# Using Markov chains to evaluate football player’s contributions

Football players can’t be ranked on the basis of simple stats such as number of passes made, tackles or pass success rate. A player who tackles a lot might do so because he is badly positioned in the first place. And a player with a high pass success rate might always choose the easy option. This is why football will never be Moneyball. We need a more nuanced view of player performance.

One solution is to look at how the actions a player takes increases and decreases the chance of his team scoring. I have recently been working on a performance metric based on this idea, using a mathematical technique called Markov Chains. The name is mathematical, but the idea is simple. Each position on the pitch is assigned a value, and that value (roughly) corresponds to the probability that a team with possession of the ball at that point scores a goal. We can then assess all player actions based on how much they increase their team’s chance of scoring and decrease the opponent’s chance of scoring.

Together with Twelve, we have implemented a system where we can use everything that happens on the ball to rank players. We take each pass, tackle, shot and dribble and assign it points. The number of points depends on how an action contributes to a team’s goal scoring chances or reduces the opposition’s chance of scoring. So although scoring a goal is the best action and worth 1000 points, all useful actions performed by team members are rewarded with between 0 and 1000 points. Players that make mistakes are punished with negative points.

I’ll go through the points system step by step, using Liverpool’s performance in the last Merseyside derby to illustrate my approach.

# Goals and shots

Scoring a goal is worth 1000 points. Below is Sadio Mane’s goal for Liverpool against Everton.

In our evaluation system all goals are worth the same, irrespective of where they were scored from or how spectacular they were. All goals are equal and are the best single action in a football match.

For shots, the number of points depends on the shot location. Shots from wide angles or from further out are worth fewer points than those in front of the goalmouth. This is because players have less chance of scoring from further out. Trent Alexander-Arnold, had two shots for Liverpool during the match against Everton, shown below.

These were both good chances and gave him a shot total of 325 points, equivalent to one third of a goal.

Missing the goal from long distance is punished by negative points (see errors below). But a shot from a good position that misses still gives a player a positive point total. This may seem counter-intuitive, because a player will be rewarded for missing a chance directly in front of goal. However, research has shown that the skill of a striker is best measured both by the number of good quality chances he generates and whether or not he scores them. Seeing goals in this way is equivalent to using an Expected Goals model (which I have looked at in earlier articles). Lots of shots in front of goal gives a player lots of points.

# Attack

A successful pass that takes the ball in to an area of the pitch that provides a better chance of scoring is worth more points. Sideways passes are worth fewer points and defensive passes don’t earn any points. Against Everton, the Liverpool player who had the greatest impact was Philippe Coutinho. Here are the passes that earned him points.

The darker circles with the arrows show his five biggest contributions, both in for take-ons and passes. These passes moved the ball near or in to the box, or created a chance for another player. In total, Coutinho received 1809 points. He scored in the match, but his contribution to the team went well beyond his goal.

Emre Can was Liverpool’s second most effective player in terms of attacking play. Here are the places on the pitch where he earned his 959 attacking points.

# Defence

According to our point system, Dejan Lovren was the Liverpool player with the best defensive performance against Everton. The heatmap below shows where he intercepted the ball won header duels and made other defensive actions.

The cross marks the area where Lovren’s contribution was highest. In total Lovren received 861 defence points.

# Press

Although successful tackles and interceptions are awarded with points to the players who make them, often it isn’t a defender who makes a tackle who deserves the greatest credit. We award points to players whenever the opposition loses the ball. We identify the three players who are typically nearest to the point where the ball is lost and award them each the same number of points. When the ball is regained from more dangerous positions on the pitch more points are awarded to the defenders. I will look in more detail at how we measure press in a future article. In the match against Everton, the Liverpool player with the best press was Lucas Leiva, who covered a lot of ground during the match. In total he received 918 press points.

# Errors

Based on attack, defence and press, Lucas would have been one of Liverpool’s best players. But he also picked up a lot of negative points for errors. Failed take-ons, a goalkeeper letting in a goal, missed interceptions, being caught offside are all punished with negative points. Lucas was punished with -410 points for losing a heading duel only 6m from his goal. Lucas ended up with -1223 negative points.

# Player rankings

We combine attack, defence, press, errors, shots and goals to rank players. Goalkeepers, defenders, midfielders and forwards all contribute to the team. The idea is to set up a system where all players are evaluated on the same scale, independent of their playing position.

Here are Liverpool’s ranking for that match. Philippe Coutinho is man of the match for Liverpool with the highest points total per minute played.

Origi was in second place, in points per minute, but only played half of the game. This is why his goal bar is twice as long as Coutinho’s.

In the next few articles I will be looking more closely at how our evaluation system can be used to find the season’s best players and to evaluate new talent.

If you want to see regularly updated top-lists and player profiles follow Twelve on Twitter.

This article first appeared on Nordic Bet’s blog.

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Professor of Applied Mathematics. Books: The Ten Equations (2020); Outnumbered (2018); Soccermatics (2016) and Collective Animal Behavior (2010).

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Professor of Applied Mathematics. Books: The Ten Equations (2020); Outnumbered (2018); Soccermatics (2016) and Collective Animal Behavior (2010).