Visualizing science

Mercurio

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I had a bit of trouble thinking of exactly how the new 5% increase per city exactly changes the game. Whether at 5% increase, the 21st city would always be useless even at even output etc. So I decided to put a few hypothethical situations in a spreadsheet and let it calculate the costs.

Not sure if anyone else has the same problems, but I thought I'd share. Image behind the spoiler. I picked the cost of the last set of techs since it's easiest to see how it works with large empires. Speed and size are both standard. The purple example is of my first completed game (science victory turn 391), cities 15-23 being hypothethical in that list/graph.
Spoiler :

What I'm not really sure of is how to calculate if a new city would be hypothethically worth it with A amount of cities and B cumulative output. How to calculate what break even point for a new city is I mean. Not that it matters much I guess since the more cities you have the smaller the impact of new cities, but it might be useful to see when a puppet is useful as a goldmine or when it is a drag.
 
I had a bit of trouble thinking of exactly how the new 5% increase per city exactly changes the game. Whether at 5% increase, the 21st city would always be useless even at even output etc. So I decided to put a few hypothethical situations in a spreadsheet and let it calculate the costs.

Not sure if anyone else has the same problems, but I thought I'd share. Image behind the spoiler. I picked the cost of the last set of techs since it's easiest to see how it works with large empires. Speed and size are both standard. The purple example is of my first completed game (science victory turn 391), cities 15-23 being hypothethical in that list/graph.
Spoiler :

What I'm not really sure of is how to calculate if a new city would be hypothethically worth it with A amount of cities and B cumulative output. How to calculate what break even point for a new city is I mean. Not that it matters much I guess since the more cities you have the smaller the impact of new cities, but it might be useful to see when a puppet is useful as a goldmine or when it is a drag.

I was thinking of doing this myself. Might still do because I have another way of looking at it. Still, thanks very much for what you've compiled!

One question: your delta in column B is constant (484). Are you sure the 5% is added to the base cost and not cumulative?

One observation: your bottom 3 scenarios, which I think are most representative, basically show there's no real improvement beyond about 6 cities. However, and I think this is something people can overlook, it doesn't have a real negative effect either - even if the cities are pretty underdeveloped (this may not hold true however if the 5% addition is cumulative rather than base-added).
 
How to calculate what break even point for a new city is I mean. Not that it matters much I guess since the more cities you have the smaller the impact of new cities, but it might be useful to see when a puppet is useful as a goldmine or when it is a drag.

Over in another thread i posted this:
... The needed comparison is the cost of building/aquiring a single new city vs NOT building a single new city, in BNW.
....

to determine the needed population of a new city to break even:
[number of cities] is the number of cities you have BEFORE building a new one, including your capitol.
[tech cost]=[base]*(1+.05*([number of cities]-1))
[base tech cost]*0.05=[increased cost]
[tech cost] / [current total beaker rate] = [turns to research]
[turns to research] * [new city beakers] = [contribution of new city]
The break even point is when [contribution of new city]=[increased cost]

This set of equations can be manipulated into [new city beakers]=[current total beaker rate]*0.05/(0.05*[number of cities]+.95)
So the population required of a new city is:
[current total beaker rate]*0.05/(0.05*[number of cities]+.95)
or if you buy a library in the new city:
[current total beaker rate]*0.0333/(0.05*[number of cities]+.95)

(It is assumed that the capital does not increase tech rates by 5%)

If you want to account for odd sources of science (e.g. mayan pyramid, Great Barrier Reef) in the new city, the amount of science you need your new city to produce to break even is:
[current total beaker rate]*0.05/(0.05*[number of cities]+.95)
 
Thanks AlazkanAssassin!
I was thinking of doing this myself. Might still do because I have another way of looking at it. Still, thanks very much for what you've compiled!

One question: your delta in column B is constant (484). Are you sure the 5% is added to the base cost and not cumulative?

One observation: your bottom 3 scenarios, which I think are most representative, basically show there's no real improvement beyond about 6 cities. However, and I think this is something people can overlook, it doesn't have a real negative effect either - even if the cities are pretty underdeveloped (this may not hold true however if the 5% addition is cumulative rather than base-added).
Yes, it keeps adding 484 science every city you found. I checked it for the first few cities and did the rest on the same pace. So each extra city adds 4,76% of the science cost at 1 or no cities or 5% of 'base cost'.
I did the same for the techs in chivalry's column, but found it kept adding either 27 or 26. So it's not always constant, but the percentages are (approximately) the same.

Yes beyond the first few core cities nothing really seems to matter much. Underdevelopped cities are still a drain though and can over the course of the game cost you a few valuable turns (especially if you warmongered like a knife through butter). The graph also goes up a lot faster if you have a couple of virtually 0 science producing hypothethical cities in the core and puppets scenario. Annoyingly puppet AI can still prioritize the wrong buildings, when you want libraries and universities asap.
 
Thanks AlazkanAssassin!
Yes, it keeps adding 484 science every city you found. I checked it for the first few cities and did the rest on the same pace. So each extra city adds 4,76% of the science cost at 1 or no cities or 5% of 'base cost'.

So the capitol does increase the cost by 5% above 'Base'. That simplifies my equations a bit.

to determine the needed population of a new city to break even:
[number of cities] is the number of cities you have BEFORE building a new one, including your capitol.
[tech cost]=[base]*(1+.05*[number of cities])
[base tech cost]*0.05=[increased cost]
[tech cost] / [current total beaker rate] = [turns to research]
[turns to research] * [new city beakers] = [contribution of new city]
The break even point is when [contribution of new city]=[increased cost]

This set of equations can be manipulated into [new city beakers]=[current total beaker rate]*0.05/(0.05*[number of cities]+1)
So the population required of a new city is:
[current total beaker rate]*0.05/(0.05*[number of cities]+1)
or if you buy a library in the new city:
[current total beaker rate]*0.0333/(0.05*[number of cities]+1)

If you want to account for odd sources of science (e.g. mayan pyramid, Great Barrier Reef) in the new city, the amount of science you need your new city to produce to break even is:
[current total beaker rate]*0.05/(0.05*[number of cities]+1)
 
Very nice man. Thanks for the effort.

I think the purple column has the most useful information because it reflects real data of one of your games.

However, I might have missed it, but you didn't take into account that each new city (besides puppets) had to grow first and needed tons of buildings (and turns) before it reached its final science output and the base of your data. Each founded city can also potentially delay the next scientific technology, so you lose science in your older cities as well.

That's why it looks like each non-puppet city you found made science victory faster. In reality, each new city decreases your overall "effective science rate " before it reaches the turning point of becoming a "positive" science city. And that can take forever unless you rushbuy buildings, especially in cities you settle rather late.

Based on your data, nobody would stop expanding at 4 cities.

So, I think the average science output per city and turn during its whole life cycle would be more significant here. So a city which produces 1-10 science for 20 turns, 10-30 for 20 turns and 60 for 40 turns, would have an average science output per turn of roughly 36 (instead of 60 like in your example). Maybe even add an additional penalty for delaying science techs. How much? I have no clue xD.

Just food for thought. :)

Edit: What it does tell us though: Conquering big cities and annexing them seems to be worth it and increases overall science value ... if science buildings weren't destroyed or you can buy or build them quickly.
 
Do puppets also subject you to the increased science cost? (I'm playing Venice and am hoping the reason for science slowness isn't that I bought out too many city states and am hoping it's just because I made too much of an effort to get Great Merchants and neglected scientists)
 
Do puppets also subject you to the increased science cost? (I'm playing Venice and am hoping the reason for science slowness isn't that I bought out too many city states and am hoping it's just because I made too much of an effort to get Great Merchants and neglected scientists)

Every city increases tech costs, including puppets.
 
I had a few questions after reading the original post, and made a spreadsheet of my own to get some answers. Thanks to Mercurio for the idea.

First sheet is a 4 city tall empire around the Atomic Era. A 30:c5citizen: capital working 3 Academies with 3 15:c5citizen: cities; all science specialist slots filled. Turn times are to research the final techs. I estimate around 30 science average from puppets.

Spoiler :



The second sheet is a wide empire, a core city of 25:c5citizen: with 3 Academies and 3 10:c5citizen: cities; with all science specialist slots filled. The rest of the cities are 5:c5citizen: with a university, working both specialists. I also calculated for a Mayan ICS with 4 additional :c5science: from Pyramids and Messenger of the Gods.

Spoiler :


The third sheet shows how long a pre-Education empire would take to research Education. It has a capital city of 12:c5citizen: with a National College and 3 cities of 6:c5citizen: with a Library. Each conquered puppet is 3:c5citizen: without a library.

Spoiler :


Basically, none of this should be all that surprising. Conquering and puppeting cities still benefits the conquerer, but does not allow them to run up a huge tech lead. Basically, to break even a puppet has to produce more :c5science: than .05*base_research_cost of the current tech in the time it takes the non-puppet cities to research the tech. If you have 4 cities producing 60:c5science: at the time that your researching Education making it an 8 turn affair, then a puppet must make more than 485*.05 = 24/8 = 3 :c5science: per turn. For Atomic Era techs this equals about 30:c5science:.
 

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To add to this, from what I'm led to believe you can't reverse the tech cost increase. If you puppet and then raze the city, the tech cost increase remains.

I don't think it's so straightforward. Over on another thread http://forums.civfanatics.com/showthread.php?p=12613354#post1261335
The findings were...
Originally Posted by Magean View Post
So, I've just made a bunch of experiments and noticed something very weird...

Before I took a city, Civil service required 333

When I immediately decided to raze it : Civil service requirements decreased to 325 And permanently so, even after the disappearing of the town.

When I decided to annex the city, then Civil service cost increased to 337 beakers. I also took the culture hit. And all of this was permanent : penalties remained even if I burnt the town (even after its complete destruction) or gifted it to somebody else.

When I puppeted it, Civil service cost increased to 337. However, if I sold/gifted the puppet : -> Civil service cost decreased to 325 !

Where does the decrease come from ?

I guess it deserves a topic.
 
This is cool. Thanks for the effort. I tried to attack it with math and think I may have gotten the same answer as AlazkanAssassin, just expressed differently (and perhaps more intuitively). In the below, I'm assuming that costs increase .05 * base cost per city, meaning the same increase between the 19th and 20th city as between the 1st and 2nd. I'm using C for the number of current cities, B for the base cost, S for current science being produced, and X for the amount of extra science that the new city needs to produce to break even -- that is, not add turns. Unless I screwed up, it is:

Current turns = Turns after adding city

or

Current cost / Current science = New cost / New science

[B * 1.05C] / S = [B*1.05C + .05B] / [S + X]

1.05CB/S = [1.05CB + .05B]/ [S + X]

Cross multiply and simplify.

1.05CBS + 1.05CBX = 1.05CBS + .05SB

1.05CBX = .05SB

1.05CX = .05S

X = .0476 (S/C), or about 4.8% of your average science production per city.

In other words, the required science contribution of the new city doesn't actually depend on the base cost of the technology. Rather it depends on how efficient your cities were already being.

I may have made a mistake here, but I've checked it multiple times and messed around with real values and it seems to work (rounding occasionally moves things 1 turn either way). My intuition for why base cost doesn't matter is that it sort of cancels out. If you currently produce a very small amount of science relative to what is needed, it will takes lots of turns. So, you don't need to add much per turn from the new city to add up to the additional cost. If you are producing a large amount relative to what is needed, the extra cost is small relative to what you are cranking out each turn, so you only need a little extra per turn.

If the increase is exponential, that is, the cost with 2 cities is 1.05 times that of 1 city and the cost with 3 cities is 1.05 times that of 2 cities, then this is obviously not right.

I hope this is useful. If I made an error, someone please call me on it.

Have fun.
 
Basically, to break even a puppet has to produce more :c5science: than .05*base_research_cost of the current tech in the time it takes the non-puppet cities to research the tech. If you have 4 cities producing 60:c5science: at the time that your researching Education making it an 8 turn affair, then a puppet must make more than 485*.05 = 24/8 = 3 :c5science: per turn. For Atomic Era techs this equals about 30:c5science:.

Current tech you're researching has nothing to do with it. Let's say I'm researching dynamite and I'm about to conquer a city. Conquering the city is bad but if switch to sailing it suddenly becomes good?

X = .0476 (S/C), or about 4.8% of your average science production per city.
 
However, I might have missed it, but you didn't take into account that each new city (besides puppets) had to grow first and needed tons of buildings (and turns) before it reached its final science output and the base of your data. Each founded city can also potentially delay the next scientific technology, so you lose science in your older cities as well.

That's why it looks like each non-puppet city you found made science victory faster. In reality, each new city decreases your overall "effective science rate " before it reaches the turning point of becoming a "positive" science city. And that can take forever unless you rushbuy buildings, especially in cities you settle rather late.
Yes very true. It's a bit harder to incorporate that into the data.

I don't think it's so straightforward. Over on another thread http://forums.civfanatics.com/showthread.php?p=12613354#post1261335
The findings were...
Wait... so when immediately razing the science cost gets actually lower than it is before taking/razing that city? Big bug?
 
[B * 1.05C] / S = [B*1.05C + .05B] / [S + X]

Your mistake is in this line. Using [B*1.05C] as the current cost assumes exponential growth of the science rate due to previous cities. What we actually see is linear growth. You need to use:
[B*(1+.05C)] / S=[B*(1+.05C)+.05B]/[S + X]

This does simplify to the same equation I had posted,
X=(0.05*S)/(1+0.05C)
 
Good to know it's 5% of the base cost, not cumulative. I can continue to go about conquering the world.

Cheers.
 
I went ahead and produced (using the _corrected_ formula thanks to AA) the attached pdf of a spreadsheet which shows how much science a new city must produce to not slow production of discoveries as a function of current science and current cities. It includes a contour graph of same. Eyeballing it, the big message seems to be that (a) new cities don't need that much science until you get over 200 or 300 total science and (b) once you have a lot of science going, even if you have a lot of cities, a new city needs to produce 30+ science to avoid slowing you down.

Happy civving, all.

View attachment 356238
 
So how many beakers did researching research cost you?

Jokes aside, it's good to know exactly how this works.
 
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