Toilets and Triple Captains

The Mathematics of the Best Toilet

Imagine you are at an outdoor event, such as a music festival, and in line for the toilets (or portapotties, add wet would say in Canada). You have had bad experiences with toilets in the past, and want to make sure you choose the best one you can. There is a queue, so you cannot pass up a toilet and then return to choose it.

At these events, the state of the toilets can vary by event and time, there isn’t a single “average toilet” you can use for comparison. A decent toilet one trip, may be a terrible one the next trip, or even a great one. Given the number of people using and, hopefully, servicing the toilets, every trip to the toilets will have a different value for “best” and “worst”. So how do you maximize your chance of selecting the best toilet?

The most simple approach would be to look at a toilet and think, “Am I ok using this toilet?”. This is an acceptable approach, but instead of maximizing your chance at the best toilet, you are minimizing your chance at a bad toilet (this is still a valid approach if that is your desired outcome).

If you want to maximize your chances at the best, or near best, toilet, you first need to establish the range of toilet quality you are dealing with.

Given the variable nature of the toilets, this is not absolute and must be determined every trip and the only way to do this is to look at, and pass up, toilets. If you are at a small event with only three toilets, you could look in the first one and then choose the next best one you see.

With six possible permiantations, using this approach would have you selecting the best one 3 times, the second best 2 times, and the worst one only once. A 50% success rate of picking the best toilet is pretty good. However, as you add toilets, the system falls apart quickly as the single data point is not enough to estimate the quality an increasing number of toilets.

To account for more toilets, you could start looking at, and not choosing, more toilets to get a better idea of the range of toilet quality. However, as you discard more and more toilets, you run the risk of discarding the best getting stuck with the last one. An ideal number to maximize your probability of choosing the best toilet must exist, and fortunately, mathematicians have taken the time to figure it out.

The ideal approach is to look at, but not choose, 37% of the toilets, and then pick the first one that is better than the best of the ones you looked at. This approach will get you the best toilet 37% of the time, while also minimizing the amount of times you get a terrible one.

The Triple Captain Chip

As FPL managers, we are faced with the decision, “which gameweek should I play my triple captain chip”. Putting aside ‘chip strategies’ and other complications, how do we determine the best week to play our chip? We are likely to look at a variety of factors when making the decision:

Player skill

Player form

Player fitness

Strength of opponent

Likelihood of minutes/rotation

Timing of the game

Some of these factors will remain constant throughout the season, while others will be variable week to week. The best information we have are for the gameweeks gone by with decent knowledge of the upcoming gameweek. Beyond that, many factors become uncertain.

In both toilet choosing and triple captain chip use, you are moving forward, evaluating the past options, and deciding whether or not to make the choice right now, or move to the next one.

We will get to the unique aspects of FPL in a bit, but let’s continue with this simplification.

Running the math, 37% of 38 [gameweeks] is approximately 14. So based on the math, the best approach would be to wait until after gameweek 14 and then choose the next best triple captain shout. While your personal choices likely differ, the most popular captain choices so far this season:

GW1: Salah home to NOR

GW2: Salah at SOU

GW3: Sterling at BOU

GW4: Sterling home to BHA

GW5: Salah home to NEW

GW6: Sterling home to WAT

GW7: Salah at SHU

GW8: Sterling home to WOL

GW9: Aguero at CRY

GW10: Sterling home to AVL

GW11: Sterling home to SOU

GW12: Abraham home to CRY

GW13: Vardy at BHA

GW14: Vardy home to EVE

Based on the math (and simplified FPL), you should use your triple captain chip in the next gameweek that is better than those 14 captain shouts.

One quick note to make, because we do not know the results of the future matches, we should not evaluate past captaincies on the results, instead we should use the pre-game expectations.

Salah (following a goal and an assist) rolling into Southampton (who lost 3-0 to Burnley the week before) in gameweek 2 was a good captain choice, regardless of the three points he ended up getting. We can use the results to help form our future evaluations, but we need to be careful to focus on expectations to keep the comparisons consistent.

The Key Differences

Of course, FPL is a bit more complicated.

First of all, there is little variability in captain quality from season to season. While players come and go from out captain choices, the expected points and strength (or weathness) of opponents aren’t that variable season to season.

The doubles also impact things significantly. Going back to the toilet analogy. Let’s say you have a friend, we will call him Ben, that says he used the toilets earlier in the day and the 37th one was a great one. Also we had heard that the 34th, “or was it the 28th?”, was good too, but he wasn’t sure how recent the information was.

So there you are at the 24th toilet thinking it looks pretty good, but you remember Ben mentioning these future toilets and it had you thinking it may be worth waiting. But you are also worried that maybe the last couple of hours has not been kind to the toilet and it isn’t worth your time now. This better represents the challenge facing FPL managers.

If we are looking for the best potential to get a big return from our captain, double gameweeks seem like a solid choice.

While the opponents may not be the “ideal” ones, having twice the number of games greatly increases (doubles?) the chances of a return. With the double gameweeks, you are typically accepting higher rotation risk and stronger opponents, but getting the opportunity of a higher floor (maybe 4 points for just showing up) and ceiling.

The biggest question is, “Does the toilet math(s) still work if we account for future double gameweeks?”.

I don’t think this question has been solved, but I have my thoughts…

With the knowledge that there will be a double gameweek at some point in the season, I would be hesitant to use my triple captain chip in the first better gameweek after gameweek 14. I would use it in the first double game week (while still maintaining the “better than past captains” to rule out a bad captain shout) after 14. In fact, I would be tempted to use the chip in the first decent double game week regardless of when it falls.

You may ask, “but what if there is a better week in the future?” This is a risk, and a risk I am willing to take. Let’s go back to our list of captain picking factors:

Things that won’t change:

Player skill

Things that vary week to week or day to day:

Player form

Player fitness

Strength of opponent (not highly variable, but double gameweek opponents are not set)

Things dependant on outside factors:

Likelihood of minutes/rotation

Timing of the game (league wrapped up? Champions/Europa League?)

Double Gameweek 24

So here we are. We have just moved past the 23rd toilet (based on some of the gameweek scores, this is an appropriate comparison), and we stand at the 24th with the door open, do we take it?

I see two games for one of the best attacking and defending teams with some of the top FPL assets in good/great form. No known fitness concerns and the league lead is comfortable, but not certain. The only negatives I see are:

Two away fixtures against medium teams (I put WOL in top 5 home defenses and WHU in bottom 5)

Limited rest days before and after the gameweek (games on: Jan 19, Jan 23 , Jan 26, Jan 29 , Feb 1)

I look at what we know and I struggle to see a better gameweek in the future. There are some prime fixures for big teams, but the uncertainty of the double, player form, player fitness, motivation, etc. are too many unknowns for me.

A triple Manchester City week sounds tempting, but the league may be wrapped up and Champions League Quarter and Semi finals may be happening, plus fitting three games into a week will be prime for resting/rotation.

Wrapping things up…

In conclusion, the 24th toilet seems like a pretty good toilet. There may be a better one out there, but I will regret not taking this one more than missing out on the slightly better one.

I say go with the math and play your chip. Now I just need to decide on which Liverpool asset I am choosing.

You can check out our Double Gameweek 24 Captain Metric to help you decide that…

If you are interested in the math behind this, check out: https://youtu.be/ZWib5olGbQ0