## Super Rugby - Elo Ratings |

### Introduction

This tool calculates ratings for each Super Rugby team using the Elo rating system. You can edit the calculations by changing the default parameters. Explanations of the methodologies used can be found at the bottom of the page. Learn more about the Elo rating system.

Note that the earliest record on file for the Super Rugby is March 7, 2009.

### Variables

### Super Rugby Elo Ratings

To learn about the differences between each methodology, refer to the notes below this table.

Rank | Team | Elo Rating Methodology | ||||
---|---|---|---|---|---|---|

Basic | AST^{1/2} | AST^{1/3} | Average | |||

1 | Crusaders | 1,774.57 | 1,958.42 | 1,882.27 | 1,871.75 | |

2 | Hurricanes | 1,722.63 | 1,900.77 | 1,832.69 | 1,818.70 | |

3 | Chiefs | 1,702.44 | 1,802.59 | 1,760.44 | 1,755.16 | |

4 | Highlanders | 1,657.86 | 1,739.52 | 1,702.39 | 1,699.93 | |

5 | Lions | 1,635.74 | 1,714.81 | 1,680.22 | 1,676.92 | |

6 | Stormers | 1,593.20 | 1,630.49 | 1,616.61 | 1,613.43 | |

7 | Sharks | 1,560.08 | 1,575.66 | 1,568.57 | 1,568.10 | |

8 | Brumbies | 1,553.17 | 1,569.92 | 1,562.37 | 1,561.82 | |

9 | Waratahs | 1,512.53 | 1,516.53 | 1,506.99 | 1,512.02 | |

10 | Blues | 1,500.06 | 1,501.45 | 1,502.25 | 1,501.25 | |

11 | Bulls | 1,502.40 | 1,479.97 | 1,488.88 | 1,490.42 | |

12 | Jaguares | 1,351.14 | 1,348.25 | 1,347.66 | 1,349.02 | |

13 | Reds | 1,396.13 | 1,283.53 | 1,333.85 | 1,337.83 | |

14 | Cheetahs | 1,374.90 | 1,273.25 | 1,317.95 | 1,322.03 | |

15 | Force | 1,358.74 | 1,253.43 | 1,299.25 | 1,303.81 | |

16 | Rebels | 1,315.27 | 1,157.96 | 1,222.69 | 1,231.98 | |

17 | Kings | 1,261.96 | 1,165.40 | 1,205.15 | 1,210.83 | |

18 | Sunwolves | 1,180.45 | 1,038.29 | 1,098.78 | 1,105.84 |

## Notes:

The league average Elo rating is 1500. Each team starts with a rating of 1500 and then has their rating adjusted based on match results.

Ratings for teams with fewer than 30 matches on file should be considered provisional. The general consensus is that the ratings converge on a team's true strength after approximately 30 games.^{1}

The calculated ratings will differ from the websites from which the above models were sourced from. Some of the reasons for this are:

- This tool primarily applies just the margin of victory multiplier approach used by each reference model.
- The default parameters may vary from those used on the reference websites.
- This tool may apply a different approach to handling new franchises and promoted teams.
- The calculated ratings are dependent on the starting point used in the historical data.

### General Formulas

The basic Elo ratings change based on the formula **R _{n} = R_{o} + K(W - W_{e})**, where

R

_{n}= the team's new rating after the game

R

_{o}= the team's rating prior to the game

K = the constant weighting applied to each game (determines sensitivity of ratings to new matches)

W = the result of the match. It equals 1 for a win, 0.5 for a draw, and 0 for a loss (with the exception of the FIFA Women's ranking system)

^{2}

W

_{e}= the expected match result, which is calculated using the formula

**W**, where R

_{e}= 1/(1 + 10 ^ ((R_{opp}-R_{o}) / 400))_{o}is the team's rating before the game and R

_{opp}is the opponent's rating before the game.

With the exception of the FIFA Women's ranking system^{2}, the other Elo ratings systems introduce a margin of victory multiplier, called G, so the formula becomes **R _{n} = R_{o} + KG(W - W_{e})**. Each model has its own approach to calculating G. The margin of victory multiplier is discussed further down the page.

### Adjustable Variables

**k** determines how sensitive the Elo ratings are to new results. If **k** is set too high, the ratings will jump around too much. Conversely if it is set too low, the ratings will take too long to react to changes in team performances. One NBA Elo rating system determined that for the NBA the optimal **k** is 20.^{3} The same website also applied a **k** of 20 for their NFL Elo rating system.^{5} A popular Elo rating website for football also uses a value of 20 for **k**.^{6}

The **Previous Season Weighting** is a discounted factor for previous season data. It can take a value between 0 and 1. A higher value places greater importance on previous season data. When a new season begins the team's new rating becomes:

New Elo Rating = (Previous Season Weighting)*(End of last season's rating) + (1 - Previous Season Weighting)*(Default rating)

FiveThirtyEight use value of 0.75 for the NBA and 0.6667 for the NFL because they found NBA teams to be more consistent each year than NFL teams. Basically, the more applicable you believe previous season data to be, the higher the value you should choose. If you want to see how the ratings would come out if you ignored all previous season data, set this value equal to 0.

The **Home Advantage** variable adds points to the home team's Elo rating before calculating the expected match result, W_{e}. This is to account for the fact that in most sports playing at home is an advantage. Both the FIFA Women's World Rankings formula^{2} and a popular NBA Elo rating system^{3} use a value of 100. Other values used are 65 for an NFL Elo ratings system^{5} and 90 for a popular football ratings system.^{6} If you don't wish to adjust the model for home advantage set this value to 0.

The **New / Promoted Team Elo Rating** variable enables you to assign a lower initial rating for new or promoted teams to account for the fact that they are perceived to be at a disadvantage compared to the rest of the league. Once the new teams are assigned their initial values, the Elo ratings of the incumbent teams are adjusted upward to reset the league average to 1500. This approach is taken by popular NBA^{3} and NFL^{5} Elo rating models. It is also the approach taken by the NZ Herald for Super Rugby. If you don't wish to apply any adjustments for new teams, enter 1500 for this value.

### Margin of Victory Multiplier (G)

The **margin of victory multiplier** enables the Elo rating model to account for large versus small scale victories. Most models that use this variable incorporate diminishing returns, so the difference in the value of G between a 5-point win and a 10-point win is larger than the difference between a 25-point and a 30-point win. Some models will also adjust G based on whether the favourite or the underdog won the game.

Below are brief outlines of the approaches used by each model.

The EloRatings.net system is used for football only. It calculates G using the formula:

- G = 1 if the game is a draw or if it is won by 1 goal
- G = 3/2 if the game is won by 2 goals
- G = (11+N)/8 if the game is won by 3 or more goals, where N = the goal difference

The ClubElo system is also used for football only. It calculates G using the formula:

- G = 1 if the game is a draw
- G = the square root of N if the game is won by 1 or more goals, where N = the goal difference

The FIFA Women's ranking system doesn't use the variable G. Instead it adjusts the result of the match based on a table of values which accounts for both the winning margin and the number of goals scored by the losing team. The figures below are percentages.

Goals scored by non winning team |
Goal Difference | ||||||
---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 /+ | |

0 | 47 | 15 | 8 | 4 | 3 | 2 | 1 |

1 | 50 | 16 | 8.9 | 4.8 | 3.7 | 2.6 | 1.5 |

2 | 51 | 17 | 9.8 | 5.6 | 4.4 | 3.2 | 2 |

3 | 52 | 18 | 10.7 | 6.4 | 5.1 | 3.8 | 2.5 |

4 | 52.5 | 19 | 11.6 | 7.2 | 5.8 | 4.4 | 3 |

5 | 53 | 20 | 12.5 | 8 | 6.5 | 5 | 3.5 |

For every other model the changes to each team's Elo ratings are symmetrical, so the league average always equals 1500. This isn't the case for the FIFA Women's ranking system because draws with score lines other than 1-1 result in asymmetric changes to the Elo ratings (the **W** variables for the two teams don't sum to 1).

The **AST ^{1/2}** system is an experimental model developed by this website that calculates G using the formula:

G = (1 + N

_{game}/N

_{median})

^{1/2}where N

_{game}is the winning margin for the game and N

_{median}is the median winning margin in the league

The **AST ^{1/3}** system is an experimental model developed by this website that calculates G using the formula:

G = (1 + N

_{game}/N

_{median})

^{1/3}where N

_{game}is the winning margin for the game and N

_{median}is the median winning margin in the league

For basketball this tool applies an approach based on the NBA model developed by FiveThirtyEight:

G = ((winning team's margin of victory + 3)^{0.8}) / (7.5 + 0.006*(winner Elo - loser Elo __+__ home field advantage adjustment)

The home field advantage adjustment is positive if the home team won and negative if the away team won. This formula accounts for the fact that favourites tend to win games by larger margins than when they lose.

For the NFL, this tool applies an approach based on the NFL model developed by FiveThirtyEight:

G = log(1 + winning team's margin of victory)*2.2/((winner Elo - loser Elo)*0.001 + 2.2)

(their website states they use the natural logarithm but we weren't able to replicate their numbers until we switched to log base 10)
Like the NBA model, this formula accounts for the fact that favourites tend to win games by larger margins than when they lose. It isn't specifically discussed on their site, but we adjust (winner Elo - loser Elo) to account for home field advantage in the same way that their NBA model does.

A Survey of football ratings system found that in international football, "the best accuracy is achieved by two Elo models: the EloRatings.net system is the most accurate with respect to binomial deviance, and the Elo model applied by FIFA in ranking women's teams, when we look at the mean squared error."

### League Expansions and Promotion

Some leagues like Super Rugby have experienced expansions. In other leagues such as the NBL, one team is disbanded and a new team is created in its place. For leagues like the EPL, the weakest teams are relegated at the end of each season with new teams promoted from the division below to replace them.

New franchises are typically at an initial disadvantage relative to existing franchises, which is why a **New / Promoted Team Default Rating** (discussed above) can be chosen.

To maintain an average league Elo rating of 1500, at the start of each season the returning teams have their ratings adjusted using the following formula:

Adjusted Elo rating = old_elo_rating * (1500 * (number_of_returning_teams + number_of_new_teams) - number_of_new_teams * new_team_default_rating) / (sum_of_returning_teams_elo_ratings)

If a team is relegated and then promoted in a later season, it receives the New Promoted Team Default Rating when it is reinstated rather than its old Elo rating.

### Converting Differences in Elo Ratings into Lines / Handicaps / Point Spreads

It will vary depending on the model, but FiveThirtyEight takes the following approach for the NBA:

Take the difference of the two teams' Elo ratings, add 100 points for the home team and then divide by 28. This gives you a projected margin of victory for the game. For example, if the home team's Elo rating is 92 points higher than the away team, then the expected handicap is (92 + 100)/28 = 6.86 ~ 7 points.

For a lower scoring sport such as the NFL, FiveThirtyEight divides the difference in ratings by 25 to estimate a handicap.

### Limitations of Elo Ratings for Sports Betting

Bookmaker handicaps account for a much wider scope of information than Elo ratings do. FiveThirtyEight back tested their NFL Elo ratings and found that it got 51% of line picks correct, which is insufficient to secure a profit unless variances in bookmaker odds and lines can be found.

One of the key limitations of the Elo rating system is the constant value for k and the inability to insert arbitrary breaks into the data to account for significant personnel changes (i.e. LeBron James joining the Cleveland Cavaliers, or Sergio Aguero getting injured for Man City).

### References:

^{1}https://en.wikipedia.org/wiki/World_Football_Elo_Ratings^{2}https://en.wikipedia.org/wiki/FIFA_Women%27s_World_Rankings^{3}http://fivethirtyeight.com/features/how-we-calculate-nba-elo-ratings/^{4}http://m.nzherald.co.nz/sport/news/article.cfm?c_id=4&objectid=11678065^{5}http://fivethirtyeight.com/datalab/introducing-nfl-elo-ratings/^{6}http://old.clubelo.com/Articles/AdjustmentforGoalDifference.html^{7}http://eloratings.net/system.html