The 2018 World Cup is upon us! If you’re tempted to do a little betting, or you’re taking part in a friendly forecast competition with friends or colleagues, read on. In this tutorial, we’ll learn how to use R and the ‘elo’ package to create Elo rankings for the 32 teams in the tournament, and how to use those rankings to predict the result of football matches.

What is Elo?

In its most basic definition, Elo is a rating system which lets us rank teams (or players in individual sports or games) relative to one another, and predict the likely outcome of a given match-up. I won’t try to explain it better than Wikipedia:

The Elo rating system is a method for calculating the relative skill levels of players in zero-sum games such as chess. It is named after its creator Arpad Elo, a Hungarian-American physics professor. The Elo system was originally invented as an improved chess rating system over the previously used Harkness system, but is also used as a rating system for multiplayer competition in a number of video games, association football, American football, basketball, Major League Baseball, Scrabble, board games such as Diplomacy and other games. The difference in the ratings between two players serves as a predictor of the outcome of a match. Two players with equal ratings who play against each other are expected to score an equal number of wins. A player whose rating is 100 points greater than their opponent’s is expected to score 64%; if the difference is 200 points, then the expected score for the stronger player is 76%. A player’s Elo rating is represented by a number which increases or decreases depending on the outcome of games between rated players. After every game, the winning player takes points from the losing one. The difference between the ratings of the winner and loser determines the total number of points gained or lost after a game. In a series of games between a high-rated player and a low-rated player, the high-rated player is expected to score more wins. If the high-rated player wins, then only a few rating points will be taken from the low-rated player. However, if the lower rated player scores an upset win, many rating points will be transferred. The lower rated player will also gain a few points from the higher rated player in the event of a draw. This means that this rating system is self-correcting. A player whose rating is too low should, in the long run, do better than the rating system predicts, and thus gain rating points until the rating reflects their true playing strength.

Interestingly, the Elo ranking is featured in a scene of the movie The Social Network, where Mark Zuckerberg and Eduardo Saverin discuss how to rank women on Facemash. (While the scene is a nice introduction to Elo, the idea that Zuckerberg would need his roommate to “give him the algorithm” – as opposed to just looking it up himself, say, on the Internet – is a bit naive. But anyway.)

Getting our data

In order to build reliable Elo rankings, we’ll need a good dataset of historical results for the 32 teams in the World Cup, going as far back as possible.

Thankfully, an awesome person named Mart Jürisoo is maintaining such a dataset on Kaggle.com. (You’ll need to create an account on Kaggle to download the dataset.)

I will be using the dplyr package a lot throughout this tutorial. If you’ve never used dplyr before, you really should! It will greatly simplify your data manipulations and analyses.

library ( dplyr ) matches <- readr :: read_csv ( 'results.csv' )

As of 1 June 2018, this dataset includes data on 38,949 international football matches from 30 November 1872 to 28 May 2018.

The data is easy enough to understand. There are only 9 variables, with self-explanatory names: date, home_team, away_team, home_score, away_score, tournament, city, country, neutral (whether the match was played on neutral ground, or at the home team’s stadium).

To illustrate how this looks, here’s an anecdote: the current record for most goals in an international match belongs to Australia vs. American Samoa in 2001, with a cruel 31-0.

matches %>% select ( date , home_team , away_team , home_score , away_score ) %>% mutate ( total_goals = home_score + away_score ) %>% arrange ( - total_goals ) %>% head ( 1 )

## date home_team away_team home_score away_score total_goals ## 1 2001-04-11 Australia American Samoa 31 0 31

Preparing our data

Our historical data is stored in matches , but we’ll need to create another data.frame separately, to store each team’s Elo rating, and update it after each match.

teams <- data.frame ( team = unique ( c ( matches $ home_team , matches $ away_team )))

To start our Elo ratings, we need to assign an initial Elo value to all the teams in our dataset. Traditionally in Elo-based systems this initial value is set to 1500.

teams <- teams %>% mutate ( elo = 1500 )

For each match, we’ll also create a variable that tells us who won. Because of how the ‘elo’ package works, this variable will take the following values:

1 if the home team won;

if the home team won; 0 if the away team won;

if the away team won; 0.5 for a draw.

matches <- matches %>% mutate ( result = if_else ( home_score > away_score , 1 , if_else ( home_score == away_score , 0.5 , 0 )))

We can also get rid of a bunch of variables we won’t need, and make sure that our historical data is ordered by date.

matches <- matches %>% select ( date , home_team , away_team , result ) %>% arrange ( date )

Here’s what our two datasets look like now:

head ( matches )

## date home_team away_team result ## 1 1872-11-30 Scotland England 0.5 ## 2 1873-03-08 England Scotland 1.0 ## 3 1874-03-07 Scotland England 1.0 ## 4 1875-03-06 England Scotland 0.5 ## 5 1876-03-04 Scotland England 1.0 ## 6 1876-03-25 Scotland Wales 1.0

head ( teams )

## team elo ## 1 Scotland 1500 ## 2 England 1500 ## 3 Wales 1500 ## 4 Northern Ireland 1500 ## 5 USA 1500 ## 6 Uruguay 1500

The ‘elo’ package

Implementing the Elo algorithm from scratch would be a bit long and complex, but as for so many others things in life, there is an R package for that!

library ( elo )

We’ll only be using one function from this package to create our rankings: elo.calc() . This function takes 4 arguments:

wins.A : whether team A won or not. This is what we’ve created and stored in our result variable, with 3 possibles values (1, 0, 0.5);

: whether team A won or not. This is what we’ve created and stored in our variable, with 3 possibles values (1, 0, 0.5); elo.A : the pre-match Elo value for team A;

: the pre-match Elo value for team A; elo.B : the pre-match Elo value for team B;

: the pre-match Elo value for team B; k : this is called the K-factor. This is basically how many Elo points are up for grabs in each match. Make this too small (e.g. 1) and each match will have almost no effect on the rankings. Make it too large (e.g. 100) and each match will completely change the rankings. The Wikipedia page on Elo ratings has an entire section on this. Based on what’s written in there, a value of 30 seems reasonable for our purposes.

Computing the ratings

Now, let’s write our program. The idea is to loop over each game in matches , get the pre-match ratings for both teams, and update them based on the result. We’ll get two new ratings, which we’ll use to update our data in teams .

for ( i in seq_len ( nrow ( matches ))) { match <- matches [ i , ] # Pre-match ratings teamA_elo <- subset ( teams , team == match $ home_team ) $ elo teamB_elo <- subset ( teams , team == match $ away_team ) $ elo # Let's update our ratings new_elo <- elo.calc ( wins.A = match $ result , elo.A = teamA_elo , elo.B = teamB_elo , k = 30 ) # The results come back as a data.frame # with team A's new rating in row 1 / column 1 # and team B's new rating in row 1 / column 2 teamA_new_elo <- new_elo [ 1 , 1 ] teamB_new_elo <- new_elo [ 1 , 2 ] # We then update the ratings for teams A and B # and leave the other teams as they were teams <- teams %>% mutate ( elo = if_else ( team == match $ home_team , teamA_new_elo , if_else ( team == match $ away_team , teamB_new_elo , elo ))) }

After a few minutes, you should get a nice teams data.frame, with the most up-to-date international Elo ratings for June 2018.

teams %>% arrange ( - elo ) %>% head

## team elo ## 1 Brazil 2032.956 ## 2 Spain 1975.339 ## 3 Germany 1958.013 ## 4 France 1937.242 ## 5 Argentina 1920.864 ## 6 England 1906.218

The 2018 World Cup

We still have to do one thing: subset our teams data.frame to only keep the 32 teams that have qualified for the 2018 World Cup.

WC_teams <- teams %>% filter ( team %in% c ( "Russia" , "Germany" , "Brazil" , "Portugal" , "Argentina" , "Belgium" , "Poland" , "France" , "Spain" , "Peru" , "Switzerland" , "England" , "Colombia" , "Mexico" , "Uruguay" , "Croatia" , "Denmark" , "Iceland" , "Costa Rica" , "Sweden" , "Tunisia" , "Egypt" , "Senegal" , "Iran" , "Serbia" , "Nigeria" , "Australia" , "Japan" , "Morocco" , "Panama" , "Korea Republic" , "Saudi Arabia" )) %>% arrange ( - elo )

Finally, here are our World Cup Elo rankings, from strongest to weakeast team:

print.data.frame ( WC_teams )

## team elo ## 1 Brazil 2032.956 ## 2 Spain 1975.339 ## 3 Germany 1958.013 ## 4 France 1937.242 ## 5 Argentina 1920.864 ## 6 England 1906.218 ## 7 Belgium 1890.040 ## 8 Portugal 1883.412 ## 9 Colombia 1865.608 ## 10 Peru 1854.196 ## 11 Mexico 1834.088 ## 12 Switzerland 1825.958 ## 13 Croatia 1816.654 ## 14 Uruguay 1814.028 ## 15 Iran 1808.954 ## 16 Poland 1795.867 ## 17 Denmark 1755.689 ## 18 Senegal 1752.168 ## 19 Korea Republic 1748.506 ## 20 Japan 1747.581 ## 21 Sweden 1739.752 ## 22 Australia 1733.484 ## 23 Russia 1732.650 ## 24 Morocco 1732.194 ## 25 Serbia 1725.170 ## 26 Costa Rica 1723.551 ## 27 Iceland 1708.682 ## 28 Nigeria 1705.434 ## 29 Egypt 1695.344 ## 30 Tunisia 1691.193 ## 31 Panama 1682.273 ## 32 Saudi Arabia 1653.511

We can also look up which teams didn’t make it to the World Cup this year, despite high Elo ratings:

teams %>% filter ( elo > 1800 , ! team %in% WC_teams $ team )

## team elo ## 1 USA 1808.151 ## 2 Netherlands 1847.749 ## 3 Italy 1846.751 ## 4 Chile 1832.832

Going further

If you want to use this data for predictions and forecast competitions, here are two things you can do:

1. Calculating probabilities for individual matches

In the ‘elo’ package, the elo.prob() function lets you calculate the probability that team A will win a match against team B, given their respective Elo ratings.

For example, in the opening match of the competition (Russia vs. Saudi Arabia), the probability of Russia winning would be 61%:

russia <- subset ( WC_teams , team == "Russia" ) $ elo saudi_arabia <- subset ( WC_teams , team == "Saudi Arabia" ) $ elo elo.prob ( russia , saudi_arabia )

## [1] 0.611961

Two more examples:

# A balanced match-up: France vs. Argentina france <- subset ( WC_teams , team == "France" ) $ elo argentina <- subset ( WC_teams , team == "Argentina" ) $ elo elo.prob ( france , argentina )

## [1] 0.523551

# A very un-balanced one: Brazil vs. Iceland brazil <- subset ( WC_teams , team == "Brazil" ) $ elo iceland <- subset ( WC_teams , team == "Iceland" ) $ elo elo.prob ( brazil , iceland )

## [1] 0.8660723

2. Simulating the entire competition

I won’t show the details of this here because the code would be much longer, but essentially you can use the probability generated by elo.prob() to simulate the outcome of each match (using the sample() function and its prob argument to choose a random winner between Russia and Saudi Arabia, but with a 61% probability of choosing Russia), and update the Elo ratings throughout the competition.

This way, you can simulate the entire competition all the way from the group stage to the final. And if you repeat this process many (thousands of) times, you will get detailed probabilities for each team to make it to the each stage of the competition. This is essentially what websites like FiveThirtyEight do for their sport predictions, with probabilities based on 100,000 simulations of the rest of the season.