Cost-benefit analysis is an important tool for soccer teams deciding whether to buy a player. When they’re on the verge of shelling out millions of dollars, clubs probably ought to think about what they might get in return. A new signing could help a team to score more goals or concede fewer, but just how much would those be goals worth?
Goals translate into points, and points translate into league positions. In the English Premier League, a high finish can lead to lucrative European tournaments, and a low finish can mean relegation. Even the teams in between receive differing “merit payments” from television broadcasts.
Last year, the gap in these payments between champions Manchester United and bottom-of-the-table Queens Park Rangers was more than £12 million. But if we consider Manchester United’s income from this year’s Champions League and Queens Park Rangers’ losses from relegation, the true gap was much bigger.
Moreover, the payoffs from a single season are only one piece of the puzzle. To see the true value of a goal, we have to look further into the future. A team finishing one place higher in the table could receive millions more in prize money, which would make a high finish more likely in the next season as well.
Goal difference does not correlate perfectly with league positions, nor do goals scored or goals conceded. But the correlation is pretty high for all three, and, using historical data, we can make statements like “No team scoring X goals has finished lower than Nth position” and “No team conceding Y goals has finished higher than Mth position.”
I decided to gauge these possibilities and the resulting cash payoffs during a hypothetical five-year period. Using data from 18 seasons of the Premier League, I estimated how every team’s goals for or against might help to determine league positions. For each position, I attached a fixed payoff in the first year – here based on the actual figures for 2011-12 and 2012-13 – and then expected payoffs depending on the most likely finishes in the following four years.
I did what statisticians would call a Bayesian analysis, because I updated the probabilities of finishing in a given position in year T+1 using the expected finish in year T. Essentially, I tried to answer the question, “Given the expected payoffs over five years to each final position in the league, how much more money would one more goal for or one less goal against likely mean for the club?”
As in any simplified statistical exercise, I had to make a few assumptions:
• A given English team will advance through qualifying rounds of European tournaments 75% of the time
• A given English team will advance through group and knockout rounds of European tournaments 50% of the time
• For club owners, competing investments pay 5-10% returns (relevant to the discount rate for future payments)
• Broadcast and prize money in the Championship is negligible compared to the Premier League
It’s worth noting that I did not include merchandise and ticket sales, which may also depend on where teams finish in the table, in my calculations.
Not surprisingly, scoring one more goal in a season can make a big difference to teams on the knife-edge between positions, and much less difference to teams whose fates are clear. The following graph shows the estimated value over five years of an additional goal scored in today’s money, depending on how many goals a team already expects to score:
The value of a goal starts low for teams that are certain to be relegated; one more goal doesn’t make much difference until they are scoring more than 30. Then the value begins to rise sharply, peaking at around 50 goals, a total that starts to raise the possibility of qualifying for European football. The value stays high through about 75 goals – generally enough to gain a spot in the Champions League – and then drops off sharply. The reason for the drop off is that additional goals do not necessarily guarantee a substantial increase in prize money until a team reaches roughly 95 goals, a total that will virtually ensure winning the league.
According to the graph, teams on the cusp of qualifying for European football should spend the most to bolster their goal totals. In the rare case where a team is already close to 95 goals per season, spending more can also be worthwhile. But otherwise, teams already scoring more than 80 goals gain little by scoring more.
Note that this graph is independent of goals conceded – clearly a team conceding five goals in every game is still very likely to be relegated! The graph is most useful for assessing the value of small deviations from the expected number of goals scored.
Similar relationships hold for goals against, but the value of conceding one less goal is not necessarily the same as the value of scoring one more goal for teams at a given position in the table. Here is a graph of the value over five years of the marginal goal against, depending on the total number of goals likely to be conceded:
Teams that concede fewer than 30 goals are extremely likely to enter the Champions League. Once again, there is little to be gained by conceding fewer until a team gives up fewer than 20, which would generally clinch the top spot in the table. Also, once a team is expected to give up 80 goals per season, conceding more makes no difference; relegation is assured. Note, as before, that these calculations are independent of goals scored. A team scoring five goals a game will almost certainly be crowned champion regardless of where it falls on this graph.
What does differ here is the overall relationship. The value of conceding one less goal peaks more sharply than the value of scoring one more goal, and it also tops out at a higher figure – close to £7,500,000 over five years! But for teams in the lower and middle reaches of the table, the value of a goal scored rises much more steeply (about £5,000,000 between 30 and 50 goals for) than the value of not conceding (about £3,000,000 between 70 and 50 goals against).
Differences in the values of goals scored and conceded may seem paradoxical at first. After all, shouldn’t any change that affects goal difference by the same amount have the same value? To understand why the answer is no, we only have to think about what happens in games. For a team that tends to win most matches, an extra goal scored may not make much difference (a 2-1 win would become a 3-1 win), but an extra goal conceded might lead to dropped points (a 2-1 win would become a 2-2 draw). By contrast, an extra goal scored might mean everything to a team that tends to draw or lose by one.
To see how the results in the graphs above might apply to the Premier League, consider the teams that finished near the cutoff for European places in 2012-13:
Premier League Goals — European contenders
|West Bromwich Albion||53||57||8|
For Everton, the expected value of scoring one more goal in the coming season may be about £6,000,000 over five years, but the value of conceding one less goal could be more than £7,000,000. In other words, even though Everton’s defense is already more outstanding than its offense, putting more money into the defense might yield the greater rewards.
By contrast, West Bromwich Albion is a team that probably needs to score more. Its defense may look leaky, but the expected value of conceding one less goal is only about £4,500,000; a much bigger improvement would be needed to make a substantial difference to league position. One more goal scored, on the other hand, could be worth close to £6,000,000.
In the future, I’m going to extend this work by evaluating the two marginal changes simultaneously. I’ll put teams on a two-dimensional plane where the axes are goals scored and conceded, and then I’ll check whether horizontal or vertical moves are likely to bring them towards higher league places. This is a complicated exercise, since it can involve assessing the proximity to other points on the plane using formulas akin to those that describe gravitational fields. For example, consider the cluster of historical second- and third-place Premier League finishes in the graph below. Is a team whose goal record places it in the middle of this cluster more likely to finish second or third?
Of course, even the best statistical model can’t perfectly predict how each team’s results will lead to a given league position in the season to come. But as I’ve written here before, teams don’t just disappear after one season. In the long term, this kind of probabilistic analysis can help them to make the right decisions more often than the wrong ones.