# On FPL, Optimization, and Ownership Weights

By Alp Cay | December 9, 2020

Robert Cialdini mentions scarcity as one of the 6 principles of persuasion. The nature of seeking something of great interest is embedded inside all of us. Our first reaction when we see other people do something is to follow them. Even though it does not make too much sense, we follow the herd even in a competitive game, like Fantasy Premier League.

Last week’s (GW11) unfortunate gold rush to Jota proved that we fear being left out. Of course the potential price changes further trigger and support our knee-jerk reactions, however it is important to separate what we want to do and feel forced to do.

This topic is of great interest to me right now, because I am at the other end of the spectrum: I am religiously following the results of my optimization model this season. Up until now, I never thought of including what other people are doing into my model, but it suddenly made sense to me. Perhaps our cognitive tendencies have something that I can benefit even in a purely mathematical model.

## Analytics in FPL

Analytics is once seen as an evil in sports. Many sports unwillingly embraced the edge analytics and data bring. Soccer/Football is still behind this transformation.1

As a fantasy sports leg of the puzzle, FPL is closely involved with data. The difficult part of FPL is the vast number of decisions you can give. Currently there are over 600 soccer players and an abundant volume of data available about each of them. You might be skeptical about whether analytics and data alone can solve the problem of decision-making each week in FPL. It depends how you look at it, but I do not think analytics and data have all the answers. However, they pave the road for well informed decisions like many sectors have been using for years.

Many of the managers are well aware of how to use basic tools and follow fixture difficulty ratings. After many hours of work, most of the time we end up with a prediction of what will happen next gameweek, however it might be away from the fact. Then, the question becomes: given this information, what is my best strategy? It is easy get overwhelmed with decisions, hence oversimplifying the most important step of the puzzle: decision-making. Unfortunately, neither statistics nor Machine Learning are decision making tools. They are called descriptive and predictive analytics tools, respectively. The last step of the analytics is what we call prescriptive, and the most common tool for this step is optimization.

Before we go back to our original discussion about joining the bandwagons, let us define a simple problem to see why optimization is a great decision support tool you should be using. Using FPL Review’s estimated values for GW12 (as of 2020-12-08), let us try to pick 4 midfield players under a certain budget to maximize our expected FPL points.

Expected points of midfield players this GW are as follows:

id name team_name selected_by_percent price xP
254 Salah Liverpool 32.2 12.3 7.732
4 Aubameyang Arsenal 9.4 11.5 5.86
251 Mané Liverpool 9.5 12 5.808
272 De Bruyne Man City 24.1 11.8 4.975
390 Son Spurs 58.5 9.5 4.893
468 Jota Liverpool 29.8 7 4.69
276 Sterling Man City 4 11.4 4.479
370 Ward-Prowse Southampton 13.5 6.2 4.171
500 Havertz Chelsea 3.3 8.3 4.033
469 Podence Wolves 3.4 5.4 4.032
231 Maddison Leicester 2.7 7 3.988
37 Grealish Aston Villa 38.9 7.7 3.958
275 Mahrez Man City 9.1 8.4 3.937
570 Raphinha Leeds 0.5 5.4 3.89
198 Klich Leeds 5.4 5.5 3.879
478 Willian Arsenal 3.7 7.6 3.83
306 Rashford Man Utd 6.1 9.4 3.813
119 Pulisic Chelsea 2.1 8.2 3.79
302 Fernandes Man Utd 41 10.9 3.78
474 Neto Wolves 4.7 5.6 3.765
120 Mount Chelsea 4.9 6.8 3.764
508 Rodríguez Everton 22.2 7.7 3.761
203 Harrison Leeds 1.3 5.4 3.488
465 Traoré Wolves 6.3 6.2 3.471
24 Saka Arsenal 2.4 5.2 3.367
69 Trossard Brighton 0.9 5.9 3.322
321 Shelvey Newcastle 0.1 5.3 3.305
141 Zaha Crystal Palace 16.8 7.4 3.235
445 Bowen West Ham 2.6 6.3 3.225
368 Armstrong Southampton 0.7 5.5 3.223
228 Tielemans Leicester 2.9 6.4 3.095
57 Groß Brighton 0.3 5.8 3.009
244 Henderson Liverpool 1 5.4 2.98
449 Soucek West Ham 3.9 4.9 2.961
466 Neves Wolves 1.3 5.2 2.945
243 Wijnaldum Liverpool 1.3 5.3 2.908
450 Fornals West Ham 1.9 6.4 2.796
480 Sean Longstaff Newcastle 0.1 4.7 2.784
204 Phillips Leeds 1.3 4.9 2.77
403 Lo Celso Spurs 0.4 6.9 2.77
40 Trézéguet Aston Villa 0.8 5.3 2.769
489 Eze Crystal Palace 0.7 5.8 2.765
100 McNeil Burnley 0.4 5.7 2.761
339 Almirón Newcastle 0.4 5.6 2.752
113 Kanté Chelsea 3.4 4.9 2.725
446 Diangana West Brom 0.3 5.3 2.696
464 Dendoncker Wolves 0.9 4.8 2.643
221 Albrighton Leicester 0.1 5.3 2.626
454 Moutinho Wolves 1.2 5.2 2.623
396 Højbjerg Spurs 0.9 4.9 2.622
364 Oriol Romeu Southampton 4.5 4.5 2.616
137 Townsend Crystal Palace 1.9 5.8 2.615
38 McGinn Aston Villa 1.7 5.5 2.596
286 Rodrigo Man City 0.5 5.4 2.559
360 Berge Sheffield Utd 0.2 5 2.549
392 Lucas Moura Spurs 1.2 6.7 2.508
448 Rice West Ham 2.5 4.8 2.5
544 Gallagher West Brom 0.1 5.5 2.498
89 Westwood Burnley 0.4 5.3 2.486
65 March Brighton 1.3 5 2.472
253 Fabinho Liverpool 0.9 5.4 2.449
175 Cairney Fulham 0.2 5.3 2.445
512 Doucouré Everton 1.1 5.3 2.394
355 Lundstram Sheffield Utd 2.4 5 2.384
52 Douglas Luiz Aston Villa 0.3 4.9 2.367
98 Brownhill Burnley 0 4.9 2.357
76 Bissouma Brighton 3.2 4.5 2.313
235 Barnes Leicester 3.9 6.9 2.27
346 Fleck Sheffield Utd 0.1 5.6 2.249
107 Kovacic Chelsea 0.3 5.3 2.202
271 Gündogan Man City 0.3 5.4 2.19
148 Walcott Southampton 0.8 5.8 2.153
315 Greenwood Man Utd 2.2 7.1 2.142
133 Kouyaté Crystal Palace 0.6 5 2.141
263 Jones Liverpool 0.7 4.4 2.14
555 Krovinovic West Brom 0 5 2.14
159 Iwobi Everton 0.2 5.9 2.134
382 Djenepo Southampton 0.1 5.4 2.121
9 Xhaka Arsenal 0.5 5.2 2.088
236 Ndidi Leicester 0.6 4.8 2.077
385 Sissoko Spurs 0.5 4.8 2.064
411 Phillips West Brom 0 5.1 2.044
95 Brady Burnley 0.1 5 2.025
502 Allan Everton 0.9 5.3 1.99
400 Bergwijn Spurs 0.4 7 1.977
130 McArthur Crystal Palace 0.1 5.3 1.964
413 Sawyers West Brom 0 4.8 1.944
261 Keita Liverpool 0.2 5.2 1.939
191 Anguissa Fulham 1.9 4.5 1.924
365 Redmond Southampton 0.2 6.4 1.9
501 Ceballos Arsenal 1.1 4.8 1.87
115 Loftus-Cheek Fulham 0.2 5.9 1.832
296 Pogba Man Utd 1.1 7.7 1.826
309 McTominay Man Utd 0.3 4.9 1.818
526 Elneny Arsenal 1.3 4.4 1.814
485 Hendrick Newcastle 2.4 4.8 1.809
405 Ndombele Spurs 0.4 5.9 1.793
540 Traoré Aston Villa 0 5.9 1.783
557 Lookman Fulham 0.6 5 1.753
225 Praet Leicester 0.4 5.5 1.695
205 Costa Leeds 4.9 5.4 1.646
187 Cavaleiro Fulham 0.1 5.3 1.633
142 Schlupp Crystal Palace 0.1 5.4 1.601
349 Norwood Sheffield Utd 0.2 4.7 1.584
138 Milivojevic Crystal Palace 0.2 5.6 1.583
299 Fred Man Utd 0.2 5.3 1.52
543 Bale Spurs 0.5 9.4 1.473
373 Reed Fulham 0.6 4.4 1.403
290 Mata Man Utd 0.2 5.9 1.388
190 Kamara Fulham 0 4.8 1.368
424 Burke Sheffield Utd 1.2 4.3 1.361
106 Barkley Aston Villa 1.6 5.9 1.357
409 Livermore West Brom 0.1 4.8 1.336
33 Hourihane Aston Villa 0.1 6 1.313
322 Ritchie Newcastle 0 4.9 1.31
284 Foden Man City 5.2 6.4 1.307
336 Hayden Newcastle 0.1 4.8 1.297
150 Sigurdsson Everton 0.8 6.8 1.272
497 Murphy Newcastle 0.1 4.9 1.27
281 Bernardo Silva Man City 0.5 7.4 1.262
427 Noble West Ham 0.4 4.7 1.253
518 Mendy Leicester 3.1 4.5 1.228
144 Riedewald Crystal Palace 1.9 4.4 1.215
71 Jahanbakhsh Brighton 0 5.4 1.2
287 Torres Man City 1.9 6.9 1.161
80 Mac Allister Brighton 0 5.3 1.138
20 Willock Arsenal 0.1 4.8 1.103
149 Delph Everton 0 4.9 1.079
229 Pérez Leicester 0.8 6 1.061
439 Lanzini West Ham 0.1 6.4 1.019
495 van de Beek Man Utd 1.7 6.7 1.011
105 Jorginho Chelsea 6.5 5 0.985
493 Lemina Fulham 0.3 4.5 0.94
45 El Ghazi Aston Villa 0.1 5.7 0.882
21 Nelson Arsenal 0 5.2 0.853
258 Minamino Liverpool 0.1 6 0.824
206 Roberts Leeds 0.1 4.8 0.82
541 Ünder Leicester 0.1 5.9 0.808
295 Matic Man Utd 0.2 4.8 0.725
131 McCarthy Crystal Palace 1.2 4.4 0.694
436 Yarmolenko West Ham 0.1 5.6 0.658
90 Gudmundsson Burnley 0 5.4 0.612
158 André Gomes Everton 0.3 5.4 0.589
356 Osborn Sheffield Utd 0 4.8 0.578
507 Fraser Newcastle 0.2 5.6 0.555
338 Saint-Maximin Newcastle 4.2 5.2 0.498
587 Benrahma West Ham 0.2 6 0.473
79 Alzate Brighton 0.7 4.4 0.459
567 Partey Arsenal 0.5 5 0.447
209 Poveda-Ocampo Leeds 0.4 4.4 0.447
55 Stephens Burnley 5.3 4.3 0.434
565 Diallo Southampton 0 4.5 0.355
515 Vitinha Wolves 0 4.8 0.338
154 Bernard Everton 0.1 5.8 0.336
180 Kebano Fulham 0 4.8 0.333
535 Benson Burnley 0 4.5 0.307
394 Alli Spurs 0.6 7.4 0.301
428 Snodgrass West Ham 0.1 5.7 0.292
163 Davies Everton 0.1 5.3 0.275
421 Edwards West Brom 0 4.8 0.275
311 James Man Utd 0.2 6.2 0.204
124 Gilmour Chelsea 0.1 4.4 0.196
44 Nakamba Aston Villa 0.4 4.3 0.189
397 Winks Spurs 0.1 5.2 0.167
550 Molumby Brighton 0 4.5 0.136
122 Hudson-Odoi Chelsea 0.2 5.7 0.113
554 Ramsey Aston Villa 0.1 4.5 0.108
59 Pröpper Brighton 0 4.8 0.104
581 Otasowie Wolves 0 4.5 0.103
423 Field West Brom 0 4.8 0.102
410 Grosicki West Brom 0 5.3 0.091
72 Izquierdo Brighton 0 5.5 0.088
379 Smallbone Southampton 0.2 4.5 0.087
553 Shabani Wolves 0 4.5 0.086
23 Smith Rowe Arsenal 0.2 4.4 0.082
143 Meyer Crystal Palace 0 4.7 0.074
551 Goodridge Burnley 0 4.5 0.071
426 Harper West Brom 0.5 4.4 0.035

For a second assume that we have wildcard enabled, and trying to pick 4 out of these 179 feasible midfields. Under a budget of £28M, could you spot the best pick that will maximize your expected points (xP)?

Salah (12.3M), Podence (5.4M), Raphinha (5.4M), Soucek (4.9M)

The total xP? It is 18.61.

Well, with a limited budget this is what we can do best. I dare you to give it a try, but this is the true optimal solution under £28M.2

Somehow, it feels weird, right?

• Raphinha (5.4M) has an xP of 3.89, but only selected by only 0.5% of all managers.
• Rodríguez (7.7M) has an xP of 3.76, but selected by 22.2% of all managers.3

Total ownership ratio of these 4 players is 40%, with Salah dominating with 32.2% alone.

The picture changes as you would expect when we have more money to spare.

Budget Total xP Players
£24M 16.783 Jota, Ward-Prowse, Podence, Raphinha
£28M 18.615 Salah, Podence, Raphinha, Soucek
£32M 20.625 Salah, Jota, Ward-Prowse, Podence
£36M 21.795 Salah, Aubameyang, Ward-Prowse, Podence
£40M 22.656 Salah, Aubameyang, Son, Ward-Prowse
£44M 24.090 Salah, Mane, Aubameyang, Jota
£48M 24.375 Salah, Mane, De Bruyne, Aubameyang

## Combining Math and Intuition

Analytics methods, including optimization, become powerful weapons at the hands of an expert. If you have an intuition about a particular subject, say FPL, then running these kind of analysis can only make you a better manager. Too often we have to simplify our decisions to “should I buy X and Y, or Z and W?” in the game. There are too many options out there, and we are trying to reduce our options to 2 or 3 when giving a final decision. Often, this is what happens when you are making other decisions, like buying a car or choosing a new phone, too. Since it is much better to use the correct tool at correct task, optimization can help us to find hidden insights.

Let me go back to the original discussion. What if there is something other managers know and you don’t?

Suppose you are running an optimization already like I mentioned, but do not think Aubameyang is the right choice when you have only £36M to spend on 4 midfield players. Why is that? It is possible that you share the feelings of 91% of managers: perhaps other popular assets worth more in your opinion despite what numbers tell you.

It is indeed quite easy to involve the common belief of other 7 million FPL managers into your own optimization model. The original objective was simply this:

$$\sum_{e \in E} \text{pick}_e \cdot \text{xP}_e$$

Here, E is the set of all players (elements), pick is a binary value, either takes 1 or 0, depending on you do or do not have the player, respectively. xP is the expected points of players. This sum gives you the total objective I have mentioned above.

Assume that you are playing against a single person (consider your mini league, for example). You can calculate the points difference as

$$\sum_{e \in E} \text{pick}_e \cdot \text{xP}_e - \sum_{e \in E} \text{opp}_e \cdot \text{xP}_e$$

In this one, denote opp as a binary multiplier whether your opponent has the player e or not.

For a second, assume all FPL managers are united against you! Imagine them as single boss you need to defeat :) The objective can now be written as

$$\sum_{e \in E} \text{pick}_e \cdot \text{xP}_e - \sum_{e \in E}(1-\text{pick}_e) \cdot \text{own}_e \cdot \text{xP}_e$$

The second term here becomes how much you are being penalized compared to other players. Here, own is the percentage ownership of a player. At the extreme, if no one has a particular player, then you cannot lose anything by not choosing him. At the other extreme, if everyone has a player and you don’t, you are going to be at a disadvantage as much as xP of that particular player.4

## Deciding strategy

The final piece of the puzzle is to re-running optimization model with this objective. This means that the optimization algorithm will bring you closer to what is called template as much as possible.

Instead of running it directly, let us put a weight parameter for how much you believe other managers know the game.

• Weight=0 means that you don’t value other manager’s picks at all, so you have no faith on others.
• Weight=1 means that you value their opinions as much as your predictions for this week.
• A significantly higher value means you value their opinions more than your own predictions for the week. You can play the GW really aggressive (weight=0) or really passive and safe (weight >> 1)5

Here is how the optimal solution changes:

On the other side of the coin, our squad gets closer to the “template”. This means our decision is more “passive” as we will less likely to get affected by wild swings.

Right at this point is where your expertise should come into play. Depending on your current rank, you might want to play it safe, and go for a moderate or very passive approach. Instead of overall ownership, you can use ownership ratios of top 1000 players, or your FPL mini-league, depending on what you are trying to achieve. If you are in need for a good jump in your rank, you can take the risk and pick the “differentials”. Of course if you fall, you might fall hard.

If you are going to remember 2 things from this post, remember these:

1. Optimization is a great tool that you should have under your belt to gain an edge in FPL.
2. It is not surprising that more budget gives you more freedom, hence expected points. However, it is equally important to trust your own instincts. As you get closer to template, you are playing it safe, meaning that you will probably stuck with your current rank.

## What’s next?

While you are here, let me point out that I’m keeping a website for those wondering what is at the end of the spectrum. “FPL Optimized” (as I call it) gives you the best 11 and squad picks for the week, purely on expected points from FPL Review, updated every day:

https://sertalpbilal.github.io/fpl_optimized/

A final note: Quite a bit number of FPL websites claim they do “optimization”, but most of the time they are simplifying and approximating without actual mathematical optimization behind, as it tends to be computationally expensive. If FPL community is interested, next time I can write about ways to run mathematical optimization on tools you commonly use, such as MS Excel or Python with open-source packages.

Until next time, I wish all of you luck!

1. I’m reading Chris Anderson and David Sally’s book “The Number Game” nowadays. It is a great book that I strongly suggest. ↩︎

2. I have the optimization formulation available for this simple problem. Reach out to me if you are interested. ↩︎

3. Of course Rodríguez’s upcoming GW points are better, which could explain the higher pick rate. Still, 0.5% vs 22.2% is a huge difference. ↩︎

4. Here, we are using what is known as multi-objective optimization. We are combining two different objectives into a single one. ↩︎

5. You can also choose a negative weight, meaning that you think everyone is wrong! Might be a good way to differentiate your team from others actively. High risk / high reward. ↩︎