Sod’s Law
The buttered toast phenomenon is the observation that when you drop a slice of buttered toast, more often than not it will inconveniently land with the buttered side down. Though it has been argued in an academic study that this phenomenon has a physical explanation – a study that earned its author the 1996 Ig Nobel Prize in physics – it’s more commonly regarded as an example of Sod’s Law. This states:
If something can go wrong, it will.
The toast could land either way up, but by Sod’s law, it invariably lands butter-side down. Not because of physics; just because that’s the most inconvenient outcome, and life works that way.
A sporting example of Sod’s Law might be:
After weeks of late seas, the surf turns perfect the day you drop (and break) your surfboard while moving it out of storage.
Again, you might have broken your board any day, but you did so on the first day in ages that you could have actually used it.
A Dangerous Score
Back in the 2016-17 season, towards the start of the Klopp era, Liverpool were leading 2-0 away to Bournemouth in the Premier League, and Gary Lineker sent this tweet:
When did 2-0 up become a dangerous score? Hear it so often now. Would like to know percentage of games lost from there? Suspect not high.
— Gary Lineker (@GaryLineker) December 4, 2016
Indeed, the observation that 2-0 is a dangerous score is so pervasive it has its own wiki page. But is it true?
Well, first, let’s answer Gary’s question directly.
Based on Premier League data from the 2016-17 season onwards, there have been 1734 matches where the home team has led 2-0 at some stage, and in 26 of those matches the away team went on to win. So, the percentage of times that an away team has overturned a 2-0 deficit is around 1.5%. Similarly, there are 31 fixtures out of 1076 for which the home team has overturned a 2-0 deficit, which is approximately 2.9% of occasions. Putting things together, a 2-0 lead has been established by either team on 2810 occasions and in 57 of those matches the opposing team managed to overturn the result, implying an almost exact 2% rate. So this is the empirical answer to Gary’s question, for the Premier League over the period examined anyway: once a 2-0 lead has been obtained, it’s been lost from there on 2% of occasions. Or to put it another way, once a team has established a 2-0 lead, they’ve not lost that game on 98% of occasions.
But does this mean 2-0 is not a dangerous score? Presumably, when people say it is, they don’t really mean that a team is more likely to lose when they’re 2-0 up than when it’s still 0-0 or 1-0. What they probably mean is that teams are more vulnerable to concede – and maybe less likely to score again – when they’re 2-0 up. But is there any evidence for this?
Here’s one way of looking at that. At any scoreline, the next goal – if there is one – could be scored by either the home or the away team. If we look at all games where a particular scoreline occurred, and drop the games when there were no more goals, we can count the proportion of times the next goal was scored by the home and away team respectively.
For example, all games start at 0-0. Ignoring the games that finish 0-0, it turns out that the first goal in a game is scored by the home team on 56.9% of occasions and by the away team on 43.1% of occasions.
| Scoreline | Home goal next | Away goal next |
|---|---|---|
| 0-0 | 56.9% | 43.1% |
| 1-0 | 58.8% | 41.2% |
| 0-1 | 52.5% | 47.6% |
| 2-0 | 64.3% | 35.7% |
| 0-2 | 45.8% | 54.2% |
The table above shows the corresponding proportions for a range of scorelines. What this actually shows is that the greater the lead that a team holds, the more likely it is that they will score rather than the opposing team. In particular, when the home team is winning 2-0, the rate at which the away team scores the next goal has dropped from 43.1% to 35.7%. When the away team is winning 2-0, the rate at which the home team scores the next goal has dropped from 56.9% to 45.8%.
This seems to contradict the notion of 2-0 being a dangerous scoreline. At that score, and assuming there is a further goal, the chance of it being the opposing team that scores it is considerably lower than it was at 0-0.
But there’s a catch. Teams that manage to reach a scoreline of 2-0 are likely to do so because they are stronger than the opposing team. So, focusing on games at that scoreline means we’re looking at games where, on average, there is a bigger imbalance between the 2 sides, in favour of the leading side, compared with games when the scoreline is 0-0. As such, we’d expect the leading side to be more likely to score next – as the table suggests – because they tend to be the stronger side. The real question is whether that tendency is tempered by a “dangerous scoreline” effect at 2-0. To answer that question we need to do something that takes into account the different strengths of the sides.
Here’s one attempt to do that. Based on the assumption that teams score at a constant rate throughout a match and that the market can be used to estimate the number of goals each team will score within a match, the standard Poisson model for goal scores enables a simple calculation that either team will win a match given the current scoreline. These probabilities can be compared for different scorelines with the observed proportion of times either team wins. Where there are discrepancies it will imply that there is something wrong with the model assumptions.
| Scoreline | Lead overturned % (model) | Lead overturned % (observed) | ||
|---|---|---|---|---|
| 1-0 | 1.0% | 1.5% | ||
| 0-1 | 1.9% | 2.9% | ||
| 2-0 | 8% | 9.5% | ||
| 0-2 | 13.6% | 16.5% | ||
The above table makes these comparisons for the probability of either a 1-0 or 2-0 result being overturned. What it shows is that when the score is either 1-0 or 0-1, the model and observed probabilities are in reasonable agreement. Not perfect because the model assumptions are still not correct, but the numbers are in the same ballpark. However, when the scoreline is 2-0 or 0-2, the observed rate at which the lead is overturned is around 50% greater than what the model predicts. In other words, there is a sense in which teams do better than expected when they are losing by a 2-goal margin, but only raising their very small chance of winning to something that’s slightly less small. This is the full extent to which a 2-0 scoreline seems to be dangerous.
Back to Sod’s Law
What’s the relevance of Sod’s Law to all this? Well, despite Gary Lineker’s tweet – or maybe because of it! – Bournemouth overcame the 2-0 deficit to win the match 4-3. There might be precious little evidence to support 2-0 being a dangerous scoreline, but on the one occasion Lineker chose to tweet about the implausibility of such an effect, Liverpool obligingly allowed Bournemouth to overturn the 2-0 lead they had created. Classic Sod’s law.
And what can we say about Sod’s Law itself? Does Gary Lineker’s tweet and the outcome of the Bournemouth-Liverpool match afford it any credibility?
Clearly not, though you can see from this story where the temptation to give credence to Sod’s Law comes from. I only remembered the elements of this story from 2015-16 because of Lineker’s tweet and because Bournemouth ended up winning. I’d probably have forgotten immediately if Bournemouth hadn’t won.
Toast gets dropped all the time. But you’re much more likely to notice when it drops with the buttered side down: “Sod’s Law”. Nobody says “not Sod’s Law” when it lands the right way up (though there’s a rather funny joke about that eventuality in the wiki entry).
In summary, there’s a strong argument that Sod’s Law is really due to selection bias – noticing the things that go wrong, ignoring the things that go right – so it feels like the things that go wrong happen a disproportionate amount of the time. So, take a little time in your day to notice the good things in your life and not just the bad things. Chances are it will then turn out that the toast lands with the buttered side up around 50% of the time, there are plenty of fine days for surfing and Liverpool will usually give Bournemouth a hammering, especially once they’re 2-0 up.
