BEFORE this season’s Premier League kicked off on 16th August, bookies rated Liverpool as mere fifth favourites to win the title. This was despite it being the second-best side in country last year. The main reason for the pessimism was that over the summer the club sold Luis Suarez (pictured), its star player, to Barcelona for £75m ($124m). This made him the third most expensive signing in football history. Brendan Rodgers, Liverpool’s manager, probably had little choice in the matter: Mr Suarez had been eager to move for some time. But history suggests even when clubs re-invest such riches in their squads, the very best players are difficult to replace. Game Theory has looked at big-money transfers between English clubs to examine what effect selling a prized asset has on a club's fortunes. We have also assessed the ways the selling clubs reinvested their transfer income. The table below compares the change in average league points for the selling club in the two seasons before and after the biggest transfers by Premier League clubs. (Several expensive deals, such as Juan Mata from Chelsea to Manchester United, are too recent to be included.) We also looked at the change in the total wage bill, because the correlation between wages and league position has been shown to be strongly positive in the long term—at around 90%.

Liverpool has been here before. In 2011 it sold another star striker, Fernando Torres, to Chelsea for £50m. The club subsequently earned around 20 points a season fewer. This was despite it reinvesting the money and, in theory, making its squad stronger—evidenced by the wage bill rising by £30m. With Mr Torres acting as a talisman the side finished second. Without him it finished sixth and then eighth.

The club spent £35m of the Torres fee immediately, on a replacement striker, Andy Carroll of Newcastle. Herein lies one problem when replacing stars. The fee was no doubt inflated because Newcastle knew that Liverpool was suddenly flush with cash. Mr Carroll was wholehearted and popular with the fans, but his muscular style of play did not suit the team's system. He scored just six league goals for the club before being moved on.

Aston Villa also fared badly following the sale of James Milner to Manchester City in 2010. Mr Milner had directed the team's play from the centre of midfield in the year before he was sold, for £26m. Aston Villa received another midfielder, Stephen Ireland, in part-exchange. It then spent most of the fee on a striker, Darren Bent, six months later. Yet, as with Liverpool, the mistake was not replacing like with like. Mr Ireland proved defensively less capable, and the team started to ship goals. Mr Bent, meanwhile, couldn’t live up to his price tag and fell out of favour. Although the club's wage bill was static, it averaged 20 points fewer for the next two seasons.

More successful was Tottenham Hotspur's sale of striker Dimitar Berbatov to Manchester United in 2008. Spurs spent the £31m it received on three new forwards: Roman Pavlyuchenko, Jermain Defoe and Robbie Keane. The latter two had played for the club previously, but Mr Pavlyuchenko was new to the league and therefore a higher risk. Of the trio, only Mr Defoe was an unqualified success, but the club improved their chances of replacing Mr Berbatov's goals by signing a group of strikers. The increase in the wage bill (32%) was higher than the increase in points (14%), but it still represented a decent return. Still, it is not a foolproof strategy. The club tried to repeat the trick in 2013, after it sold Gareth Bale to Real Madrid for a world-record fee. The £90m it received helped it buy half a side’s worth of new players. Integrating so many new footballers proved tough, however, and last year, without Mr Bale’s match-winning performances, it fell down the league.

The effect of selling a star player isn’t isolated to the selling club. There is also the danger of strengthening a rival. Arsenal sold Robin van Persie, a Dutch forward, for £24m in 2012. On paper, it looked a good deal for the Gunners. The fee was high for an injury-prone player, albeit a very fine one, who was nearing the end of his contract. Indeed, the team won more points without him and maintained their fourth position in the league. However, he was sold to arch-rivals Manchester United, where he promptly helped them win the league. Few Arsenal fans saw the transfer as a positive step.

The data presented here are far from perfect. The sample is small and there is plenty of noise surrounding the figures. The cost of a player is more than a reflection of his ability; it also takes into consideration things such as his age, the length of his contract and the desperation of the buying club. What is more, a club's performance can be affected by many other things, such as a change of manager. Nonetheless, it seems fair to say that clubs are better off hanging on to star players when they can, rather than cashing them in and improving the overall squad. Footballers with the ability to change the course of games singlehandedly tend to be irreplaceable. Liverpool may discover that for a second time this season.