Spire squad size calc - we need your data!

[Deleted User - 3932582]

I went back through my notes to find my increases when chapter 16 was added to see if this bump happened from the final tech of chapter 15 becoming required, and the only thing I did that week was place the new unlocked scouting expansion. I had a 28 point increase that week, so was that only partly for the expansion and partly for the tech becoming required, or did this not apply to chapter 15?

Edit: I did also find one odd set of numbers for my city. My March 23rd post for the previous week, I had an increase of 44, but I only added 5 AW levels. That was a month or so before chapter 16 was added to the game, but if what you think is correct for current end-game techs, maybe they activated that increase for the chapter 15 AW tech at that time? Does 10-12 points for the tech sound about right?

While we can't go back in time and peek at more data that we didn't already get, here are a few things to consider. The actual research tree for chapter 16 was there before the chapter actually started - it was just hidden. So from that perspective, in all likelihood the final tech in chapter 15 already had descendants in the tech tree by the time chapter 16 was officially rolled out - we just haven't seen it. How much ahead of time the new tech tree was there in place is obviously hard to tell now. So your 28 point increase for Apr 26 is all due to expansion, there is no research component to it.

But to your second observation - this might just be it. Unlike most of the data in this thread, I was tracking my cities for much longer, and I can tell that from about mid-March and backwards my own city data is a bit off ( 8) ), as per the model. The thing is, I assume that my research was frozen at the time (as I was in end-game for a long time), so I used the current end-of-chapter-15 numbers all the way. But if I assume that around mid-March and earlier my M value was actually less by one (due to the last ch15 tech not being mandatory at the time), then my data lines up quite a bit better. Again, the difference of 1 M value for our cities is about 10 points, so it's hard to tell regardless (AW level number off by 1 would have similar impact). This might be more visible for much bigger cities by the time the next chapter rolls in, as now we know what to look for.

So it is quite possible that chapter 16 tech tree got plugged in around mid-March or so, so for most observations in this thread we would have never seen the impact of that.

 

[Deleted User - 4646370]

A thought : long ago Elves used to have 2 tiles less used (so 2 more free tiles) for mandatory buildings (MH : -6 ; Trader : -4 ; MA : +5 ; Builders : +3.) But, with chapters 9 (MH) and 11 (Trader) these buildings got upgrades, so that now Humans have 8 more free tiles... The +0.83% increase in Spire costs due to two more mandatory techs in Human tech tree looks like a compensation for these 8 tiles

 

[Deleted User - 3932582]

A thought : long ago Elves used to have 2 tiles less used (so 2 more free tiles) for mandatory buildings (MH : -6 ; Trader : -4 ; MA : +5 ; Builders : +3.) But, with chapters 9 (MH) and 11 (Trader) these buildings got upgrades, so that now Humans have 8 more free tiles... The +0.83% increase in Spire costs due to two more mandatory techs in Human tech tree looks like a compensation for these 8 tiles

Probably just a coincidence. All these (including tech tree differences) were in place long, long before the Spire was even on the horizon. Unless you think it was all planned like this several years in advance

 

[Deleted User - 4646370]

Yes, I think too that it's just a coincidence but it's funny that itbgoes the right way...
PS : last row of your chart is a bit confusing. By recounting I saw you didn't count last chapter 16 techs (which are currently optional).

 

[sam767]

Wow! You do good work. Are we declaring success?

 

[Deleted User - 3932582]

By recounting I saw you didn't count last chapter 16 techs (which are currently optional).

That's right, and that's how they behave as far as I can tell. I mean, there are not that many cities that finished chapter 16 yet, but the one that I have data for indicates quite unambiguously that these last 3 techs do not count. Not yet at least.

 

[Deleted User - 3932582]

Wow! You do good work. Are we declaring success?

Well, that might be a bit too soon, we'll validate results for some period of time and see if they still line up. But the existing results look too good to be just a fluke, so I kind of expect it to continue to work. With that, I don't expect much more in terms of insight from the advanced cities. Well, except if it turns out this model is also wrong. Which is always an option

What I am really looking for as a next (and final?) step is the data on the tiniest of cities - chapter 3, ideally no AWs, no premium expansions, minimum regular expansions. So if you guys have some of these (or can make some ), that's probably what I will be looking for next. Because <100 SSSo data still looks a bit weird.

 

[michmarc]

Neat formula. The one thing that jumped out at me is that 3.67V term: it screams to me of rounding and really wants to be (3+2/3)V which is 11V/3. And the other term in that same row for premium is 11P/4 in fraction form. The common numerator satisfies my sense of mathematical elegance.

(Maybe you did this already but preferred the decimal representation for your page.)

 

[Deleted User - 3932582]

(Maybe you did this already but preferred the decimal representation for your page.)

Indeed. Given that formula is a product of terms, coefficients can be rearranged in many different ways (e.g. 0.003*2.75 = 0.03*0.275 etc). The important part is that P:V coefficients are in 3:4 ratio. I wanted to keep +1 term for AW factor, and didn't want any extra constant factors, so everything else went into expansion coefficients. I played around with different combinations to see what would look like the most realistic formula that was designed.

Another neat form for the expansion term is this:

(2.75P + 3.67V - 18) = (2.75P + 3.67(V-5))

This makes it obvious that expansion term is always positive, as V >= 6 at all times (V includes initial 6 expansions).

 

[Pheryll]

(2.75P + 3.67V - 18) = (2.75P + 3.67(V-5))

Replacing A with P to avoid confusion.

 

[Deleted User - 3932582]

Replacing A with P to avoid confusion.

Fixed, thanks.

 

[Deleted User - 1178646]

Indeed. Given that formula is a product of terms, coefficients can be rearranged in many different ways (e.g. 0.003*2.75 = 0.03*0.275 etc). The important part is that P:V coefficients are in 3:4 ratio. I wanted to keep +1 term for AW factor, and didn't want any extra constant factors, so everything else went into expansion coefficients. I played around with different combinations to see what would look like the most realistic formula that was designed.

Another neat form for the expansion term is this:

(2.75P + 3.67V - 18) = (2.75P + 3.67(V-5))

This makes it obvious that expansion term is always positive, as V >= 6 at all times (V includes initial 6 expansions).

The use of a fixed -18 of V-5 still sounds weird to me and not something a developer would make.
I would believe a focus should be on somehow eleminating those.

Are you certain only mandatory researches count?

 

[Deleted User - 4646370]

Another neat form for the expansion term is this:

(2.75P + 3.67V - 18) = (2.75P + 3.67(V-5))

This makes it obvious that expansion term is always positive, as V >= 6 at all times (V includes initial 6 expansions).

3.67×5 = 18.35 not 18

The use of a fixed -18 of V-5 still sounds weird to me and not something a developer would make.

It doesn't surprise me. It looks similar to the base map/training squad size of 6 (even though base squad size with the 6 starting expansions here is 4.)
One a bit more surprising thing is that T term uses exponential while other terms use linear increase.

 

[Deleted User - 4646370]

"Seemingly high -23 error is almost exactly corresponding to just 1 difference in M. So in all likelihood it is a measurement error (other points from the same city work quite fine)." This one may be my fault.

16/35/622/449/4140/151 [48:35] --> 6071/2696. (Researched L3 Senate, city expansion 48, L3 and L4 Dwarven embassy, Meditation circle, L3 Fairy embassy and L4 Senate, and L28 manufactory upgrades. Placed 1 research expansion.)

Just realised I forgot one tech in the list

 

[Deleted User - 3932582]

The use of a fixed -18 of V-5 still sounds weird to me and not something a developer would make.
I would believe a focus should be on somehow eleminating those.

Umm, why? Yes, when you develop a model like this one, you define a general shape of the dependencies, then calibrate individual coefficients to get a difficulty ramp up that is deemed reasonable. So you don't have issues with 2.75 or 3.67 numbers, but somehow you have an issue with 18 term? Why?

(V-5) form also has a very simple intuitive meaning - you get the first 5 expansions (out of initial 6) for free.

The focus should be on making the model representing reality close enough, and that part works really well on the data we've got. It is also as parsimonious as they get in this form. So not sure what exactly do you want to eliminate here.

 

[Deleted User - 3932582]

3.67×5 = 18.35 not 18

I am aware of that. This is a low-sensitivity coefficient, so there is no real difference in fit between 18 and 18.35. Your research term varies from 1 to ~5.5, your AW term varies from 1 to ~3.5, so their product varies in 1 to 20 range at most, at this moment (and about 1-10 for most cities). 0.35 difference in the expansion term will make 0.35-7 (but mostly 0.35-3.5) points of difference to the Spire SS calc. This is negligible.

Remember, I already round up coefficients so that they look more reasonable.

One a bit more surprising thing is that T term uses exponential while other terms use linear increase.

Nothing particularly surprising here. This is all part of a model design process, you choose a dependency that gives you the ramp up that you want to see. If they decide that as chapters progress an average city's ability to handle the Spire grows in the exponential fashion, then exponential term is the right approach. There is no rule that all variables have to have the same impact - not just in terms of coefficients, but in terms of functional dependencies as well.

On small increments both look very similar as (1+m)^x ~ 1+xm, but it doesn't work over big increments, obviously. I have tried some linear terms with respect to research, and it just doesn't fit the data across the whole spectrum. Unlike exponential term.

"Seemingly high -23 error is almost exactly corresponding to just 1 difference in M. So in all likelihood it is a measurement error (other points from the same city work quite fine)." This one may be my fault.

Heh, you figured out the problem city When you look at model error outliers, and you can detect issues with input rather than with model, that's when you know that the model is right. And as I mentioned, after updating M with +1 you're within 1 point of the actual SSSo. I am really happy that we figured it out

A few of the other errors can be explained by differences in 1 AW level (e.g. AWs upgrading close to the Spire spawning). Regardless, most of these are within 10 points already (outside of tiny cities).

 

[Deleted User - 1178646]

3.67×5 = 18.35 not 18

It doesn't surprise me. It looks similar to the base map/training squad size of 6 (even though base squad size with the 6 starting expansions here is 4.)
One a bit more surprising thing is that T term uses exponential while other terms use linear increase.

Umm, why? Yes, when you develop a model like this one, you define a general shape of the dependencies, then calibrate individual coefficients to get a difficulty ramp up that is deemed reasonable. So you don't have issues with 2.75 or 3.67 numbers, but somehow you have an issue with 18 term? Why?

(V-5) form also has a very simple intuitive meaning - you get the first 5 expansions (out of initial 6) for free.

The focus should be on making the model representing reality close enough, and that part works really well on the data we've got. It is also as parsimonious as they get in this form. So not sure what exactly do you want to eliminate here.

Because it looks like someone made it with a brain...and we all know what some employees in germany lack

 

[Deleted User - 3932582]

@MinMax Gamer What's the data look like so far on the AW side of things? Does there seem to be anything indicating that the AW being leveled matters or are they all treated the same regardless of size, level, etc? Is this something that hasn't really been closely examined yet? I've got 11 AWs ready to upgrade now that I've been holding off on for the sake of providing single increments each week, so before I go leveling all of them this week I'm hoping to hear what you've discovered so far in that regard!

Almost missed that. Based on the most recent results, it doesn't seem that anything AW-specific - outside of the total AW levels - matters. Given where we're at right now, I'd say go ahead with upgrades or anything else that you wanted to do in your city. The model is supposed to cover all the simultaneous changes, and we should be able to verify it (just keep in mind that changes close to Sunday may get reflected in either week). There is nothing at the moment that I need to investigate (except for the tiny cities), all we need to do is to validate new results. If we see something unexpected, then that's when any additional incremental analysis/investigation may be needed. But for now, it seems that we're good to continue with our regular lives in the advanced cities

I am still looking for chapter 3 and/or <150 Spire squad size cities, to take a closer look at the low end.

 

[Crow Last Elf]

Chpt. 3 screenshots, taken today

 

[Crow Last Elf]