The world’s most popular sport has more than 250 million players in over 200 countries. Every week it fills stadiums with many millions more. This sport, soccer, is played by two teams of 11 players who cooperate to score goals while preventing their opponents from doing so. For each team, a game can have one of three outcomes — win, draw or lose. 

This kind of game is usually part of a bigger business to determine the best team. The most common process is a league structure in which teams play each other and are then ranked by results. The team at the top of the rankings at the end of the season is the winner. 

This kind of ranking seems pretty straightforward but it actually hides some subtle but important questions. Chief among these is the question of the value of a win. In most professional leagues, the answer is that a win is 3 times as valuable as a draw while a loss is worth nothing, written as (3-1-0).  

 It hasn’t always been this way. Before 1981, a professional football win was worth just two points in a (2-1-0) system. So a win was twice as valuable as a draw. However, the sport’ governing body increased the value of a win to increase the motivation to win. 

Nevertheless, these numbers are essentially arbitrary, decided by committee in the (metaphorically) smoke-filled boardrooms of the past. But if winning is the objective, why isn’t it 5 times more valuable than a draw, or 10 times, or 100?  

Play To Win

And that raises an interesting question. Is there an objective way of determining the value of a win and if so, what is its value? 

 Today we get an answer thanks to the work of Leszek Szczecinski at the Institut National de la Recherche Scientifique in Montreal, Canada. Szczecinski has developed a probabilistic model of game results and the rankings they produce.  

In this model, there is a free parameter that corresponds to the value of a win relative to a draw. His idea is to use results from the real world of professional football to determine its value. And when he does that, the answer, he says, is 5. 

Szczecinski’s model is relatively simple. It is based on the idea that the teams have intrinsic capabilities that determine whether they beat other teams. However, this intrinsic capability is hidden in the real world and can only be determined by repeatedly comparing the teams. In other words, by playing games to see who wins.   

Furthermore, games have an element of luck — sometimes, the lesser team can win due. Nevertheless, the league process should eventually rank the teams according to their hidden intrinsic ability.  

So in his model, Szczecinski allocates each team a probability of winning that is related to its hidden intrinsic ability. The teams then “play” each other with these probabilities determining whether the game ends in a win, loss or a draw. 

 The question that Szczecinski examines is what scoring system ensures that the league ranking best reflects the hidden intrinsic capabilities that the process is designed to reveal.  

Of course, there is no way of knowing the intrinsic ranking in real life so Szczecinski uses data from professional football leagues since 1981 in England, Germany and Spain to help calibrate his model.   

It turns out that the ideal scoring scheme varies across these leagues. For example, in the English premier league, arguably the toughest league in the world, a win is 3.9 times more valuable than a draw. In the top Spanish league, La Liga, which is largely dominated by just two teams, Real Madrid and Barcelona, a win is 7.5 times more valuable than a draw.  

Sore Point

 “The results indicate that the nominal scoring rules in football do not match the empirical data,” says Szczecinski. Instead, the data suggests the value of a win is close to 5 points. 

Of course, Szczecinski recognizes that changing the value of a win itself influences the nature of the game. He points out that exactly this happened when the football authorities changed the value of a win from 2 to 3 points in 1981 in England and in 1994 in Germany. At that time, the data suggests that a draw was worth nothing and the scoring system should have been (1-0-0). 

 “It seems that the governing body, by introducing the scoring rule (3-1-0), managed to change the pattern of the results: the (conditional) probability of wins is now smaller than the probability of draws,” he says.  

That suggests further changes should be made with caution. Szczecinski says this should involve an iterative process—make a change and use the results over time to calculate a new value for a win and so on. “For example, the rule may be first set to (4-1-0),” he suggests 

The message is that if the aim of these leagues is to determine the best team, then the scoring system needs to be based on continuous assessment of the data, rather than on arbitrary numbers.  An interesting idea, for sure. Five points for a win, anyone? 

Ref: Why winning a soccer game is worth 5 points : arxiv.org/abs/2303.15305