I didn't play for the last few ages but I'm thinking about going back for the next one as a t/m. Does anyone know the diminishing returns on the science? All I have is some raw data from KD mates but it's not enough for accurate calculations.
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I didn't play for the last few ages but I'm thinking about going back for the next one as a t/m. Does anyone know the diminishing returns on the science? All I have is some raw data from KD mates but it's not enough for accurate calculations.
I am working on a curve right now; not a huge variety of info in it yet, so not sure of the reliability, and the rounding on science page is annoying too!
y = 1.018054e^-0.006045x (ONLY ACCURATE UP TO ~60 points over the cap ... I have an updated curve below)
where y is the effectiveness of the category in question, and x is the number of skill points over the cap in that category.
recruit = 0 skill points
novice = 1 skill point
graduate = 1.5 skill points
professor = 2 skill points
in a few days ill probably create more data points by moving/stacking a lot of science in couple categories
Stealing some info from the Utopia Talk thread, I had to change from exponential to a polynomial description... Obviously this can't be right because it starts to curve back up eventually- but it should be fairly accurate up to at least the point where X = 180. This 180 value represents 90 (ninety) Professors over the category's cap... If you get substantially beyond this point let me know :P
Y =[1.9*(10^-5)]X^2 - [6.35*(10^-3)]X + 1.02
Where;
X = # of Skill Points above the cap of the category in question
Y = The effectiveness of the category in question
Thanks for the posts.
I used the last formula you provide with 5 profs and got 0.988725 but does this apply to all categories? Different categories have different multiplayer is it ignored?
Did you plot them separately? I'm thinking it's much easier and based on % over cap. That would give a slightly different curve for each category.
I've been plotting them all together, and it seems more to more or less follow the same curve from what I can see. You could be right that the curve is slightly different for each category, but I am pretty lazy, and this seems fairly accurate so far :)
Each category is separate, going over the cap in one does not change effect in any others. Category in Question only
Okay, so based on what you've said, the 10th professor past the cap in a category is still getting about 85% of the effect of the "below-cap" professors, (your two functions plots very close together).
Am I reading this right? I thought the returns diminished much quicker than this!
Really, because if so, I'm dumping 45 scientists into my Dwarf Sage as we speak.
I started to try and figure this out last age by looking at only military on my undead sage. I went up to 44 profs in that catagory and then reassigned them one at a time to other catagories and noted the percentages each time. Do you want these numbers? I gave up on trying to do anything with it.
Thanks, but I am confident I have enough data, up to the x = 180 mark anyways... If you get more than that this Age drop me some info at that point!
which formula is accurate up to 180 skill points?
They diverge a lot at 180. (90 professors). one shows about 50% effect and the other 34%.
I repeat, are you sure these are correct? I didn't do any math last age but I thought I was seeing a drop of 50% effectiveness as early as the 10th professor. That contrasts sharply with the formulas you've posted.
Gift me blocks of data and I'll sort it out! It's surely some variety of root function.
C/T * (T-C) + C = overall science effect is probably a good place to start. they like simple stuff like that.
C=cap
T=total
May I kindly ask the people who have data to share the raw data they had? It'll make this process a lot easier :P
Especially because even if we find the exact formula, when you actually play the game, you'll probably say something like "Ok, the Xth professor after the max will be 50% effective" to yourself and that will probably be the only thing you will remember and need to remember as you play the game. If the formula is indeed too weird (although I doubt that would be the case), at least the raw data can give us the value of X.
The second one, where it mentions the 180 in the post along with the formula(I've edited my original post). I definitely can confirm 10 profs over does not result 50% effect. I made direct observations from my province, and KD mates at/around 30 profs over the cap, and the categories come out at about 70% of what they would be if those professors were giving full effects.
@Wilmington I did not try to use the other in-game examples of diminishing gains, like you suggests, so its entirely possible.
I'm just using simple excel trendline to make the curve... I would imagine its a cubic function... around the 180 point it has more or less leveled out, I expect with info that went beyond there we would see the rate of diminish pick up again.
@yootohpeega I am not sure what you mean... every 2X = 1 Professor
Yes, pleas share some data. One question I have that would clear a lot of things up is the 31st scientist in tools. The hard cap used to be 20, truncating the last scientist. If 31 profs is now more than 20, but lees than 20.46 it's figured on effect not scientists
I didn't even understand that the way I wrote it. 31 profs in tools would be 20.46 without the cap. Diminishing returns based on the number of scientists would have the 31st at full strength making the bonus 20.46. If it's based on the ratio of overage (what it would be/soft cap), it would be slightly less. Unfortunately, we don't get that accurate of info unless it diminishes enough to drop it below 20.45 and that's doubtful.
Thanks Rattlehead. You really... rattled... my conception of the science diminishing returns and I was definitely playing it wrong last age! Thanks for the works you've done on this. And thanks also for the work on the power rankings sheet. It's a very tricky and subjective thing to attempt, and the category breakdown is very instructive.
You are using different X's. Yootohpeega's X is the professor where the additional benefit of that prof is 50% the full amount. Your X is the amount of skill points over the cap you are.
It would be great if someone could translate the formula into something like a rule of thumb (which is what I think Yootohpeega was trying to get at). ie I need to decide how far over the cap to go. How many extra skill points can I have before the gain drops to (a) 75% of normal for that extra skill point (b) 50%, (c) 25% (d) 10%. Or something like that. A chart from 90% down to 10% at 10% intervals would be even more wonderful ;)
I am not using # of scientists at all... and if you check this thread out; http://forums.utopia-game.com/showth...e-Skill-Points
You can see where the caps actually are, and that they aren't necessarily related to "# Profs" ... such as Thievery the exact soft cap of which cannot be landed on with any combination of scientists, Or channeling which needs at least 1 Graduate because it has a decimal figure.
@Wilmington the other thing is that the first couple points after the soft caps, effects are actually INCREASED for the first 3-4 points beyond that cap, and then start to diminish after that... You will see that my graphs start just above 100%!
I think something like y = -1E-7 (x-175)^3+0.5 would be a rough approximation of the curve I am expecting to see.
Here are some graphical representations, first my data with the trendline, and then one that shows a comparison of the trendline I've come to, and my approximation, just above here.
https://preview.ibb.co/iSuVVw/Diminish.png
The trendline based on my data is shown here, in green. You can see, in red, an example of what I am expecting to see if I had more data (I pulled the red function out of thin air more or less, there is no data supporting it)
https://image.ibb.co/f5paxb/dimcube.png
If someone wants my actual data, I prefer you to DM me so I can send it that way- I'm not just going to post a large series of numbers in this thread.
Thanks. This is incredibly useful.
What are the axis? y axis I am guessing is the % (or fraction in 2nd graph) of the full amount of benefit received by each additional SP. X is the number of skill points you have over the number of SP needed to cap?
Does this one curve apply to every category, even though they need a different number of skill points to soft cap?
So If i am understanding all this correctly, you can go a very long way above the soft cap (something like 160-200 SP above the soft cap) and still be getting 50% of the normal benefit for each additional SP?
Yes, X is the number of SP over the required cap for a given category, and Y is the effectiveness of the category in question (not all your science). It also appears to me that, yes, one curve applies to all categories, but each category's diminish is calculated individually.
I wouldn't look much beyond that 160-180 range where it gets flat, maybe it even stays flat beyond there but I don't have info... I think your understanding is a bit off on the value per additional SP, explanation below.
I'll pull out a few numbers here;
20 SP (10 Profs) over the cap - That category works at ~90% of what it would be, if science continued to accumulate at base rates
40 SP (20 Profs) over the cap - ~80%
60 SP (30 Profs) over the cap - ~70%
90 SP (45 Profs) over the cap - ~60%
150 SP (75 Profs) over the cap - ~50%
It starts out much much like Wilmington has postulated, but you start to see the curve once you exceed ~30 profs over (I would call that roughly ~2x the cap).
So, you hit soft cap for Alch, 60SP = +30% Income
If you got up to 210SP (150 over the cap in Alch), the base would be;
210SP * 0.5 Alch multiplier = 105 = +105% Income
But your actual effect(according to my curve) would be ~+52.5% Income... So you have the first 60 SP giving you +30%, and then the next 150 SP only gives you +22.5%
The base rate for Alchemy is 0.5 per SP, by the time you reach 150 SP, you are getting only (22.5/150=0.15) 0.15 effect per SP, which is actually 30% effect for each additional SP at that point.
Ah - so the y axis is showing me the % benefit of the category as a whole as compared to if there were no caps and it continued in a linear way. I had thought it was showing the % of the normal benefit from each additional SP beyond the cap), hence why
So alchemy at 20 SP over the cap, according to your graph is about 85% of non-capped benefit. So non-capped would be +40% income, but capped is +34%, so an extra 0.2 benefit per SP (on average), as compared to the normal 0.5, ie only 40% of the normal benefit.
So when Dvorak said "I was seeing a drop of 50% effectiveness as early as the 10th professor. That contrasts sharply with the formulas you've posted.", then this would be correct (and in fact underestimate the speed of drop off) if Dvorak's reference to 50% was to the effectiveness of each additional prof/SP rather than the category as a whole.
Am I understanding this right now or still missing the point?
If I've got it right, I think it would be helpful to have a graph that shows the % of normal benefit of each SP over the soft cap on the y axis (rather than the % of the benefit the whole category is giving). I have no idea how to do it but it must be possible to draw that graph from the data /formula we already have.
Yep, I think you are onto it now!
Ya I should be able to pull that info, it will be different for each category... but I do try to disclaim my laziness as often as I can :P
So what I've come up with is this;
Alchemy, Tools, Production, follow the same pattern (they have the same SP cap)...
20 SP (10 Profs) over the cap - each SP is worth 60% of base
40 SP (20 Profs) over the cap - each SP is worth 50% of base
60 SP (30 Profs) over the cap - each SP is worth 40% of base
90 SP (45 Profs) over the cap - each SP is worth 33% of base
150 SP (75 Profs) over the cap - each additional SP is worth 30% of base
Crime and Channeling are very similar to the above, within 1%, but they are not the same, even as each other, because their SP cap differs.
Housing and Military you get this;
20 SP (10 Profs) over the cap - each SP is worth 52.5% of base
40 SP (20 Profs) over the cap - each SP is worth 42.5% of base
60 SP (30 Profs) over the cap - each SP is worth 32.5% of base
90 SP (45 Profs) over the cap - each SP is worth 26.7% of base
150 SP (75 Profs) over the cap - each additional SP is worth 25% of base
This table plots Alchemy, Tools, Production, Crime, and Channeling as the blue dots, and Housing and Military as the orange dots.
X axis = # of SP over the cap
Y axis = % effect of your SP in addition to the cap... roughly this tells you what your 'next' SP will be worth as a percent of the base SP in that category.
https://image.ibb.co/bsBGLw/perSP.png
These numbers are rough, as they are based on my rounded estimates in post #24
Thank you Chris for digging deeper. This distinction is really important.
Rattlehead and I were obviously were talking about different things and I misunderstood the formula. I was talking exactly as you say, % effect for the next professor, or you could say "marginal scientist effect".
I wonder then, if you did a curve-fit would the curve come out to a more... simple formula? if the final "y" variable was marginal scientist effectiveness instead of total effectiveness of all scientists?
y=(x-1)/(x+1) +1 x>1
y*cap=effect
x= raw total/cap (effect, not scientists)
This would give a limit of 2 times the cap
yx=(x-1)/(x+1) +1 would give your curve
Actually, I'm thinking it might be just (x-1)/x +1
should be negative, I think?
y = -[(x-1)/(x+1)]+1 (oic urs is yx= :P nvvvm)
or ya your final option looks like the right shape as well... im not sure if this lines up quite with my observations though, its quite possible I'm missing something in your description though, my mind is currently in an altered state
Thanks everyone! Rattlehead, that last graph is really useful. I don't understand these equations enough to do anything meaningful with them in game but that graph is perfect and with that, relatively simple even to teach a new player how this complicated science system works!
Glad to be of service!
Thanks RattleHead and Chris 121. This is quite valuable :)
Just an update here, I have done the numbers with categories separated, and the results are the same from category to category. I know some were asking for my data before, but were disappointed that it was just all mashed together :P
Anyway, here is a chart I made with my new set; Hopefully it contains graphically everything we covered, mostly the Prof% is new here...
https://image.ibb.co/e2f53b/dimg.png
The yellow trendline represents Total Category Effect, as a % of what the base would be, while the white trendline shows Professor Effect as a % of a base-value Professor. The X axis values are the # SP over the 'soft cap' (1 prof = 2 SP).
EDIT: I do have this data separated by category now if people want to get their hands on it. DM me!
This is nice. Imma take this. Don't tell nobody and we wont have any trouble!
https://i.giphy.com/media/q4uzbjzovloWc/giphy.gif
Based on the orange data your last chart, it looks like there is a random factor for each scientist?
This is due to the rounding that takes place on the Science page... in certain categories when you get far enough over the cap, there is of course some change for each SP, but it doesn't register on the science page due to rounding.
Thanks for the clarification.
This is great RattleHead, thanks again.
I'm not 100% sure if this is a stupid question (every time I come back to look at this it takes a while to get back into it...), but presumably it would be easy from your data to draw a graph that shows the actual % science benefit for a given number of SPs. EG if I think "I want a 50% income bonus, how many SPs do I need", or "I could put my 400 scientists all into military, what % bonus would I get?".
So that would be slightly complicated by the fact that a SP in each category provides a different actual bonus, so we would need either 6 graphs or presumably it is possible to draw a line that fits each one but where the user will need to then multiply in the effect modifier for that category (I haven't expressed this last part very clearly as I haven't worked out quite how to say what I mean yet..sorry).
You can use the yellow trendline to predict what science effects will be with X SP over the cap.
The number of SP over the cap becomes X in the trendline equation. You then multiply that result by what the base effect of that much SP would be for the prediction.
That said, this equation will start to curve back up if you go over ~130 SP in excess(bottoming out around 55% of base effect on this curve), the older version has this same issue... I had a War right up to EoA or I could have stacked deeper to test this, but you can see in post #22 of this thread what I think the actual curve might look like, where it starts to rapidly drop in effectiveness again after ~150 extra SP or so... But it could be that it just flattens right out at some point too, perhaps. But I certainly wouldn't expect it to actually tick back up.