1. PSG deserved it. Let’s get that out of the way first: across the two legs, they were the better team. They won the xG score 4.41 vs 1.86 in total, and they look like a team that can go deep in this competition. In fact, I would not be surprised if they will win it. As for Liverpool, yes, they had their chances on Anfield, but c’est la vie!
2. The little details count. “In a game now packed with the latest modern technology and VAR, it’s curious that an old-fashioned coin toss can still have such an impact on the outcome of such an important encounter“. In the beginning of the game, Van Dijk won the coin toss but gave away the chance to attack in front of the Kop in the second half (he changed his mind the second time around, before the extra time). But PSG then won the two important coin tosses: to have the shoot-out in front of their fans and to shoot first. There is statistical evidence showing that the team kicking first wins in 60% of the cases, so PSG started the penalty shootout with two small, but important advantages. Finally, another little (big) detail was that two of the Liverpool players were observing the Ramadam fasting tradition: Salah and Konate. This means their game preparation was impacted, being only able to eat or drink after sunset (only a bit more than an hour before the game started).
3. Speaking of little details, having a weekend to rest helped PSG massively, especially with the game going into extra time. After a Klopp-like pressing in the beginning of the two halves, Liverpool got tired, and they lost the midfield battle after Szoboszlai and MacAllister went off. PSG on the other hand looked more fresh and controlled the game better after absorbing the initial Liverpool pressure. In retrospect, the penalty shootout was Liverpool’s best chance to go through.
4. It was one of the few games when the Liverpool substitutes were really poor and could not influence the game at all. Quansah was the only one having a decent game (and perhaps Endo too, despite playing only 10 minutes), but the other 4 – Nunez, Jones, Gakpo and Elliott – had a game to forget, especially the first two. Gakpo might have the excuse of coming back from an injury and was likely not 100% fit to play. Perhaps this is an opportunity to reflect on the squad depth and how things could be improved in the summer.
5. Alisson was outstanding, but his penalty record is average. He made 16 saves across the two PSG legs, with his performance in Paris described as ‘best of his life’. But he could not save any of the four PSG penalties, and looking at his penalty record, his stats are not more than average: he saved two penalties in the PL (Chelsea and West Ham) and one in the CL (Napoli) out of 12. He was involved in three penalty shootouts: two on Wembley (against City and Chelsea) and the one on Anfield last night. He only won one (against Chelsea, in the FA Cup final in 2022), after saving one penalty (against Mason Mount). Until now, Alisson has only saved one penalty on Anfield (Bowen, 2022).
6. Slot got his penalty takers wrong. I would have gone with Salah, Virgil, Gakpo, Endo and Eliott, in this order. Would have kept Nunez as the 11th penalty taker. Wrong end and megaphones aside, there was sense of inevitability that we will miss (not a big fan of stuttering penalties, BTW).
7. Liverpool depend too much on Salah. When Salah is having a bad day (and against PSG he had two), and he doesn’t score, chances are Liverpool will not win. This season there were only 5 games when Liverpool managed to win despite Salah having no goal involvement (Milan and PSG away, Palace, Forest and Brentford away). There were of course three 0-1 losses – against Forest (PL), Spurs (EFL) and PSG last night when Salah had no impact. This is not encouraging for next season, so
8. Liverpool will probably look for offensive players in the summer. There is uncertainty regarding the new contract of Salah. Nunez is likely to be sold (and Liverpool should sell him, if you ask me). Jota is decent as a squad player, but he’s frequently getting injured (and a bit unreliable, recently). Chiesa is a mystery and I can only hope that he will have the same trajectory as Gravenberch, who only played a few games in his first season but then he became irreplaceable in his second. I also hope that Diaz will stay, but you never know. It could be a long and complicated summer for Liverpool.
9. All done and dusted, if you’re going to exit from the CL, better do it early. I am aware of the financial implications, but losing two CL finals hurt a lot. Despite having a brilliant season so far, I could not see Liverpool going deep in the CL (it would have been different if they were drawn on the other side of the table, playing Benfica and Lille/Borussia for fun until the semifinals). It’s been a long season and players like Virgil, Salah, Gravenberch, Mac Allister or Szoboszlai have a lot of minutes in their legs.
10. There’s still a final to win on Sunday, and more importantly, a Premier League advantage of 15 points with 9 games to go. Only 10 games to be played this season – there’s hope that the players and the supporters will enjoy them.

With 9 games to go, Liverpool are top of the Premier League with a 15 points advantage:

Arsenal now have less than 1% chances, and the bookmakers offer 41 to 1 odds if you bet on Arsenal. It’s been a bit of a rollercoaster though, just watch the fall from 3.75 on 19 February to 26 just one week later:

Liverpool navigated very well the fiery February, but, with two draws that felt like two losses (at Everton and Villa), they needed Arsenal to drop some points. Arsenal dropped more than expected, and now it’s simply a matter of time until Liverpool are crowned PL champions.

I wanted to find a scientific way to determine when this could happen. And the answer is this:

How to read it?

The chart says that the earliest that Liverpool can win the PL is in match day 32 (with a 15.3% chance), when they will play West Ham at home. The most likely is the next match day (Leicester away, 35.9%), and by the time they play Arsenal (MD 36), there is a 90% chance that the PL title is already won. Arsenal only have a 0.35% chance to win the PL (not shown in the chart).

How does it work?

I used the xG data from fbref.com, averaging the xG at home and away for every PL team. If we take the next Liverpool game, against Everton: Liverpool have a 2.18 average xG at home, while Everton have a 0.99 average xG away. I used a Poisson distribution function to turn these xG numbers into actual goals, and, ultimately, into winning percentages: 65% Liverpool win, 19% draw, 16% Everton. This is how it looks like:

I ran 100.000 simulations of the remaining PL games for both Liverpool and Arsenal and I came up with the percentages in the chart above. Interesting fact: running 10.000 simulations on an Intel Core i5 8th generation took about one minute. The same number of simulations on a MacBook M1 Pro took 13 seconds; so I could afford to run 100.000 simulations. However, running 10x more simulations did not change the data more than a rounding error, so for all intents and purposes, 10.000 simulations are sufficient.

More data

10 PL teams can no longer catch Liverpool. 2 more can follow if Liverpool win their next game against Everton at home, and another one (Bournemouth) if they don’t win their next game:

According to my simulations, 80 points are sufficient to win the PL in more than 80% of cases. So 10 more to go!

See also:

• https://www.reddit.com/r/LiverpoolFC/comments/1j2vj0j/when_can_liverpool_win_the_premier_league/
• https://www.reddit.com/r/LiverpoolFC/comments/1j87tzr/when_can_liverpool_win_the_premier_league_update/

Update after GW 29

In the Seinfeld episode ‘The Kiss Hello’ (S06E17), Jerry’s Nana remembers that Uncle Leo was supposed to give Jerry’s mom 50 dollars 53 years before. When Morty (Jerry’s father) finds out, he starts to calculate the amount that Uncle Leo would owe his wife, Helen:

MORTY: Do you know what the interest on that fifty dollars comes to over fifty-three years?

HELEN: Oh, Morty, please.

MORTY: Six hundred and sixty-three dollars and forty-five cents. And that’s figuring conservatively at five percent interest, over fifty-three years, compounded quarterly. #

Using the 5% interest rate that Morty mentions, the best case scenario for the $50 over 53 years would be for continuous compounding:

Total = P * e ^ rt, where P=50, r=0.05 and t=53, so Total ~= $707

The value that Morty came up with – $663.45 is extremely close to what he would have received for a yearly compound:

Total = P * ( 1 + r/n) ^ nt, where P=50, r=0.05, t=53 and n=1, so Total ~= $663.75

However he does mention “compounded quarterly“, so the final expected amount should be:

Total = P * ( 1 + r/n) ^ nt, where P=50, r=0.05, t=53 and n=4, so Total ~= $696.17

In conclusion, Morty underestimated the amount that Helen should have received by $32.42. Quarterly compound is better than yearly compound, and the continuous compound is the best. Also, $696.17 in 1995 (when the episode aired) is worth $1,451.14 today.

Intuition about the e number

Imagine an initial investment of $1, earning 100% annual interest. So P=1, r=1, t=1.

• If interest is compounded once per year, you end up with: (1+1)^1 = 2
• If interest is compounded twice per year, you end up with (1 + 1/2) ^2 = 2.25
• Quarterly compounding: (1 + 1/4) ^ 4 = 2.44
• Monthly compounding: (1 + 1/12) ^ 12 = 2.61
• Weekly compounding: (1 + 1/52) ^ 52 = 2.69
• Continuous compounding: e = 2.718

Discrete vs continuous growth

At its core, e measures the idea of continuous growth, which is different from the discrete growth. Discrete growth means that a given population only grows at certain moments (example at the hour mark). Continuous growth is different: instead of waiting until the end of the hour, the population is growing all the time, at every moment, like a plant that grows gradually instead of in sudden jumps.

The discrete growth is modelled by the (1 + r)^t formula, while the continuous growth is modeled by the e^rt formula. The second one results in a slightly higher number because you’re adding a little bit of growth continuously.

Assuming a population of 1000 astrophage, and a growth rate of 5% per hour, here is how it will grow over a 10-hours period: