04-21-2020, 10:43 AM
(This post was last modified: 04-21-2020, 10:43 AM by IthicaHawk.)
Introduction
First off, what is Elo? Elo is a rating and ranking system designed for head-to-head matchups. It is designed to take opinion out of the rating process and only measure based on actual results.
It is designed to form a ranking system less influenced by human biases. It is, of course, not free from bias as we have to decide what values form the rating itself.
Elo cannot account for wild changes, if top players in one team all decide to retire between weeks, for example. Unless manual intervention is used.
A raw Elo is a zero-sum system where the winners recieve a boost to their ranking equal to the drop recieved by the losers. However, the system can be modified so underdogs get more credit for a win against a stronger opponent and, likewise, a strong opponent gets a smaller gain for beating a low ranked team.
Every team starts with an Elo rating of 1500. There's no mathematical reason it can't be 1000 or 0 or 540055. But 1500 is a nice number that scales well and it seems to be the norm.
The Basics
First, see the image in Fig 1 below. It shows the stats involved for one particular week of one particular season. In this case, S21 week 7. I will walk you through the various elements of the calculations referencing this image.
![[Image: nXI1t8B.png]](https://i.imgur.com/nXI1t8B.png)
Fig 1. S21 Week 7
Update Factor - This is the base value for wins and loses. Before any other factors are included, a team that wins gains 20 points and the other team loses 20 points. Simple. We can
scale this factor for more important games if we want to, for example, the ultimus.
Next, we have the matches themselves. Each team is given their own line, this is purely for simplicities sake. I made the sheet so I simply fill in the opponent of each team, home/away
and how many points that team scored, everything else auto-fills from that.
E(h) - Estimated chance of winning for A team. Originally I had planned on doing 4 rows with Home and Away rather than 8 rows but it worked out simpler this way. E(h) is a
throwback to E(home) and E(away) and I just haven't bothered changing them. This is a % representation of the W(e), win expectancy which I'll explain later.
E(a) - Estimated chance of winning for B team. Opposite of E(h) above.
Elo - Current Elo rating for that team.
Opp Elo - Current Elo rating for the opponents.
Checkvalue - A simple sum and compare of each matchup to catch data entry errors. Helped me a couple of times.
W/L - If the game is a win, this is 1, if a loss, it's -1. Nothing complicated.
D(e) - Difference in Elo between the two teams. In this case, we can see that (in the first row of Fig 1) Hawks have an Elo 16.51 higher than Yeti going into the game.
D(h) - Difference in Elo after Home Field Advantage is taken into account. As it happens, based on Some analysis I did the
Home Field Advantage is strong and present in the NSFL. It works out about 60% of the time,
the home team wins. Based on that the Elo difference is modified by 65 depending on if that
team is home or away. In Fig 1 you can see that due to Yeti being at home, the modified Elo difference is now favours Yeti by 48.49
65 was chosen because, all other things equal, two 1500 Elo teams seperated only by home/away of 65 Elo will result in the home team winning roughly 60% of the time, in line with what we'd expect based on our Home Field Advantage findings.
W(e) - The statistical win probability of that team. Calculated by using the following approximation fomula: W(e) = 1/(10^(D(h)/400)+1)
The next two columns, M(p) and M(e) are factors for the third column M. This is now into calculating the updated Elo based on the results of the game. M is a multiplier that is dependant on the points difference between the two teams. Specifically, M(p) is the natural log of the absolute difference between points(w) and points(l) plus 1. ln(|Points(w) - Points(l)|+1). A tied game would be ln(1) = 0 so no multiplier. Winning by a touchdown would be ln(7+1) = 2.08. This system gives decreasing returns for win point difference. ln(15) = 2.71, ln(22) = 3.09, ln(29) = 3.37 and so on.
M(e), we start with a multiplier of 2.2 and adjust it based on the difference between the two teams Elo values (before home and away modifier is applied). This is shown as 2.2/(2.2+(D(e)/1000)). This causes the multiplier to start at 1 and decrease as the elo ratings get further apart.
M is simply M(e)*M(p), the product of the previous two factors.
K(n) is M times our update value. The winning team recieves a boost to their Elo of a proportion of K(n) while the other team loses the remaining K(n)
New Elo - The old Elo plus 1-W(e)*K(n) or -W(e)* K(n) depending on win or loss.
Diff - The Elo difference after that game compared to before it.
PPS - Predicted points spread. In the NFL, a difference of 26 Elo points is worth about 1 point in the spread. I've used the same value here. Thus, this value is simply the Elo difference (including HFA) divided by 25 and rounded to the nearest 0.5. It's a rough approximation.
APS - The actual point spread after the game is concluded.
D(ps) - The difference between the predicted points spread and actual result.
P(w) - Was that team predicted to win? Based on if E(h) is higher than 50%
W - Did that team win?
P(ac) - Was our prediction accurate?
In terms of prediction accuracy, I've fed in all matches from S18 to S21 and got an average accuracy of 68%, including 79% accuracy in the playoffs.
So now you know how it all works. Lets look at some pretty graphs.
Results
First off, I have included all seasons from S21 back to S18. The more data included, the more accurate this sheet will become. All teams started with an Elo of 1500 at the start of S18 and everything else is calculated from that.
After each season is completed. I regress all teams by 33% towards the 1500 value again. That is to say, teams above 1500 move down by 33% of the difference and teams below 1500 move up by 33% of the difference. This is to account for teams drafting new players and losing vets.
![[Image: kHPu6wj.png]](https://i.imgur.com/kHPu6wj.png)
Fig 2. S18 to S21 Elo Rankings
The large bumps in Fig 2 are mostly the 33% normisation after each seaon. The better/worse a team is at the end of the season, the more impactful this is. We can see quite clearly the relative domination of both Wraiths and Otters over the last four seasons along with the rise of Second Line in S21. You can equally see the fall of the Outlaws over the last few seasons from a high of 1631 during S18 down to 1422 going into S22.
![[Image: kvK80Je.png]](https://i.imgur.com/kvK80Je.png)
Fig 3. S21 Elo Rankings
Fig 3 shows a close up of S21 specifically, from initial week 0 rankings through to the end of the Ultimus.
Going into S22 (i.e. after the end-of-season regression) we have the following initial Elo values and rankings:
Disclaimer: I am no stats expert. There may be mistakes, errors, data entry or otherwise. I spotted a number during my sanity checking and this seems to give an output that matches reality somewhat closely.
First off, what is Elo? Elo is a rating and ranking system designed for head-to-head matchups. It is designed to take opinion out of the rating process and only measure based on actual results.
It is designed to form a ranking system less influenced by human biases. It is, of course, not free from bias as we have to decide what values form the rating itself.
Elo cannot account for wild changes, if top players in one team all decide to retire between weeks, for example. Unless manual intervention is used.
A raw Elo is a zero-sum system where the winners recieve a boost to their ranking equal to the drop recieved by the losers. However, the system can be modified so underdogs get more credit for a win against a stronger opponent and, likewise, a strong opponent gets a smaller gain for beating a low ranked team.
Every team starts with an Elo rating of 1500. There's no mathematical reason it can't be 1000 or 0 or 540055. But 1500 is a nice number that scales well and it seems to be the norm.
The Basics
First, see the image in Fig 1 below. It shows the stats involved for one particular week of one particular season. In this case, S21 week 7. I will walk you through the various elements of the calculations referencing this image.
Fig 1. S21 Week 7
Update Factor - This is the base value for wins and loses. Before any other factors are included, a team that wins gains 20 points and the other team loses 20 points. Simple. We can
scale this factor for more important games if we want to, for example, the ultimus.
Next, we have the matches themselves. Each team is given their own line, this is purely for simplicities sake. I made the sheet so I simply fill in the opponent of each team, home/away
and how many points that team scored, everything else auto-fills from that.
E(h) - Estimated chance of winning for A team. Originally I had planned on doing 4 rows with Home and Away rather than 8 rows but it worked out simpler this way. E(h) is a
throwback to E(home) and E(away) and I just haven't bothered changing them. This is a % representation of the W(e), win expectancy which I'll explain later.
E(a) - Estimated chance of winning for B team. Opposite of E(h) above.
Elo - Current Elo rating for that team.
Opp Elo - Current Elo rating for the opponents.
Checkvalue - A simple sum and compare of each matchup to catch data entry errors. Helped me a couple of times.
W/L - If the game is a win, this is 1, if a loss, it's -1. Nothing complicated.
D(e) - Difference in Elo between the two teams. In this case, we can see that (in the first row of Fig 1) Hawks have an Elo 16.51 higher than Yeti going into the game.
D(h) - Difference in Elo after Home Field Advantage is taken into account. As it happens, based on Some analysis I did the
Home Field Advantage is strong and present in the NSFL. It works out about 60% of the time,
the home team wins. Based on that the Elo difference is modified by 65 depending on if that
team is home or away. In Fig 1 you can see that due to Yeti being at home, the modified Elo difference is now favours Yeti by 48.49
65 was chosen because, all other things equal, two 1500 Elo teams seperated only by home/away of 65 Elo will result in the home team winning roughly 60% of the time, in line with what we'd expect based on our Home Field Advantage findings.
W(e) - The statistical win probability of that team. Calculated by using the following approximation fomula: W(e) = 1/(10^(D(h)/400)+1)
The next two columns, M(p) and M(e) are factors for the third column M. This is now into calculating the updated Elo based on the results of the game. M is a multiplier that is dependant on the points difference between the two teams. Specifically, M(p) is the natural log of the absolute difference between points(w) and points(l) plus 1. ln(|Points(w) - Points(l)|+1). A tied game would be ln(1) = 0 so no multiplier. Winning by a touchdown would be ln(7+1) = 2.08. This system gives decreasing returns for win point difference. ln(15) = 2.71, ln(22) = 3.09, ln(29) = 3.37 and so on.
M(e), we start with a multiplier of 2.2 and adjust it based on the difference between the two teams Elo values (before home and away modifier is applied). This is shown as 2.2/(2.2+(D(e)/1000)). This causes the multiplier to start at 1 and decrease as the elo ratings get further apart.
M is simply M(e)*M(p), the product of the previous two factors.
K(n) is M times our update value. The winning team recieves a boost to their Elo of a proportion of K(n) while the other team loses the remaining K(n)
New Elo - The old Elo plus 1-W(e)*K(n) or -W(e)* K(n) depending on win or loss.
Diff - The Elo difference after that game compared to before it.
PPS - Predicted points spread. In the NFL, a difference of 26 Elo points is worth about 1 point in the spread. I've used the same value here. Thus, this value is simply the Elo difference (including HFA) divided by 25 and rounded to the nearest 0.5. It's a rough approximation.
APS - The actual point spread after the game is concluded.
D(ps) - The difference between the predicted points spread and actual result.
P(w) - Was that team predicted to win? Based on if E(h) is higher than 50%
W - Did that team win?
P(ac) - Was our prediction accurate?
In terms of prediction accuracy, I've fed in all matches from S18 to S21 and got an average accuracy of 68%, including 79% accuracy in the playoffs.
So now you know how it all works. Lets look at some pretty graphs.
Results
First off, I have included all seasons from S21 back to S18. The more data included, the more accurate this sheet will become. All teams started with an Elo of 1500 at the start of S18 and everything else is calculated from that.
After each season is completed. I regress all teams by 33% towards the 1500 value again. That is to say, teams above 1500 move down by 33% of the difference and teams below 1500 move up by 33% of the difference. This is to account for teams drafting new players and losing vets.
Fig 2. S18 to S21 Elo Rankings
The large bumps in Fig 2 are mostly the 33% normisation after each seaon. The better/worse a team is at the end of the season, the more impactful this is. We can see quite clearly the relative domination of both Wraiths and Otters over the last four seasons along with the rise of Second Line in S21. You can equally see the fall of the Outlaws over the last few seasons from a high of 1631 during S18 down to 1422 going into S22.
Fig 3. S21 Elo Rankings
Fig 3 shows a close up of S21 specifically, from initial week 0 rankings through to the end of the Ultimus.
Going into S22 (i.e. after the end-of-season regression) we have the following initial Elo values and rankings:
- 1 - Second Line - 1,623.43
- 2 - Otters - 1,576.06
- 3 - Wraiths - 1,530.21
- 4 - Hawks - 1,529.01
- 5 - Yeti - 1,516.74
- 6 - Copperheads - 1,514.69
- 7 - Hahlua - 1500
- 8 - Sailfish - 1500
- 9 - Butchers - 1,429.89
- 10 - Outlaws - 1,422.12
- 11 - SaberCats - 1,415.98
- 12 - Liberty - 1,402.34
Disclaimer: I am no stats expert. There may be mistakes, errors, data entry or otherwise. I spotted a number during my sanity checking and this seems to give an output that matches reality somewhat closely.
Hamish MacAndrew #20 - Safety - Austin Copperheads - [Player Profile - Update Thread - Wiki page - Twitter]
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