The Mathematics of Progress

Or if you prefer, The Progress of Mathematics

Well, I had a nice long post all typed up, and ready to submit, when the Failbetter Games Community forum decided to log me off. Apparently, for no reason other than I had spent a good deal of time typing the post up. Of course there was no indication I had been logged off until it just decided not to allow me to create the thread.

Amused, I was not.

So in the interest of not being logged off in mid-posting again, a much more concise and far less in-character post.

To determine the number of change points to reach a certain level, when you currently have no levels in a quality:

b/2[/b]

Where Level is the desired level of the quality.

But if you already have some levels in the quality, you can instead use:

b*(LAST-FIRST+1)/2[/b]

Where FIRST is the level you will achieve… first (i.e. current level + 1) and LAST is the desired level (functionally identical to Level in the prior equation).

So if you want 20 in a quality, but currently have zero, you would need:

(20^2+20)/2
210

But if you already had a 15 in the same quality it would only take you:

(16+20)*(20-16+1)/2
90.

Hopefully this will help my fellow Fallen Londoners to be more scientific in their Running Battles and Casings.

Note: The Change Points needed to raise a level seem to cap at fifty (at level… fifty). I have never actually gotten any quality past fifty other than Attributes, so that may actually just be a quirk… I mean property… of Attributes.

Edit: It is possible that my being logged out was not FGC’s fault, as apparently Facebook had decided to log me out at the same time. My anger can still be accurately characterized as ‘definitely more than none’, I am merely uncertain where to direct my wrath.

ReEdit: It would seem I was logged out of just about everything all at once, including Google Docs and Wikidot, which kinda points to Chrome as the culprit.

ReReEdit: Apparently the problem was a corrupted cookies file, which has now been fixed.
edited by Abraham Bounty on 1/8/2012

I’m afraid I don’t understand any of this… but then again, I really really suck at math…

Well, so long as you understand that I was logged out mid-post, apparently by my browser, that’s really the important bit.

Yeah, that’s the bit I understood ;)
Happened to me once, too… but that’s why, when writing really long posts, I always copy everything I’ve written before I hit “reply”. Safety first…

A man of science, and a mathematician at that, not bad, not bad indeed. This research will certainly save me a lot of time, as I usually do all sorts of little calculations while playing, related to qualities of one sort or the other, it will indeed be much quicker than doing “1+2+3+4+5+…”. For that you have my gratitude, I hope to be able to repay the good deed at some point.

Perhaps I can start by adding that I’m pretty sure connections do not cap at 50 (I will check to make sure once my actions recharge).
edited by Malt Jones on 1/8/2012

My original post detailed how I came up with this while awaiting my turn at the family computer and that it was a variation of an (until now useless) theorem I came up in high school that proved that consecutive odd numbers starting at 1 will always add up to a perfect square.

It also contained speculation that a Casing of 100 might be enough for stealing Fallen London from the masters as a straightforward challenge.

Yes, Chrome has taught me that lesson. From now on when I need to do a post of more than 10 lines or so, I’ll post the ten lines, then edit in the subsequent batches afterwords.

I look forward to your findings with great curiosity and eagerness.
edited by Abraham Bounty on 1/8/2012

Connections do not cap at 50. What does cap at 50 is the number of change points required to advance a trait. From 45 to 46 will require 46cp. From 55 to 56 will require 50cp, as will all future advances.

Speaking of change points, I had a question. What do you guys recommend I aim for when it comes to doing content? Should I aim for higher difficulty storylets, so that I advance my qualities faster? Or should I go for straightforward and low-risk storylets to ensure I have many things when I reach the point where I need them?

I used to go after Chancy storylets, as that’s apparently the highest you can go and still get the same number of change points whether you succeed or fail, but lately I’ve switched to modest ,as it’s still twice as many change points as the easier ones, but still has quite a good chance of success. Though, one downside is finding storylets that fit into that range.

Ah, that was what I found for Attributes. So it is not an exclusive property of attributes, which is good to know.

Thank you.

I am currently doing a great many Progress storylets (Running Battle, Casing, etc.) and loosing progress when a bad result occurs gets tiresome quite rapidly. While I am doing these progress storylets, I do not go above straightforward, as a bad result means wasting two turns (at the least) as well as granting the usual crop of menaces. Instead, I save the greater challenges for the cards I happen to draw that day (which I might well fail, but will leave my progress intact).
edited by Abraham Bounty on 1/9/2012

[quote=Urthdigger]Speaking of change points, I had a question. What do you guys recommend I aim for when it comes to doing content? Should I aim for higher difficulty storylets, so that I advance my qualities faster? Or should I go for straightforward and low-risk storylets to ensure I have many things when I reach the point where I need them?

I used to go after Chancy storylets, as that’s apparently the highest you can go and still get the same number of change points whether you succeed or fail, but lately I’ve switched to modest ,as it’s still twice as many change points as the easier ones, but still has quite a good chance of success. Though, one downside is finding storylets that fit into that range.[/quote]

You can always build up a complete gear collection: having equipment granting any bonus from +1 to the max you can get, and some negative bonuses too, so you can literally tailor your Quality. Say you’ve got a +1 bonus hat, gloves, clothes, weapon and pet, you can deselect one or more to cover any value up to your original one, even less if you’ve got a negative item, and then just select/deselect what you need to obtain your desired outcome before trying the Action ^^

It seems I was wrong, my Connected: Bohemian, still only requires 50 cp to go from 53 to 54, but at least we know that it caps.

Relating to what risk level to aim for I usually go for modest challenges. 2 change points per action gives you a fair growth-rate and you still have pretty good odds of getting a success. Usually you can remove points of menace per action if you die/get exiled etc. so the higher risk compared to straightforward seems worth it. Chancy could work as well, but it’s the only risk level that you’re not entirely sure of how many cp you’ll get (it gives either 2/2 or 3/3), which means that sometimes you’ll end up taking a bigger risk for the same amount of cp as a modest challenge. Same thing about low-risk, bigger risk for the same amount of cp should be avoided (except if it tests a progress quality and you can save a lot of actions by not going for straightforward).

I will agree with Abraham that for building progress it’s not worth taking anything but straightforward if you lose progress by failing the challenge, but if I only gain menace (preferably only 1 point) for failing I’ll still go moderate.

As a last point I’ll mention that sometimes you come across storylets that have no penalty for failing the challenge (such as “Track Down A Spifer” in Ladybones Road). These are ideal to boost your attributes. For these I recommend a high-risk challenge, awarding you 3 points if you fail and 4 if you win. The chance of success should be about 30%. Almost impossible awards you with 3/5 instead of 3/4, but only has about 10% chance of success which means that in the long run it should yield less cp than a high-risk.

I have my own mathy speculations to add for your edification or misinformation as the case may be:

Firstly, the game text hints that we receive differing amounts of change points (cp) if our actions are more, or less, difficult. Allow me to make some wild and evidence-free speculations as to those numbers, as well as the probability of success:

Straightforward: 100% success, 1 cp success
Low-risk: 90% success, 1 cp success / 1 cp failure.
Modest: 70% success, 2 cp success / 2 cp failure.
Chancy: 50% success, 2.5 cp success / 2.5 cp failure.
High-Risk: 30% success,4 cp success / 3 cp failure.
Almost Impossible: 10% success, 5 cp success / 3 cp failure.

Now, given these, we can calculate how quickly we can hit the next level in a given quality by pursuing challenges precisely fitted to our competence. For instance, if we attempt a high-risk challenge, we can expect to get 4 cp 30% of the time, and 3 cp 70% of the time. Assuming the quality in question is 50 or higher (and thus takes 50 cp to raise), here is how long it takes to reach a new level:

Straightforward: 50 actions
Low-risk: 50 actions
Modest: 25 actions
Chancy: 20 actions
High-Risk: 15.15 actions
Almost Impossible: 15.63 actions

Now, all of that is easy math. Here’s where it gets fun. Further speculate that you have a reliable companion, with whom you can swap the “advantage” actions - hastily scrawled warning notes and the like. If said companion gets you, on average, 2 of these advantages per action - and you factor the time cost of these social interactions into your formula, the times we calculated above are somewhat increased with the advantage of greatly improved success rate. Assuming (once again) that your second try has the same odds as the first for success, you get the following table:

Straightforward: 50 actions, 100% success rate
Low-risk: 52.5 actions, 99% success rate
Modest: 28.75 actions, 91% success rate
Chancy: 25 actions, 75% success rate
High-Risk: 20.45 actions, 51% success rate
Almost Impossible: 22.6 actions, 19% success rate

Well, there you have it. Inductions built on speculations deduced from numbers pulled from the air. I obviously make no warranty as to their accuracy; in fact this whole thing may be some honey-dazed journal entry. One specifically calculated to justify my showing up to request various goods drunk of my socks and wearing a ridiculous hat. Nonetheless, I hope you find it useful.

I do not believe that straightforward challenges are an absolutely guaranteed 100% chance of success. I know I did fail at least one early on in my entry to Fallen London, and the mouse-over text still proclaims ‘success is never certain’. I will admit that failing a straightforward challenge has not happened to me in several thousand actions, though not all of them were straightforward ones. It is possible that the success rate for straightforward challenges were increased in one of the multiple updated Echo Bazaar has experienced since I failed that straightforward challenge so long ago.

Just a note on your little formula there: The social actions for second chances DO give some change points. However, I’m not sure how much.

Taking in to account Urthdigger’s comment, I’ve redone my work to take into account that forgotten parameter. I have also reworked the loot percentages to take into account the fact that you don’t get loot on turns spent gaining social actions. I’ve set my manservant to cranking my computational engine. After some time produced this spreadsheet where you can tinker with various settings. Simply make a copy of the sheet and have at it.

…I really wish the forums made links visible without needing to mouse over them. I looked at the post for a little bit wondering “…WHAT spreadsheet, I don’t see one.”

I wish the forums were robust enough that we could construct the spreadsheet (or even a decent mockup of one) within the forum post itself without the need for links (visible or otherwise). This is rather a common complaint of mine though. Barring wikidot, I’ve only found one forum that lets you construct tables let alone full fledged spreadsheets.

A habit I picked up from an equally annoying forum years ago: ctrl-c, ctrl-a, return. Once the habit exists, you forget that you’re doing it: until you’re logged out and fury is replaced by the knowledge that you have a copy.

On the maths: for working out what you need, Pascal developed a neat trick:
To add 1+2+3 etc, the trick is: add your lowest number to your highest, then multiply by 1/2 the highest. Thus:
To get to 20:
(1+20) x (20/2) = 21 x 10 = 210 change points.
(To understand why: you need to add all the numbers of course. But the lowest plus the highest is the same as the 2nd lowest and the 2nd highest, and so on. So you have a number of pairs that all, in this case, add up to 21. Twenty = 10 pairs of numbers).

In Echo Bazaar, this trick only works up to 50, as above that every level takes 50 change points. Thus 50 requires (51x25) = 1275 change points. Add 50 change points for every level above that you need to work out.

There was a point in the game when it was possible to fail at a straightforward challenge, but I believe they adjusted things after folks complained, which gave us the variable change points based on difficulty instead. So I’m pretty sure it is now 100% at straightforward, else someone in the past 6 months or so would have mentioned somewhere about failing one.

Quite often while playing I find myself faced with straightforward challenges with 98% chances. Yesterday I failed at one of those.