.682
2022-10-30	New Orleans Saints	Las Vegas Raiders	24-0	👍	0.996	0.583
2022-10-30	Minnesota Vikings	Arizona Cardinals	34-26	👍👍	0.863	0.569
2022-10-30	Dallas Cowboys	Chicago Bears	49-29	👍	0.990	0.585
2022-10-30	Atlanta Falcons	Carolina Panthers	37-34	👍👍👍	0.666	0.540
2022-10-30	Philadelphia Eagles	Pittsburgh Steelers	35-13	👎👎👎	0.994	0.486
2022-10-30	Miami Dolphins	Detroit Lions	31-27	👍👍👍	0.715	0.680
2022-10-30	Denver Broncos	Jacksonville Jaguars	21-17	👍👍👍	0.715	0.648
2022-10-27	Baltimore Ravens	Tampa Bay Buccaneers	27-22	👎👎👎	0.760	0.377
2022-10-24	Chicago Bears	New England Patriots	33-14	👎👎👎	0.988	0.388
2022-10-23	Miami Dolphins	Pittsburgh Steelers	16-10	👍👍	0.799	0.555
2022-10-23	Seattle Seahawks	Los Angeles Chargers	37-23	👍	0.962	0.526
2022-10-23	Kansas City Chiefs	San Francisco 49ers	44-23	👍	0.992	0.575
2022-10-23	New York Jets	Denver Broncos	16-9	👎👎👎	0.834	0.390
2022-10-23	Las Vegas Raiders	Houston Texans	38-20	👍	0.984	0.591
2022-10-23	New York Giants	Jacksonville Jaguars	23-17	👍👍	0.799	0.542
2022-10-23	Washington Commanders	Green Bay Packers	23-21	👎👎	0.613	0.338
2022-10-23	Dallas Cowboys	Detroit Lions	24-6	👍	0.984	0.672
2022-10-23	Tennessee Titans	Indianapolis Colts	19-10	👍	0.888	0.505
2022-10-23	Baltimore Ravens	Cleveland Browns	23-20	👍👍👍	0.666	0.521
2022-10-23	Carolina Panthers	Tampa Bay Buccaneers	21-3	👎👎👎	0.984	0.263
2022-10-23	Cincinnati Bengals	Atlanta Falcons	35-17	👍	0.984	0.649
2022-10-20	Arizona Cardinals	New Orleans Saints	42-34	👎👎👎	0.863	0.450
2022-10-17	Los Angeles Chargers	Denver Broncos	19-16	👍👍👍	0.666	0.581
2022-10-16	Philadelphia Eagles	Dallas Cowboys	26-17	👎👎👎	0.888	0.382
2022-10-16	Buffalo Bills	Kansas City Chiefs	24-20	👎👎	0.715	0.486
2022-10-16	Seattle Seahawks	Arizona Cardinals	19-9	👍	0.909	0.538
2022-10-16	Los Angeles Rams	Carolina Panthers	24-10	👍	0.962	0.810
2022-10-16	Pittsburgh Steelers	Tampa Bay Buccaneers	20-18	👎👎	0.613	0.309
2022-10-16	New York Giants	Baltimore Ravens	24-20	👎👎👎	0.715	0.280
2022-10-16	Minnesota Vikings	Miami Dolphins	24-16	👎👎👎	0.863	0.448
2022-10-16	New York Jets	Green Bay Packers	27-10	👎👎👎	0.980	0.175
2022-10-16	Indianapolis Colts	Jacksonville Jaguars	34-27	👍👍👍	0.834	0.802
2022-10-16	New England Patriots	Cleveland Browns	38-15	👍	0.995	0.558
2022-10-16	Cincinnati Bengals	New Orleans Saints	30-26	👍👍	0.715	0.513
2022-10-16	Atlanta Falcons	San Francisco 49ers	28-14	👎👎👎	0.962	0.273
2022-10-13	Washington Commanders	Chicago Bears	12-7	👍👍	0.760	0.524
2022-10-10	Kansas City Chiefs	Las Vegas Raiders	30-29	👍👍	0.557	0.732
2022-10-09	Baltimore Ravens	Cincinnati Bengals	19-17	👎👎	0.613	0.426
2022-10-09	Dallas Cowboys	Los Angeles Rams	22-10	👎👎👎	0.941	0.428
2022-10-09	Philadelphia Eagles	Arizona Cardinals	20-17	👎👎	0.666	0.479
2022-10-09	San Francisco 49ers	Carolina Panthers	37-15	👍	0.994	0.764
2022-10-09	Tennessee Titans	Washington Commanders	21-17	👍👍👍	0.715	0.665
2022-10-09	New Orleans Saints	Seattle Seahawks	39-32	👍👍	0.834	0.533
2022-10-09	New York Jets	Miami Dolphins	40-17	👎👎👎	0.995	0.230
2022-10-09	Houston Texans	Jacksonville Jaguars	13-6	👍👍	0.834	0.612
2022-10-09	New England Patriots	Detroit Lions	29-0	👍	0.999	0.692
2022-10-09	Los Angeles Chargers	Cleveland Browns	30-28	👍👍👍	0.613	0.515
2022-10-09	Minnesota Vikings	Chicago Bears	29-22	👍👍	0.834	0.624
2022-10-09	Buffalo Bills	Pittsburgh Steelers	38-3	👍	1.000	0.708
2022-10-09	Tampa Bay Buccaneers	Atlanta Falcons	21-15	👍👍👍	0.799	0.782
2022-10-09	New York Giants	Green Bay Packers	27-22	👎👎👎	0.760	0.188
2022-10-06	Indianapolis Colts	Denver Broncos	12-9	👍👍👍	0.666	0.648
2022-10-03	San Francisco 49ers	Los Angeles Rams	24-9	👎👎👎	0.969	0.455
2022-10-02	Kansas City Chiefs	Tampa Bay Buccaneers	41-31	👍	0.909	0.557
2022-10-02	Green Bay Packers	New England Patriots	27-24	👍👍👍	0.666	0.601
2022-10-02	Las Vegas Raiders	Denver Broncos	32-23	👍	0.888	0.548
2022-10-02	Arizona Cardinals	Carolina Panthers	26-16	👍👍	0.909	0.676
2022-10-02	New York Jets	Pittsburgh Steelers	24-20	👎👎👎	0.715	0.290
2022-10-02	Philadelphia Eagles	Jacksonville Jaguars	29-21	👍👍	0.863	0.737
2022-10-02	Los Angeles Chargers	Houston Texans	34-24	👍	0.909	0.660
2022-10-02	Seattle Seahawks	Detroit Lions	48-45	👍👍👍	0.666	0.686
2022-10-02	Dallas Cowboys	Washington Commanders	25-10	👍	0.969	0.684
2022-10-02	Tennessee Titans	Indianapolis Colts	24-17	👍👍	0.834	0.507
2022-10-02	New York Giants	Chicago Bears	20-12	👎👎👎	0.863	0.400
2022-10-02	Buffalo Bills	Baltimore Ravens	23-20	👍👍👍	0.666	0.679
2022-10-02	Atlanta Falcons	Cleveland Browns	23-20	👎👎	0.666	0.381
2022-10-02	Minnesota Vikings	New Orleans Saints	28-25	👎👎	0.666	0.437
2022-09-29	Cincinnati Bengals	Miami Dolphins	27-15	👍	0.941	0.507
2022-09-26	Dallas Cowboys	New York Giants	23-16	👍👍👍	0.834	0.781
2022-09-25	Denver Broncos	San Francisco 49ers	11-10	👎👎	0.557	0.324
2022-09-25	Green Bay Packers	Tampa Bay Buccaneers	14-12	👎	0.613	0.463
2022-09-25	Los Angeles Rams	Arizona Cardinals	20-12	👍👍	0.863	0.645
2022-09-25	Atlanta Falcons	Seattle Seahawks	27-23	👎👎👎	0.715	0.333
2022-09-25	Jacksonville Jaguars	Los Angeles Chargers	38-10	👎👎👎	0.998	0.246
2022-09-25	Philadelphia Eagles	Washington Commanders	24-8	👍	0.975	0.572
2022-09-25	Tennessee Titans	Las Vegas Raiders	24-22	👍👍👍	0.613	0.622
2022-09-25	Baltimore Ravens	New England Patriots	37-26	👎👎👎	0.926	0.472
2022-09-25	Minnesota Vikings	Detroit Lions	28-24	👍👍👍	0.715	0.661
2022-09-25	Indianapolis Colts	Kansas City Chiefs	20-17	👎👎	0.666	0.355
2022-09-25	Cincinnati Bengals	New York Jets	27-12	👍	0.969	0.765
2022-09-25	Chicago Bears	Houston Texans	23-20	👍👍👍	0.666	0.547
2022-09-25	Carolina Panthers	New Orleans Saints	22-14	👎👎👎	0.863	0.264
2022-09-25	Miami Dolphins	Buffalo Bills	21-19	👎👎	0.613	0.342
2022-09-22	Cleveland Browns	Pittsburgh Steelers	29-17	👍	0.941	0.504
2022-09-19	Philadelphia Eagles	Minnesota Vikings	24-7	👎👎👎	0.980	0.491
2022-09-19	Buffalo Bills	Tennessee Titans	41-7	👍	1.000	0.606
2022-09-18	Green Bay Packers	Chicago Bears	27-10	👍	0.980	0.754
2022-09-18	Denver Broncos	Houston Texans	16-9	👍👍	0.834	0.583
2022-09-18	Arizona Cardinals	Las Vegas Raiders	29-23	👍👍	0.799	0.540
2022-09-18	Dallas Cowboys	Cincinnati Bengals	20-17	👍👍👍	0.666	0.588
2022-09-18	San Francisco 49ers	Seattle Seahawks	27-7	👍	0.990	0.541
2022-09-18	Los Angeles Rams	Atlanta Falcons	31-27	👍👍👍	0.715	0.736
2022-09-18	New England Patriots	Pittsburgh Steelers	17-14	👍👍👍	0.666	0.567
2022-09-18	Tampa Bay Buccaneers	New Orleans Saints	20-10	👍	0.909	0.604
2022-09-18	Miami Dolphins	Baltimore Ravens	42-38	👍👍	0.715	0.545
2022-09-18	Detroit Lions	Washington Commanders	36-27	👎👎👎	0.888	0.437
2022-09-18	Jacksonville Jaguars	Indianapolis Colts	24-0	👎👎👎	0.996	0.198
2022-09-18	New York Jets	Cleveland Browns	31-30	👎👎	0.557	0.287
2022-09-18	New York Giants	Carolina Panthers	19-16	👎👎	0.666	0.465
2022-09-15	Kansas City Chiefs	Los Angeles Chargers	27-24	👍👍👍	0.666	0.688
2022-09-12	Seattle Seahawks	Denver Broncos	17-16	👍👍👍	0.557	0.593
2022-09-11	Tampa Bay Buccaneers	Dallas Cowboys	19-3	👍	0.975	0.570
2022-09-11	Los Angeles Chargers	Las Vegas Raiders	24-19	👍👍	0.760	0.543
2022-09-11	New York Giants	Tennessee Titans	21-20	👎👎	0.557	0.239
2022-09-11	Minnesota Vikings	Green Bay Packers	23-7	👎👎👎	0.975	0.288
2022-09-11	Kansas City Chiefs	Arizona Cardinals	44-21	👍	0.995	0.642
2022-09-11	Baltimore Ravens	New York Jets	24-9	👍	0.969	0.741
2022-09-11	Miami Dolphins	New England Patriots	20-7	👎👎👎	0.952	0.428
2022-09-11	Washington Commanders	Jacksonville Jaguars	28-22	👍👍👍	0.799	0.703
2022-09-11	Philadelphia Eagles	Detroit Lions	38-35	👍👍👍	0.666	0.718
2022-09-11	Houston Texans	Indianapolis Colts	20-20	👎👎	0.500	0.232
2022-09-11	Pittsburgh Steelers	Cincinnati Bengals	23-20	👎👎	0.666	0.431
2022-09-11	Chicago Bears	San Francisco 49ers	19-10	👎👎👎	0.888	0.351
2022-09-11	Cleveland Browns	Carolina Panthers	26-24	👍👍👍	0.613	0.629
2022-09-11	New Orleans Saints	Atlanta Falcons	27-26	👍👍👍	0.557	0.667
2022-09-08	Buffalo Bills	Los Angeles Rams	31-10	👍	0.992	0.552