Every season I consider trying to characterise the selection of my Ppremier League (FPL) starting line up mathematically as a constrained-optimisation problem.
However, it’s a very difficult problem to model. Even if I pin down expected payoffs, for example using the Rate My Team (RMT) tool in the Members Area, it is difficult to account for rotation from the bench, captaincy rotation, injury risk and the ability to make transfers as the season unfolds. A comprehensive model remains a work in progress. Today I modelled an extremely simplified version of the optimisation problem to get a feel for where 100 million will give me the best return. It was a quite enlightening exercise and the results challenge the conventional wisdom about how much of our budget we should invest in defenders.
The Approach
I set about identifying the 15 players with the highest expected points over the first six Gameweeks, subject to the budget and positional constraints imposed by FPL.
For simplicity, I included the expected points for all 15 players each Gameweek and I ignored captaincy. I fully accept that this squad will not be optimal as a result, but the exercise was nevertheless informative and the potential effects of accounting for these complexities are discussed later in this article.
I approached the problem by taking a squad of the highest-scoring, low-budget players over Gameweeks one to six based on the RMT tool as at July 22. This initial squad cost a total of 69.0. I then calculated the increase in Gamweek 1-6 points per 1.0 spend (points/£m) that would be achieved by upgrading each of those 15 players to any of the other players in the game. I then made the player upgrade that offered the maximum points/£m across all possible player upgrades. I repeated this process until the budget constraint was binding.
The Results
The squad arrived at through this process was:
Myhill, Lloris
Clyne, Smalling, Ivanovic, Terry, Fonte.
Mane, Ritchie, Henderson, Hazard, Johnson.
Sakho, Kane, Wilson.
One thing in particular stood out about this squad.
First, there is quite a lot of investment in defence relative to the teams I have seen people putting on the boards. While the squad includes lower-cost midfielders and forwards, namely Matt Ritchie, Bradley Johnson, Diafra Sakho and Callum Wilson, it does not include any defenders from the budget category. The process of optimising the squad revealed that there is a greater return on upgrading budget defenders than there is on upgrading the aforementioned midfielders and forwards.
Given this result is somewhat contrary to the conventional wisdom I have seen on the boards so far in pre-season, I checked the optimisation by looking for combinations of upgrades and downgrades that might increase the overall score. The top 10 marginal benefits from upgrading players in the squad are (in points/£m):
2.04 Ritchie – Silva
1.91 Johnson – Silva
1.82 Clyne – Koscielny
1.76 Wilson – Benteke
1.74 Mane – Silva
1.74 Fonte – Koscielny
1.70 Henderson – Silva
1.67 Wilson – Rooney
1.65 Ritchie – Chadli
1.58 Sakho – Benteke
The bottom 10 marginal costs from downgrading players in this team are (in points/£m):
1.76 Terry – Koscielny
1.79 Hazard – Silva
1.79 Terry – Yoshida
1.80 Clyne – Yoshida
1.84 Fonte – Yoshida
1.87 Terry – Rose
2.01 Ivanovic – Yoshida
2.05 Hazard – Chadli
2.12 Terry – Skrtel
2.14 Hazard – Mahrez
Notice that the highest marginal benefits from upgrades are similar to the lowest marginal costs from downgrades. This is what we want to see in our teams, since it means there are no gains to be made by changing things around. I couldn’t find a way to improve it. In relation to the option of shifting investment from defence to forwards, we can see that the best value forward upgrade is Wilson to Benteke at 1.76 points/£1m. There is no downgrade or set of downgrades that is less costly than 1.76 points/£1m. Ivanovic to Yoshida costs 2.01 points/£1m. Hazard to Chadli costs 2.05 points/£1m
The Conclusion
Can we conclude that we need to resist the temptation to pile our funds into expensive attacking options and instead invest in less sexy options in defence?
Is the popular approach of buying Ivanovic/Azpilicueta plus four 4.5m defenders suboptimal?
Perhaps. But there are obviously some qualifications needed.
First, you may not agree with the RMT predictions. Second, I haven’t accounted for captaincy. I don’t expect that incorporating captaincy would change much about the squad, since the highest scoring player (Eden Hazard) is already included in the team. However, the ability to rotate captaincy may justify trading two mid-price options for a budget option and another heavy hitter.
Third, I have not accounted for the fact that one goalkeeper and three outfield players will be on the bench each Gameweek.
Presumably, the players likely to be on the bench, such as Wilson, Boaz Myhill (four Gameweeks), Sakho, Ritchie, and Nathaniel Clyne (three Gameweeks), would be downgraded to fund upgrades elsewhere. Players that rotate well with each other may be elevated to the squad. It’s hard to say whether these players would be more likely to be defenders, though I do tend to agree that fixtures are a more important consideration for defenders than for midfielders or forwards.
These are more complex sets of trade-offs to analyse and something I am trying to work into my model.
In the meantime, this is food for thought. I’d be interested in hearing your views.
