View source for World Football Elo Ratings

{{Short description|Ranking system for men's national association football teams}}
{{Use dmy dates|date=November 2020}}
{{Sports rankings table|World Football Elo Ratings|1|20<!--RANKINGS ARE UPDATED AT: Module:SportsRankings/data/World Football Elo Ratings-->
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|header2='''Top 20 rankings as of INSERT_UPDATE_DATE'''INSERT_REFERENCE
|footer1=<small>*Change from one year ago</small>
|footer2=<small>[https://www.eloratings.net/ Complete rankings at eloratings.net]</small>
}}
The '''World Football Elo Ratings''' are a ranking system for men's national [[association football]] teams that is published by the website eloratings.net. It is based on the [[Elo rating system]] but includes modifications to take various football-specific variables into account, like the margin of victory, importance of a match, and home field advantage. Other implementations of the Elo rating system are possible and there is no single nor any official Elo ranking for football teams.

Since being developed, the Elo rankings have been found to have the highest predictive capability for football matches.<ref name="Laesk et. al."/> FIFA's official rankings, both the [[FIFA World Rankings]] for men and the [[FIFA Women's World Rankings]] are based on a modified version of the Elo formula, the men's rankings having switched away from FIFA's own system for matches played since June 2018.<ref>{{cite web |url=https://www.fifa.com/about-fifa/news/y=2018/m=6/news=2026-fifa-world-cuptm-fifa-council-designates-bids-for-final-voting-by-the-fifa-.html|archive-url=https://web.archive.org/web/20180612143611/https://www.fifa.com/about-fifa/news/y=2018/m=6/news=2026-fifa-world-cuptm-fifa-council-designates-bids-for-final-voting-by-the-fifa-.html|url-status=dead|archive-date=12 June 2018|title=2026 FIFA World Cup: FIFA Council designates bids for final voting by the FIFA Congress|work=fifa.com|publisher=FIFA|date=10 June 2018|access-date=2 July 2018}}</ref>

== History and overview ==
The Elo system, developed by [[Hungary|Hungarian]]-American mathematician [[Arpad Elo|Árpád Élő]], is used by [[FIDE]], the international chess federation, to rate [[chess]] players, and by the [[European Go Federation]], to rate [[Go (game)|Go]] players. In 1997, Bob Runyan adapted the Elo rating system to international [[Association football|football]] and posted the results on the Internet.<ref>{{cite web |last1=Lyons |first1=Keith |title=What are the World Football Elo Ratings? |url=https://theconversation.com/what-are-the-world-football-elo-ratings-27851 |publisher=The Conversation |access-date=16 July 2018}}</ref> He was also the first maintainer of the World Football Elo Ratings web site, currently maintained by Kirill Bulygin. Other implementations of the Elo rating system are possible.<ref name="Laesk et. al.">J. Lasek, Z. Szlávik and S. Bhulai (2013), [http://lasek.rexamine.com/football_rankings.pdf The predictive power of ranking systems in association football], {{webarchive |url=https://web.archive.org/web/20140726212402/http://lasek.rexamine.com/football_rankings.pdf |date=26 July 2014 }}, Int. J. Applied Pattern Recognition1: 27–46.</ref>

The Elo system was adapted for football by adding a weighting for the kind of match, an adjustment for the home team advantage, and an adjustment for goal difference in the match result.

The ratings consider all official international matches for which results are available. Ratings tend to converge on a team's true strength relative to its competitors after about 30 matches.<ref>{{cite web|url=http://www.eloratings.net/about |title=The World Football Elo Rating System |publisher=Eloratings.net |access-date=26 February 2012}}</ref> Ratings for teams with fewer than 30 matches are considered provisional.

===Comparison with other systems===
A 2009 comparative study of eight methods found that the implementation of the Elo rating system described below had the highest predictive capability for football matches, while the men's FIFA ranking method (2006–2018 system) performed poorly.<ref name="Laesk et. al." />

The [[FIFA World Rankings]] is the official national teams rating system used by the [[FIFA|international governing body of football]]. The [[FIFA Women's World Rankings|FIFA Women's World Rankings system]] has used a modified version of the Elo formula since 2003. In June 2018, the FIFA ranking switched to an Elo-based ranking as well, starting from the current FIFA rating points.<ref>FIFA Council, [https://web.archive.org/web/20180612143611/https://www.fifa.com/about-fifa/news/y=2018/m=6/news=2026-fifa-world-cuptm-fifa-council-designates-bids-for-final-voting-by-the-fifa-.html 2026 FIFA World Cup: FIFA Council designates bids for final voting by the FIFA Congress], 10 June 2018</ref> The major difference between the World Football Elo Rating and the new men's FIFA rating system is that the latter does not consider goal differential and counts a [[penalty shoot-out (association football)|penalty shoot-out]] as a win/loss rather than a draw (neither method distinguishes a win in [[Overtime (sports)#Association football|extra time]] from a win in regular time).<ref name=fifarev>FIFA council, [https://web.archive.org/web/20180701075927/https://resources.fifa.com/image/upload/revision-of-the-fifa-coca-cola-world-ranking.pdf?cloudid=fzltr4s8tz3v3vy0aqo1 Revision of the FIFA / Coca-Cola World Ranking]</ref>

==Calculation principles==
<!--This text is a tad unclear and as it is doesn't add to the understanding
The basic principle behind the Elo ratings is only in its simplest form similar to that of a league; unlike the FIFA rankings, who effectively run their table as a normal league table but with weightings to take into account the other factors, the Elo system has its one formula which takes into account the factors mentioned above. There is no first step as in the FIFA system where a team immediately receives points for the result, there is just one calculation in the Elo system.
 -->
The ratings are based on the following formula:

:'''<math>R_n = R_o + P</math>'''
where
:'''<math>P = K G (W - W_e)</math>'''

Where;
{| style="width:50%;"
|-
|width=4%|
|-
|'''<math>R_n</math>'''||= The new team rating<br>
|-
|'''<math>R_o</math>'''||= The old team rating<br>
|-
|'''<math>K</math>'''||= Weight index regarding the tournament of the match<br>
|-
|'''<math>G</math>'''||= A number from the index of goal differences<br>
|-
|'''<math>W</math>'''||= The result of the match<br>
|-
|'''<math>W_e</math>'''||= The expected result<br>
|-
|'''<math>P</math>'''||= Points Change<br>
|}

"Points Change" is rounded to the nearest integer before updating the team rating.

===Status of match===
The status of the match is incorporated by the use of a weight constant. The constant reflects the importance of a match, which, in turn, is determined entirely by which tournament the match is in; the weight constant for each major tournament is:
{| class="wikitable" style="font-size: 90%;"
|-
!Tournament or Match type!!K
|-
|[[FIFA World Cup|World Cup]], [[Football at the Olympic Games|Olympic Games]] (1908–1980) ||60
|-
|[[Football Continental Championships|Continental championship]] and intercontinental tournaments||50
|-
|World Cup and Continental qualifiers and major tournaments||40
|-
|All other tournaments||30
|-
|[[Friendly match]]es||20
|}
The FIFA adaptation of the Elo rating features 8 weights, with the knockout stages in the World Cup weighing 12 times more than some friendly matches.<ref name=fifarev/>

===Number of goals===
The number of goals is taken into account by use of a goal difference index.

If the game is a draw or is won by one goal
:<math>G = 1</math>

If the game is won by two goals
:<math>G = \frac{3}{2}</math>

If the game is won by three or more goals:
*Where '''N''' is the goal difference (<math> \forall </math> N ≥ 3)
:<math>G = \frac{11+N}{8}</math>

Table of examples:

{| class="wikitable" style="font-size: 90%;"
|-
!Goal Difference
|style="text-align: center;"|0 ||style="text-align: center;"|+1 ||style="text-align: center;"|+2 ||style="text-align: center;"|+3 ||style="text-align: center;"|+4 ||style="text-align: center;"|+5 ||style="text-align: center;"|+6 ||style="text-align: center;"|+7 ||style="text-align: center;"|+8 ||style="text-align: center;"|+9 ||style="text-align: center;"|+10
|-
!G 
|style="text-align: center;"|1||style="text-align: center;"| 1 ||style="text-align: center;"|1.5 ||style="text-align: center;"| 1.75 ||style="text-align: center;"| 1.875 ||style="text-align: center;"|2 ||style="text-align: center;"|2.125 ||style="text-align: center;"|2.25 ||style="text-align: center;"|2.375 ||style="text-align: center;"|2.5 ||style="text-align: center;"|2.625
|-
|}

===Result of match===
W is the result of the game (1 for a win, 0.5 for a draw, and 0 for a loss). This also holds when a game is won or lost in extra time. If the match is decided on penalties, however, the result of the game is considered a draw (W = 0.5).

===Expected result of match===
W<sub>e</sub> is the expected result (win expectancy with a draw counting as 0.5) from the following formula:

:<math>W_e = \frac{1}{10^{-dr/400} + 1}</math>

where ''dr'' equals the difference in ratings (add 100 points for the home team). So ''dr'' of 0 gives 0.5, of 120 gives 0.666 to the higher-ranked team and 0.334 to the lower, and of 800 gives 0.99 to the higher-ranked team and 0.01 to the lower.

The FIFA adaptation of the Elo rating does not incorporate a home team advantage and has a larger divisor in the formula (600 vs 400), making the points exchange less sensitive to the rating difference of two teams.<ref name=fifarev/>

===Examples for clarification===
{{unsourced-section|date=April 2021}}
The same example of a three-team friendly tournament on neutral territory is used as on the [[FIFA World Rankings]] page. Beforehand team A had a rating of 630 points, team B 500 points, and teams C 480 points.<br>The first table shows the points allocations based on three possible outcomes of the match between the strongest team A, and the somewhat weaker team B:

{| class="wikitable" style="font-size: 90%;"
|-
| ||'''Team A'''||'''Team B'''||'''Team A'''||'''Team B'''||'''Team A'''||'''Team B'''
|-
!Score
|colspan="2" style="text-align: center;"|3–1||colspan="2" style="text-align: center;"|1–3||colspan="2" style="text-align: center;"|2–2
|-
!'''<math>K</math>'''
||20||20||20||20||20||20
|-
!'''<math>G</math>'''
||1.5||1.5||1.5||1.5||1||1
|-
!'''<math>W</math>'''
||1||0||0||1||0.5||0.5
|-
!'''<math>W_e</math>'''
||0.679||0.321||0.679||0.321||0.679||0.321
|-
!Total (P)
||+9.63||-9.63||-20.37||+20.37||-3.58||+3.58
|}

When the difference in strength between the two teams is less, so also will be the difference in points allocation. The next table illustrates how the points would be divided following the same results as above, but with two roughly equally ranked teams, B and C, being involved:

{| class="wikitable" style="font-size: 90%;"
|-
| ||'''Team B'''||'''Team C'''||'''Team B'''||'''Team C'''||'''Team B'''||'''Team C'''
|-
!Score
|colspan="2" style="text-align: center;"|3–1||colspan="2" style="text-align: center;"|1–3||colspan="2" style="text-align: center;"|2–2
|-
!'''<math>K</math>'''
||20||20||20||20||20||20
|-
!'''<math>G</math>'''
||1.5||1.5||1.5||1.5||1||1
|-
!'''<math>W</math>'''
||1||0||0||1||0.5||0.5
|-
!'''<math>W_e</math>'''
||0.529||0.471||0.529||0.471||0.529||0.471
|-
!Total (P)
||+14.13||-14.13||-15.87||+15.87||-0.58||+0.58
|}

==See also==
*[[Statistical association football predictions]]

==References==
{{reflist}}

==External links==
*{{Official|http://www.eloratings.net/}}

{{International Football}}
{{Sport world rankings}}

[[Category:Association football rankings]]
[[Category:Tournament rating systems]]