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Mark Needham

Elo Rating System: Ranking Champions League teams using Clojure

As I mentioned in an earlier blog post I’ve been learning about ranking systems and one of the first ones I came across was the Elo rating system which is most famously used to rank chess players.

The Elo rating system uses the following formula to work out a player/team’s ranking after they’ve participated in a match:

R’ = R + K * (S – E)

  • R’ is the new rating
  • R is the old rating
  • K is a maximum value for increase or decrease of rating (16 or 32 for ELO)
  • S is the score for a game
  • E is the expected score for a game

I converted that formula into the following Clojure functions:

(defn ranking-after-win
  [{ ranking :ranking opponent-ranking : opponent-ranking importance :importance}]
  (+ ranking (* importance (- 1 (expected ranking opponent-ranking) ))))
(defn ranking-after-loss
  [{ ranking :ranking opponent-ranking : opponent-ranking importance :importance}]
  (+ ranking (* importance (- 0 (expected ranking opponent-ranking) ))))
(defn expected [my-ranking opponent-ranking]
  (/ 1.0
     (+ 1 (math/expt 10 (/ (- opponent-ranking my-ranking) 400)))))

which would be called like this to work out the new ranking of a team ranked 1200 that beat a team ranked 1500:

> (ranking-after-win { :ranking 1200 : opponent-ranking 1500 :importance 32 })

The way it works is that we first work out the likelihood that we should win the match by calling expected:

> (expected 1200 1500)

This tells us that we have a 15% chance of winning the match so if we do win then our ranking should be increased by a large amount as we aren’t expected to win. In this case a win gives us a points increase of ’32 * (1-0.15)’ which is ~27 points.

I kept things simple by always setting the importance/maximum value of increase or decrease to 32. The World Football Rankings took a different approach where they vary it based on the importance of a match and the margin of victory.

I decided to try out the algorithm on the 2002/2003 Champions League season. I was able to grab the data from The Rec Sport Soccer Statistics Foundation and I’ve written previously about how I scraped it using Enlive.

With a lot of help from Paul Bostrom I ended up with the following code to run a reduce over the matches while updating team rankings after each match:

(defn top-teams [number matches]
  (let [teams-with-rankings
    (apply array-map (mapcat (fn [x] [x {:points 1200}]) (extract-teams matches)))]
      (take number
        (sort-by (fn [x] (:points (val x)))
                 (seq (reduce process-match teams-with-rankings matches))))))

(defn process-match [ts match]
  (let [{:keys [home away home_score away_score]} match]
     (> home_score away_score)
     (-> ts
         (update-in  [home :points]
                     #(ranking-after-win {:ranking % : opponent-ranking (:points (get ts away)) :importance 32}))
         (update-in  [away :points]
                     #(ranking-after-loss {:ranking % : opponent-ranking (:points (get ts home)) :importance 32}))) 
     (> away_score home_score)
     (-> ts
         (update-in  [home :points]
                     #(ranking-after-loss {:ranking % : opponent-ranking (:points  (get ts away)) :importance 32}))
         (update-in  [away :points]
                     #(ranking-after-win {:ranking % : opponent-ranking (:points (get ts home)) :importance 32})))
     (= home_score away_score) ts)))

The matches parameter that we pass into top-teams looks like this:

> (take 5 all-matches)
({:home "Tampere", :away "Pyunik Erewan", :home_score 0, :away_score 4} {:home "Pyunik Erewan", :away "Tampere", :home_score 2, :away_score 0} {:home "Skonto Riga", :away "Barry Town", :home_score 5, :away_score 0} {:home "Barry Town", :away "Skonto Riga", :home_score 0, :away_score 1} {:home "Portadown", :away "Belshina Bobruisk", :home_score 0, :away_score 0})

And calling extract-teams on it gets us a set of all the teams involved:

> (extract-teams (take 5 all-matches))
#{"Portadown" "Tampere" "Pyunik Erewan" "Barry Town" "Skonto Riga"}

We then mapcat over it to get a vector containing team/default points pairs:

> (mapcat (fn [x] [x {:points 1200}]) (extract-teams (take 5 all-matches)))
("Portadown" {:points 1200} "Tampere" {:points 1200} "Pyunik Erewan" {:points 1200} "Barry Town" {:points 1200} "Skonto Riga" {:points 1200})

before calling array-map to make a hash of the result:

> (apply array-map (mapcat (fn [x] [x {:points 1200}]) (extract-teams (take 5 all-matches))))
{"Portadown" {:points 1200}, "Tampere" {:points 1200}, "Pyunik Erewan" {:points 1200}, "Barry Town" {:points 1200}, "Skonto Riga" {:points 1200}}

We then apply a reduce over all the matches and call the function process-match on each iteration to update team rankings appropriately. The final step is to sort the teams by their ranking so we can list the top teams:

> (top-teams 10 all-matches)
(["CF Barcelona" {:points 1343.900393287903}] 
 ["Manchester United" {:points 1292.4731214788262}] 
 ["FC Valencia" {:points 1277.1820905112208}] 
 ["Internazionale Milaan" {:points 1269.8028023141364}] 
 ["AC Milan" {:points 1257.4564374787687}]
 ["Juventus Turijn" {:points 1254.2498432522466}] 
 ["Real Madrid" {:points 1248.0758162475993}] 
 ["Deportivo La Coruna" {:points 1235.7792317210403}] 
 ["Borussia Dortmund" {:points 1231.1671952364256}] 
 ["Sparta Praag" {:points 1229.3249513256828}])

Interestingly the winners (Juventus) are only in 5th place and the top 2 places are occupied by teams that lost in the Quarter Final. I wrote the following functions to investigate what was going on:

(defn show-matches [team matches]
  (->> matches
       (filter #(or (= team (:home %)) (= team (:away %))))
       (map #(show-opposition team %))))

(defn show-opposition [team match]
  (if (= team (:home match))
    {:opposition (:away match) :score (str (:home_score match) "-" (:away_score match))}
    {:opposition (:home match) :score (str (:away_score match) "-" (:home_score match))}))

If we call it with Juventus we can see how they performed in their matches:

ranking-algorithms.parse> (show-matches "Juventus Turijn" all-matches)
({:opposition "Feyenoord", :score "1-1"} 
 {:opposition "Dynamo Kiev", :score "5-0"} 
 {:opposition "Newcastle United", :score "2-0"} 
 {:opposition "Newcastle United", :score "0-1"} 
 {:opposition "Feyenoord", :score "2-0"} 
 {:opposition "Dynamo Kiev", :score "2-1"} 
 {:opposition "Deportivo La Coruna", :score "2-2"} 
 {:opposition "FC Basel", :score "4-0"} 
 {:opposition "Manchester United", :score "1-2"} 
 {:opposition "Manchester United", :score "0-3"} 
 {:opposition "Deportivo La Coruna", :score "3-2"} 
 {:opposition "FC Basel", :score "1-2"} 
 {:opposition "CF Barcelona", :score "1-1"} 
 {:opposition "CF Barcelona", :score "2-1"} 
 {:opposition "Real Madrid", :score "1-2"} 
 {:opposition "Real Madrid", :score "3-1"})

Although I’m missing the final – I need to fix the parser to pick that match up and it was a draw anyway – they actually only won 8 of their matches outright. Barcelona, on the other hand, won 13 matches although 2 of those were qualifiers.

The next step is to take into account the importance of the match rather than applying an importance of 32 across the board and adding some value to winning a tie/match even if it’s on penalties or away goals.

The code is on github if you want to play around with it or have suggestions for something else I can try out.

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5 years ago

How to write this code in Java, PHP or JavaScript ?