After the 100 day shutdown of football and consequently the Fantasy Premier League competition the return to football was greatly anticipated. Even without fan presence at games people were desperate for the return of the beautiful game. The return of football was interesting as allowances had to be made so that the season could be finished as swiftly and as uninterrupted as possible. This resulted in a ‘festival of football’ in which there was games almost every day for six weeks. To account for the delay and how the season has changed over the last 100 days Fantasy Premier League (FPL) allowed unlimited free transfers before the first game of the return. This, of course, was a major opportunity.

足球停工100天后，幻想超级联赛比赛结束了100天，人们重新期待了足球的恢复。 即使没有游戏迷的参与，人们也渴望获得精美游戏的回归。 足球的回归很有趣，因为必须进行补贴，这样才能尽可能快地完成赛季。 这导致了“足球节”，其中几乎每天都有六周的比赛。 考虑到延误以及过去100天内的赛季变化，幻想英超(FPL)允许在回程的第一场比赛前无限次免费转会。 当然，这是一个重大机会。

There is no difficult maths involved in this, it’s something anyone with a very basic understanding of any programming language (even excel) could do in an afternoon.

不需要复杂的数学运算，对任何编程语言(甚至是excel)都有非常基本了解的任何人都可以在下午进行。

游戏的当前状态 (The current state of the game)

On the return to FPL there were nine game-weeks remaining with the first being a double gameweek for four teams (Arsenal, Aston Villa, Sheffield United, and Manchester city) which means they all play twice. The double gameweek arose out of the need to get these four teams on to the same amount of fixtures as the rest of the league. This confused things slightly but also presented an opportunity.

回到FPL后，剩下的比赛时间为九周，第一周是四支球队(阿森纳，阿斯顿维拉，谢菲尔德联和曼彻斯特城)的两场比赛周，这意味着他们都上场两次。 双人比赛周的产生是出于使这四支球队拥有与联盟其他球队相同的固定装置的需要。 这使事情有些混乱，但也带来了机会。

Therefore, I had one main objective and that is to simply maximise the points in the remaining nine gameweeks. The double gameweek complicates things as there is a trade-off between picking players in the four teams mentioned above to maximise points in the first week but then having to potentially transfer them out afterwards. For example, Aston Villa players who sit 19th are not a good choice going forward but are likely going to be profitable in the double gameweek. This is a problem as FPL only allows one transfer per week without taking a points deduction. Transfers can be saved and added cumulatively, but only a maximum of two transfers can be held at the same time.

因此，我有一个主要目标，那就是在剩余的九个游戏周内简单地最大化积分。 双重比赛周使事情变得复杂，因为上述四支球队的选拔球员之间需要权衡取舍，以在第一周最大化积分，但随后必须将他们转移出去。 例如，坐在第19位的阿斯顿维拉球员不是前进的好选择，但可能会在双倍游戏周中获利。 这是一个问题，因为FPL每周只允许一次转账，而不会扣除积分。 可以累计保存和添加传输，但是一次最多只能保存两个传输。

Therefore the main objective I had was to maximise the total points over the remaining 9 gameweeks while taking advantage as much as possible the opportunity that the double gameweek presented.

因此，我的主要目标是在剩下的9个游戏周内最大化总积分，同时尽可能多地利用双游戏周带来的机会。

理论 (The theory)

The logic is as follows, if we maximise return on investment (ROI) within our budget constraint (can you tell I’m an Economist?) and use our full budget (or as close to as possible) we will maximise points in the long run.

逻辑如下，如果我们在预算限制内(您能告诉我我是经济学家)使投资回报率(ROI)最大化，并使用全部预算(或尽可能接近)，则我们将长期跑。

You can think about it in terms of two potential teams (Team A and Team B) with identical budget constraints of £100m. If we calculate the ROI for both teams such that the ROI of team a is greater than the ROI of team b:

您可以考虑两个潜在的团队(团队A和团队B)，预算约束为1亿英镑。 如果我们计算两个团队的ROI，以使团队a的ROI大于团队b的ROI：

Which shows that the if we maximise our budget, the team with the highest ROI will be the team that produces the highest points. With a few caveats that we will discuss shortly.

这表明，如果我们最大化预算，则ROI最高的团队将是得分最高的团队。 我们将在短期内提出一些警告。

守门员 (Goalkeepers)

Therefore, getting stuck in and downloading the data we have a data set which looks like this (Arsenal players are there due to nothing more than alphabetical order). The only real edits I had to make was to play around with the indexes and create some new variables. This example shows the head of the data set (the first five rows).

因此，卡住并下载数据后，我们得到了一个看起来像这样的数据集(阿森纳球员在那里只是出于字母顺序)。 我唯一需要做的真正编辑就是使用索引并创建一些新变量。 本示例显示了数据集的头(前五行)。

I decided to start off with choosing a goalkeeper as it tends to be a position that isn't changed much. This actually turned out to be a bit of a disaster but I feel like my methodology was sound — I was just unfortunate. Luckily the rest of the team made up for it. So as mentioned, the main stat I was going to look for in players was ROI.

我决定从选择一个门将开始，因为它的位置变化不大。 事实证明，这确实是一场灾难，但是我觉得我的方法学不错–我只是不幸。 幸运的是，团队的其他成员对此进行了弥补。 因此，如上所述，我要在球员中寻找的主要数据是投资回报率。

Therefore, I simply organised players by ROI and highlighted the top players.

因此，我只是按投资回报率来组织球员，并突出显示了顶尖球员。

As can be seen, Nick Pope has the highest ROI of any player let alone goalkeepers. His only potential rival is Dean Henderson. There was an argument for Dean Henderson as he does play for one of the teams that starts with a double gameweek. However, the following gameweek Sheffield play against Manchester United — the club who have loaned him to Sheffield — and is therefore ineligible to play in this game. Therefore, the double gameweek advantage is cancelled out and I decided to stick with Nick Pope as he has a superior points total as well ROI. Just for clarification I also graphed each teams remaining fixtures in terms of difficulty. I calculated the average current league position of every teams remaining opponent as can be seen below.

可以看出，尼克·波普的投资回报率是所有球员中最高的，更不用说守门员了。 他唯一的潜在竞争对手是迪恩·亨德森。 迪恩·亨德森(Dean Henderson)确实存在争议，因为他确实为一支以双周比赛开始的球队效力。 但是，在接下来的比赛周谢菲尔德对阵曼联的比赛中(曼联已将他借给谢菲尔德)，因此没有资格参加这场比赛。 因此，双周比赛的优势被抵消了，我决定坚持尼克·波普，因为他的总积分和投资回报率都很高。 为了澄清起见，我还按照难度绘制了每个团队剩余的固定装置。 我计算了剩下的每支球队的平均当前联赛排名，如下所示。

As can be seen, Manchester United on paper have the easiest fixtures of any team with an average position of 12th while Bournemouth have the most difficult. This is obviously not perfect as some mid table teams have little to play for and some teams at the bottom are fighting for their lives. As can be seen, Burnley’s remaining fixtures are slightly easier than Sheffield’s which bodes well for Nick Pope again. Therefore, the final decision to make him the first player in the team was a relatively easy one.

可以看出，曼联在纸面上的得分是所有球队中最简单的，平均排名为第12，而伯恩茅斯则是最困难。 这显然是不完美的，因为一些中端桌的球队几乎没有比赛可玩，而一些底层球队则在为自己的生命而战。 可以看出，伯恩利剩下的固定装置比谢菲尔德的固定装置要容易一些，这对尼克·波普来说又是一个好兆头。 因此，最终决定让他成为团队中的第一名球员是一个相对容易的决定。

长椅 (The Bench)

Due to the fact that the first week was a double gameweek, I decided to use my bench boost chip to double down on the gameweek possibilities. For anyone who doesn't know, the bench boost chip is a token that you can use once per season that allows you to gain points for your bench players as well as your starting 11. The issue with the bench is that after this gameweek they are unlikely to make it to the first team (unless mass injuries occur) and therefore it wouldn't be wise to spend a lot of money on them. I therefore set three main criteria for the players I would chose for the bench. Firstly, that they were in double gameweek teams, secondly that they were averaging above 60 minutes a game and finally that they were less than £5 million. This would give a robust bench of guaranteed starters. With four bench players both playing twice, we would expect a minimum of 16 extra points plus any other points they achieve over their combined eight games (a player receives one point for making an appearance and another for playing at least 60 minutes as well as big bonuses for goals, assists and clean sheets). I therefore, filtered the data set to players below five million pounds and to players who are playing at least 60 minutes a game on average . This also turned out to be an easy decision as in the top 10 there was only three players who fit into this category (Reina, Lundstrum and Mings). These players were then added to the team (bench) while I decided to keep the last bench spot free for some flexibility at this stage.

由于第一周是一个双周比赛的事实，我决定使用板凳助推器芯片来提高游戏周的可能性。 对于任何不知道的人来说，替补席筹码都是一个令牌，您每个赛季可以使用一次，这样您就可以为替补席球员和开始的11点获得积分。替补席的问题是在本周比赛之后，不太可能进入一线队(除非发生大规模伤病)，因此花很多钱在他们身上是不明智的。 因此，我为选择替补席的球员设定了三个主要标准。 首先，他们参加了两周的比赛，其次，他们场均上场时间超过60分钟，最后，他们的收入不到500万英镑。 这样可以保证有充足的启动器。 如果有四名替补球员都上场两次，我们希望他们至少能获得16分的加分，再加上他们在总共八场比赛中所获得的任何其他分值(一名球员出场获得1分，而出场至少60分钟又获得另一分)进球，助攻和床单奖金)。 因此，我将数据集过滤到了500万英镑以下的球员以及平均每场比赛至少60分钟的球员。 事实证明，这也是一个容易的决定，因为在前10名中，只有三名球员(Reina，Lundstrum和Mings)适合该类别。 然后，我将这些球员添加到了团队中(长凳)，而我决定在此阶段保留最后一个长凳的位置，以提供一些灵活性。

捍卫者 (Defenders)

I followed an almost identical approach to what I had done before, I found the list of the players with the highest ROI and the easiest upcoming fixtures. The results here were strange with the three star Liverpool defenders (Van Dijk, Robertson and Trent Alexander-Arnold) coming out on top in terms of ROI. They however, are all extremely expensive (>7m) so at most it would make sense to choose two of them and possibly only one. Referring back to the fixtures graph it can be seen that Liverpool don’t exactly have the easiest fixtures and at this point had effectively won the league. Due to fear of rotation I decided to only go for one Liverpool defender. I chose Virgil Van Dijk due to him simply having the highest ROI. I followed on this by also choosing Doherty from Wolves as he was the best of the rest as well as Harry Maguire due to his reasonably high ROI, the fact that he had played every minute of every game, and Manchester United’s easy fixture list.

我采用了几乎与以前相同的方法，发现了投资回报率最高且即将推出的灯具最简单的球员。 利物浦三星级后卫(范迪克，罗伯逊和特伦特·亚历山大·阿诺德)在ROI方面名列前茅，结果令人惊讶。 但是，它们都非常昂贵(> 7m)，因此最多选择两个，也许只有一个是有意义的。 回顾赛程图，可以看出利物浦并不具备最简单的赛程，并且在这一点上已经有效地赢得了联赛冠军。 由于担心轮换，我决定只选一名利物浦后卫。 我之所以选择Virgil Van Dijk，是因为他的投资回报率最高。 接下来，我还从狼队中选择了多赫蒂(Doherty)，因为他是其他人中最好的；由于他的高投资回报率(ROI)，他每场比赛的每一分钟都打过比赛以及曼联的简易赛程名单，哈里·马奎尔(Harry Maguire)也是。

明星球员(队长和副队长) (Star players (Captains and Vice Captains))

The only problem with the ROI model is that in FPL you have a choice of one player to be your captain who receives double points and one to be the vice captain who receives the armband if your captain doesn’t play any minutes that week. Therefore, having a team of mid-low value players all with high ROI might not be optimal since it will not take advantage of the double points. Therefore, I decided to have at least two players who were high value, high return players that would be captain choices each week and sacrifice a little of the budget.

ROI模型的唯一问题是，在FPL中，如果您的队长那一周没有上场任何时间，您可以选择一名球员作为获得双倍积分的队长，以及一名将获得臂章的副队长。 因此，拥有一支具有高投资回报率的中低价值球员团队可能不是最佳选择，因为它不会利用双倍积分的优势。 因此，我决定至少要有两名高价值，高回报的球员，这些球员每周都会成为队长的选择，并且会牺牲一些预算。

Therefore, I sorted the data to show the ROI and total points for only the 15 top points scoring players.

因此，我对数据进行了排序，以仅显示15个得分最高的球员的投资回报率和总得分。

Therefore, looking at the top 15 point scorers in terms of ROI, we already have the top 3 players in the team, which is reassuring. We can also exclude the other defenders as we have sorted the defence already. I made the decision to cut Aubamayang and Firminho immediately as their ROI is significantly worse than everyone else and their total points isn't high enough to compensate. I also did not feel that Rashford’s, Richarlison’s or Abraham’s total points were high enough so they were also removed. This left Vardy, Mane, Salah, and De Bruyne. De Bruyne was the first pick and an easy one. In terms of fixtures we have already seen that city had extremely easy fixtures, he also has second highest ROI and total points. City were also one of the double gameweek teams therefore he is an ideal pick for captain in the first week back. I also eliminated Mane from the choice at this point due to the fact that he had lower total points and ROI than Salah and of course both play for Liverpool.

因此，按照投资回报率排名前15名的得分手，我们已经拥有团队中排名前三的球员，这令人放心。 我们也可以排除其他防御者，因为我们已经对防御进行了排序。 我决定立即裁掉Aubamayang和Firminho，因为他们的投资回报率明显差于其他所有人，而且他们的总积分还不够高，无法弥补。 我也不认为拉什福德，里查利森或亚伯拉罕的总分足够高，因此也被删除了。 这留下了瓦尔迪，鬃毛，萨拉赫和德布鲁因。 德布吕恩(De Bruyne)是首选，也是一个轻松选择。 在固定装置方面，我们已经看到城市拥有极其简单的固定装置，他的投资回报率和总积分也位居第二。 曼城也是每周比赛的两支球队之一，因此他在首周就成为队长的理想之选。 由于他的总积分和投资回报率均低于萨拉赫，因此我也从选择中剔除了曼恩，当然他俩都为利物浦效力。

I couldn't make my mind up about who to chose between Vardy and Salah at this point and decided to carry on with the rest of the team and come back to this choice later when It was clearer about how much budget I would have left.

此时我无法决定在Vardy和Salah之间进行选择，并决定继续与团队中的其他成员合作，等到我能剩下多少预算时再回到这个选择。

离群值和隐藏的宝石 (Outliers and hidden gems)

As I’ve already mentioned, this analysis is very straight foreword. I was also concerned that it may overlook some players. For example, players that came in January (I’m sure all premier league fans can think of one player who has done particularly well after arriving in January) as well as players who have longstanding injuries who are now fit again after the break. Therefore, I decided to try and source these players.

正如我已经提到的，此分析是非常直接的前言。 我还担心它可能会忽略一些球员。 例如，一月份到来的球员(我确定所有英超联赛的球迷都可以想到一位在一月份到来后表现特别出色的球员)，以及长期受伤的球员，现在在休息后又能恢复健康。 因此，我决定尝试寻找这些玩家。

I also wanted to look for players who weren't common picks, players that were performing well but just under the radar. This is beneficial because if these players do succeed and little or nobody else has them then it’s really beneficial. In FPL (especially towards the end of the season) players teams start beginning to look awfully similar to each others. People at the top of the table often attempt to mirror players below them so it makes it impossible for them to be caught.

我还想寻找不常见的选秀球员，表现不错但在雷达之下的球员。 这是有益的，因为如果这些参与者成功并且很少或没有其他人拥有他们，那么这确实是有益的。 在FPL中(尤其是在赛季结束时)，球员团队开始看起来彼此非常相似。 桌上最上方的人经常试图镜像下方的玩家，因此不可能被他们抓住。

Firstly, I created a new metric ROM (return on minutes) which is the same as ROI but instead of investment its how many points they get per minute — pretty self explanatory. This is useful in conjunction with ROI for the reasons previously mentioned, a player might have only played three or four games so their total points and resulting ROI will be really low. They may however have played extremely well in those three of four games and ROM will capture this. If a player has a high ROM and is expected to get a lot more game time than they previously have (recovered from injury etc.) then they may be a good choice. Therefore, I gathered the top 15 players in terms of ROM.

首先，我创建了一个新的度量ROM(分钟返回)，该ROM与ROI相同，但是与其投资而不是投资每分钟获得的点数，这很容易解释。 由于前面提到的原因，这与ROI结合使用非常有用，因为玩家可能只玩了3或4场游戏，所以他们的总积分和产生的ROI确实很低。 但是，他们可能在四场比赛中的三场中表现非常出色，ROM将抓住这一点。 如果玩家的ROM高，并且预计将获得比以前更多的时间(从受伤等状态中恢复过来)，那么他们可能是一个不错的选择。 因此，就ROM而言，我排名前15位。

As can be seen there is some of the usual suspects there but also some new names that haven't appeared on many lists so far. A fair few of these players are substitute players who are going to continue getting the same minutes and are not worth considering (Ighalo, Batshuayi etc). There is one player who stands out (you might have guessed from the previous hint) — Bruno Fernandes. He was certain to play every game going forward and his ROM was not inflated by very scarce substitute appearances as he already had around ten games under his belt.

可以看出，那里有一些通常的嫌疑人，但到目前为止还没有出现在许多名单上的一些新名字。 这些球员中有相当一部分是替代球员，他们将继续获得相同的上场时间，因此不值得考虑(Ighalo，Batshuayi等)。 有一个选手脱颖而出(您可能已经从前面的提示中猜到了)—布鲁诺·费尔南德斯。 他肯定会继续进行每场比赛，并且他的ROM并不会因为很少出现的替补出场而膨胀，因为他已经有大约10场比赛了。

I also created similar lists for other factors such as time played. I created a top ROI for players who played almost every minute of the league so far, so called indispensable players — players that when fit will play the full 90 and rarely get injured. These would act as stable players in the squad that would not have to be rotated often. Players I could rely on to gather a good haul of points each week who I didn’t have to worry about them not playing.

我还为其他因素(例如比赛时间)创建了类似列表。 我为迄今为止几乎在联赛中每一分钟都在比赛的球员创造了最高的投资回报率，这些球员被称为必不可少的球员-体形好的球员将打满90杆而很少受伤。 这些将成为球队中稳定的球员，而不必经常轮换。 我可以依靠的球员每周收集大量积分，而我不必担心他们不参加比赛。

I created a similar list with players that have been picked by less than 10% of people. As mentioned this was a good technique to benefit from hidden gems. Neil Maupay was an outstanding choice from this list and Brighton have some easier fixtures earlier on. Maupay was added to he team alongside Fernandes and Mahrez (high ROI, extremely high ROM and great fixture list).

我创建了一个类似的列表，其中被不到10％的人选中的玩家。 如前所述，这是一种受益于隐藏宝石的好技术。 尼尔·莫帕(Neil Maupay)是该列表中的绝佳选择，布莱顿(Brighton)早些时候也有一些简单的装置。 Maupay与Fernandes和Mahrez一起加入了他的团队(高ROI，极高的ROM和出色的灯具清单)。

填补空白 (Filling the gaps)

At this point I decided that Salah was going to be too expensive considering who I had just added and the likelihood of rotation with Liverpool (this was proved correct as Salah was rested a few times in the run out). So I decided to add Vardy to the Team (he didnt last long in the team after a poor run of form). I complemented him with Rual Jiminez, the next highest ROI forward. I also added John Fleck as he was a low cost-high ROI player.

在这一点上，考虑到我刚刚添加的人以及利物浦的轮换可能性，我认为萨拉赫将变得太昂贵了(事实证明，萨拉赫休息了几次，这是正确的)。 因此，我决定将Vardy加入团队(在状态不佳的情况下，他没有在团队中维持很长时间)。 我以Rual Jiminez(他的投资回报率第二高)作为他的补充。 我还添加了John Fleck，因为他是低成本高投资回报率的公司。

With only one space left I created a very simple function that took a price ceiling and a price floor as inputs and produced the players in this bracket with the highest ROI. Luckily Adama Traore fit perfectly into this with a really high ROI of 1.89. He was therefore the final piece of the jigsaw and his price maximized the budget. The final team can therefore be seen below. The team here probably didn’t maximise ROI as mentioned earlier. However, it maximised ROI at every possible point without making clear and obvious (shoutout to VAR) errors.

仅剩一个空间，我创建了一个非常简单的功能，即以价格上限和价格下限作为输入，并在此支架中产生了具有最高ROI的播放器。 幸运的是，Adama Traore非常适合此应用，其1.89的很高的投资回报率。 因此，他是拼图的最后一部分，他的价格使预算最大化。 因此，最终的团队如下所示。 正如前面提到的，这里的团队可能没有使ROI最大化。 但是，它可以在每个可能的点上最大化ROI，而不会产生明显的错误(对VAR的喊叫)。

决赛队伍 (The final team)

As can be seen the final team and their gameweek scores can be seen above. The only major disappointment was Pepe Reina, who didn’t start either of the two games for villa and a younger goalkeeper was chosen instead of him. This is something that couldn't have been predicted and was just unfortunate. Pope also blanked as City gave Burnley a good old thumping but Pope bounced back and proved to be a beast for the remainder of the season.

可以看出最终球队和他们的比赛周得分可以在上面看到。 唯一令人失望的是佩佩·雷纳(Pepe Reina)，他没有参加维拉的两场比赛，而是选择了一位年轻的门将代替他。 这是无法预料的，​​只是不幸的。 教皇也蒙混了，因为曼城给了伯恩利一个不错的老old，但教皇反弹了起来，并被证明是本赛季剩余时间的野兽。

结果 (Results)

The final points were more than double the average and resulted in the team being in the top 1% (of more than 7 Million players) for the week. I feel that I could have increased this point total more in this individual week by making other choices but I didn’t feel I could do it without sacrificing the team going forward. So a result of 118 points with a team that is also heavily influenced by upcoming fixtures is, I feel, a strong result.

最终得分是平均水平的两倍还多，这使得本周(超过700万名球员)的球队排名前1％。 我觉得我本可以通过做出其他选择来在本周内增加更多的积分，但是我觉得在不牺牲团队前进的情况下我无法做到。 因此，我认为一支同样受到即将来临的比赛影响很大的球队的得分为118分，这是一个不错的成绩。

Throughout the remaining weeks I totalled 676 compared to the average 482 points which is more than 40% higher than average.

在剩下的几周中，我总计676分，而平均得分为482分，比平均水平高出40％以上。

向前走 (Going forward)

The next step is to design a proper selection algorithm centred around ROI and calculating expected goals and clean sheets for teams each gameweek. I feel like a selection algorithm which constantly transfers players in who have the highest ROI (moving average to account for form) which is also includes a weighting factor that controls for fixture difficulty. I plan to try and build this before the next season starts. I think this will take a little longer than an afternoon this time..

下一步是设计一个以投资报酬率为中心的适当选择算法，并在每个游戏周为团队计算预期目标和清理表。 我感觉就像一个选择算法，该算法不断转移具有最高ROI(移动平均值以解决形式问题)的球员，其中还包括一个控制夹具难度的加权因子。 我计划在下个季节开始之前尝试构建它。 我认为这将比这次下午花费更长的时间。

I hope you enjoyed the read and please reach out if you have any questions or would like to follow or see any of the notebooks.

希望您喜欢阅读，如果有任何疑问或想关注或查看任何笔记本，请与我们联系。