Strategy for Underdogs: Does Fortune Favour the Brave?

NB: code for this post can be found here.

What is the best strategy to avoid relegation? You can categorize a team’s style of play in many different ways, but often we talk about defensive and offensive sides: those that prioritize scoring and those that prioritize not conceding. The best teams have the quality to be good at both, but weaker sides may have more of a decision to make. In order to survive, should the manager of a newly promoted club prioritize attack or defence? It clearly makes no sense for Newcastle and Brighton get involved in a shoot-out with Man City; they must try to shut up shop. But what about their matches against Bournemouth, Crystal Palace, Burnley and West Brom?

In this post, I’m going to build a simple model of underdog strategy, based on the superiority of the opposition and some basic statistical properties of football. I’ll demonstrate that most teams in the bottom half of the league should be adopting an offensive strategy in the majority of their games; only the very worst teams should be predominantly defensive.

Thought Experiment

Imagine a match between two teams, let’s call them Team A and Team B. The only thing we know is that Team A is a factor of X better than Team B, where X is their expected goal ratio: the average number of goals Team A scores per game divided by the average of Team B. In this experiment, X is always greater than 1, i.e. Team A is always better.

The number of goals a team scores in a match is well described by a Poisson distribution, with the average depending on how good each team is. The distribution of goals scored minus goals conceded is therefore the difference between two Poisson distributions (a Skellam distribution). Given the average goals/game of Team A and Team B, we can easily calculate the probability of either team winning, or a draw.

Now – here’s the important bit – let’s assume that Team B can choose their strategy; that is, vary the number of goals they aim to score in the match. They could decide to attack like Keegan's Newcastle, or to defend like Chelsea at the Nou Camp. However, the key assumption is that the goals ratio between the sides always remains the same. Whether it be a high scoring or low scoring game, Team A is always expected to score X more goals than Team B (I told you this was a simple model). The question I want to answer is: how offensively should Team B play to maximize the number of points they take from the match?

I’ll make something clear from the outset: Team A – the superior team – should always try to score as many goals as possible to maximize their chances of winning.

When to attack, when to defend?

Figure 1 shows the number of points Team B (the underdog) is expected to gain from their match against Team A, based on how offensively they play -- parameterized by their ‘expected’ goals (x-axis) -- and the superiority of Team A, which is determined by the expected goal ratio. I've plotted the results for four different goal ratios: 1.1 (blue line), 1.2 (green line), 1.4 (black line) and 1.6 (red line). The circles indicate the peak of each line: the point at which Team B’s ‘strategy’ maximizes their expected points.

For reference, I’ve also marked the average number of goals that the team finishing 18th (1 goal/match), 10th (1.25 goals/match) and 5th (1.5 goals/match) have scored, as measured from the last 22 EPL seasons. So for a team in the lower half of the table, aiming to score less than one goal/match can be viewed as very defensive; 1-1.5 goals/match is roughly normal, and above 1.5 would be a very offensive strategy. As it happens, all three relegated teams last season scored less than one goal/game.

Figure 1: The number of points the underdog can expect to take from a match against a superior team based on how offensively they play, as measured by their expected goals (x-axis). The quality of the opponent is determined by the expected goals ratio between the two teams, which varies from 1.1 (blue line: a marginally superior opponent) to 1.6 (red line: significantly superior). The dots mark the maximum expected points for each line.

Let’s start with the easy case: when Team A is substantially better (black and red lines). This typically corresponds to matches between teams separated by at least 10 places in the league table. The lines rise towards the left of the plot (lower expected goals), which indicates that the number of points the underdog can expect to take from the match increases the more defensively they play. In these matches, Team B should therefore park the bus and play for a draw. 

Things are more interesting when Team A is only slightly better than Team B. The blue line rises towards the right of the plot, indicating that the underdog should go for goals, attacking with the fervor of a top 5 side. They certainly shouldn’t play defensively. As I’ll demonstrate, this situation is applicable to many of the matches between teams in the lower half of the table. 

A goal ratio of 1.2 (green line) is roughly the dividing line between these two cases. Above this, the underdog should prioritize defence. Below this they should either attack, or at least adopt a strategy that is balanced between attack and defence.

Why is it that there are occasions when the weaker team should plan to attack? Playing offensively always reduces the probability of a draw, and increases the probability of one or either team winning. When the opponent is only marginally better, a high-scoring game increases the win probability of both teams (at the expense of a draw). With three points awarded for a win, this is the best strategy for the underdog. 

However, when the opponent is substantially better, a high-scoring game only benefits the stronger team. In this case, the best course of action for the underdog is to force a low scoring game and try to sneak a lucky win, or settle for a draw. One of my previous posts demonstrates how difficult it can be to score against an ultra-defensive team.

Should relegation candidates focus on defensive or offensive strategies?

It’s clear that the strategy adopted by weaker teams over a season depends on the degree to which their opponents are better than them. In Figure 2 I plot the average number of goals per game scored for each league position in the EPL, La Liga, Serie A and the Bundesliga over the last 20 years (13 for Serie A). The dots indicate the average, the 1-sided error bars indicate a 20% increase on the average: this is meant to represent a superior opponent with a goal ratio of 1.2. 

Figure 2: The average number of goals per game scored as a function of league position in the EPL, La Liga, Serie A and the Bundesliga over the last 20 years (13 for Serie A). The upward error bars indicate a 20% increase in goals/game: i.e. the expected goals of an opponent with a goal ratio of 1.2

In all four leagues, nearly half of the teams score between 1 and 1.4 goals/game (shaded region), on average. Furthermore, in the EPL and La Liga, the profile flattens substantially toward the bottom half of the table, with many teams bunched into an even narrower range: there is clearly not a huge difference in quality between the teams that finish in these positions. When they play one another the goal ratio (stronger/weaker team) is typically less than 1.2, and that means that neither team should adopt a defensive strategy. In fact, most teams in the lower half of the EPL, Bundesliga and La Liga should be adopting an offensive, or at least a balanced, strategy in around half of their matches (the more uniform distribution of teams in Serie A does suggest a more defensive strategy for the weaker sides in Italy than the other leagues).

To avoid relegation you must look to beat all the teams in the lower half of the table, and not simply rely on a few lucky victories in tight games. Furthermore, the games against the top teams don’t even matter that much. If we remove all the points gained against the top 7 teams in the EPL over the last 20 seasons, you find that nearly three quarters of the teams that were relegated would still have gone down. So lucky victories and hard-fought draws against the top third of the table are not what saves teams.

The Twittersphere Debate.

There has been much discussion on Twitter recently on this topic. In particular Luis Husier wrote a very nice piece (excluding his accusation that no one else can count) in which he directly measures the success of offensive or defensive strategies in terms of points accumulated over a season. He shows that there is a statistically identifiable difference between the success of attacking and defensive teams. I have been able to exactly reproduce his results (code here).

However, he does not actually test the hypothesis that he sets out to: should relegation candidates prioritize defence? That is because it's the better teams in the league that drive the result he obtains (again, code here). It’s definitely true that if you want to finish high up the table you should prioritize attack. But if you remove the ‘high quality’ teams from the analysis – limiting yourself to those in the lower half of the table -- you find a much smaller difference between the success of defensive and offensive teams (a result that doesn't change if you increase the sample size). I’m definitely not saying that the hypothesis is wrong, just that it hasn’t really been thoroughly tested. 

Summary

The model I’ve described here is very theoretical and massively over-simplified. It does not account for home advantage, in-game tactics, important one-off matches, or the psychological benefits of avoiding defeat. I assume that the weaker team determines the style of game by deciding how much to defend: that can’t really be true. 

However, I think the conclusions are reasonable and fairly intuitive: teams in the lower half of the table should aim to glean maximum points from their encounters by adopting a more offensive strategy. Therefore, to avoid relegation, they must play offensively in a significant proportion of their matches. A good defence alone will not save you. 

NB: code for this post can be found here.