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"The second leg at home is normally a slight advantage" – Arsène Wenger
Indisputably the greatest mind to grace the beautiful game, if Arsène Wenger raises the possibility of Second Leg Home Advantage (SLHA), then it is surely worth investigating. Particularly on the eve of Arsenal’s home leg in their first Champions League Quarter Final appearance for 14 years.
The idea of second leg home advantage (SHLA) is that, on average, teams are more likely to win a two-stage knock out competition when they play the second leg at home. Whilst the general home advantage effect has been studied extensively – quantifying size and reasons, there has been less empirical investigation into whether the perceived extra advantage of playing at home specifically in the second leg of a two-stage knock-out competition even exists – let alone why.
On the face of it, the effect should reveal itself simply – assuming the draw is random, the SHLA implies that the probability of victory for the second leg home team is higher than 50%. But it isn’t quite that simple.
For a start, in the last 16 of the Champions League, the team progressing top of their group is granted a home second leg (seeding bias). Then there is the mess of extra-time & penalties and, until recently, away goals. Isolating the marginal second-leg advantage above and beyond differences in team strength and longer game length is challenging.
Table and Figure taken from Eugster, M. J., Gertheiss, J., & Kaiser, S. (2011) who concluded that SLHA could be fully accounted for by teams' general strengths
One such study, an investigation into 6,084 two-stage knock-out rounds from three major European Cup competitions between 1955-2006, found that even after accounting for seeding bias using UEFA calculated coefficients – the team at home second had a 54.3% probability of winning a two-legged tie (significantly different from 50% at p<0.001). This effect persisted even when accounting for extra time and penalties.
They do show, however, that SHLA has been decreasing over time and no longer exists if looking at just the Champions League era. This is supported by a more recent study concluding that apparent second-leg home advantage in the Champions League can be completely explained by the teams' performances in the group stage and – more importantly – by the teams’ general strength.
We had consensus before a new paper challenged the methodology underpinning these studies and stated that in European cups between 2009 and 2015, holding everything else constant, the team at home second had a 54% probability of winning.
If you want a job done properly…
...I have spent my lunch break noting down the scores of every ‘non-seeded’ two-legged tie in the Champions League since 1995 – all 324 games. The table below provides the breakdown results won by the second leg home team in different game states (and the p-values against a null hypothesis of no SLHA).
Breakdown of 324 Champions League games since 1995. Overall, the team at home second is not significantly more likely to win the tie
We find no evidence for SHLA (in accordance with a study looking at the same fixtures between 2010 and 2017 which I hadn’t seen when I embarked on my data collection journey). 52.5% of ties were won by the second leg home team, which isn't significantly different from 50% - and this drops to 51% when only looking at ties that finished in 'normal time'.
But that isn’t the end of it – all the focus looking for second-leg home advantage (perhaps cognitively imprinted through the impact of biased seedings) means we have been oblivious to the absence of what we should expect to find: first-leg home advantage (FLHA).
FLHA is predicted by both established psychological theory and evidence from elsewhere in football itself.
Prospect theory, a model of how we make decisions under risk and uncertainty, highlights our inherent loss aversion: losses loom larger than gains (perhaps even twice as much!). If the team at home first uses their home advantage effectively, the increased pressure for the second leg home team chasing the game to avoid losing should interfere with players’ ability to perform optimally.
If you think this sounds far-fetched – this isn’t just academic theory – we see this in the real world, in knock-out football, in penalties. The team that shoots first in a penalty shoot-out wins 60.5% of the time, a huge first-mover advantage.
Taking first puts the taker in a gain-framed mindset (trying to get ahead) and taking second in a loss-framed mindset (avoiding going behind). This effect is clearest with decisive penalties – players with an opportunity to win a shoot-out are 30% more likely to score than players who must score to avoid losing the shoot-out.
The first-mover advantage is so well recognised that UEFA have trialled strategies to overcome it (ABBA penalty order), and clubs have even turned to neuroscientists to help cognitively train players to reduce anxiety and optimise performance at critical game moments.
Dan James misses the crucial penalty having needed to score to keep Wales in the contest. Penalties taken under these conditions are more likely to be missed than penalties to win shootouts
Although there may not be a SLHA outright – there must be something going on to nullify the first-mover advantage you would expect.
The crux lies in the condition: If the team at home first uses their home advantage effectively. There are significantly fewer goals in the first leg of a two-legged Champions tie than the second (~20%; p<0.01), and the team at home first scores significantly fewer goals at home than the team at home second (~20%; p<0.01).
This points to an overly-conservative, ‘cagey’ (in the football cliché lexicon) first leg where the home team fails to make the most of their first-mover advantage. The game states follow a pattern observed in the Premier League, where there are 31% fewer goals in the first half than the second.
Together this suggests that the problem is that the team at home first views the two-legged tie as precisely that, a two-legged tie. To maximise first-mover advantage, the team at home first needs to view their home leg in isolation – and go out all guns blazing.
Arsenal are in a good position to do this – scoring only 46% of their goals in the first half (the fifth highest proportion in the league) – let’s hope the occasion and thoughts of Munich doesn’t buck this trend.
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