Tech costs are FAQed up a bit

[mueller]

Since there are many active threads at the moment dealing with technology costs, particularly for large countries, I've decided to run a few tests to supply some hard data on the relationship between # of provinces and tech costs. These tests are by no means exhaustive, so please feel free to supply some additional data.

Testing procedure:
1419 scenario, Tuscany, catholic, latin tech group.
Treasury edited to 50,000d. Sliders were set to innovativeness 5, centralization 5, so no modifiers apply.
Tech was edited to 18/2/2/2.
I wait till January 1420 to eliminate "ahead of time" penalty for infra and go on an annexation spree.
Catholic religion has a techspeed (TS) of -1 (as of 1.07beta). I have run two tests for techspeed=-1 and techspeed=0, tracking infra costs as I captured provinces.

Here are results:

Size	TS=0	Mult.	TS=-1	TS penalty
1	3025	1.000	3085	60
2	3630	1.200	3690	60
3	4235	1.400	4295	60
4	4537	1.500	4598	61
5	4840	1.600	4900	60
6	5445	1.800	5505	60
7	5748	1.900	5808	60
8	6050	2.000	6111	61
9	6126	2.025	6186	60
10	6201	2.050	6261	60
11	6277	2.075	6338	61
12	6352	2.100	6412	60

1st column -> # of provinces
2nd -> infra 3 cost at techspeed=0 (edited religion.csv)
3rd -> countrysize multiplier rounded to 3 decimal points
4th -> infra 3 cost at techspeed=-1 (unmodified religion.csv)
5th -> (4th column)-(2nd column).


Comments:
Average discovery date for infra 2 is 1425 (from infra.csv).
Average discovery date for infra 3 is 1450.

You do not get hit with "ahead of time" penalty if current year >= average discovery for current level - 5. Ie Tuscany suffers from ahead of time penalty (about 60%) in 1419 but not in 1420.

First, look at the tech cost for size=1. Tech cost is proportional to the difference between nearby discovery dates. 1450-1425=25 years or 300 months, ie it is tempting to conclude that each month of the expected discovery period contributes exactly 10d to research costs. Where extra 25d come from, I have no idea. I assumed that base cost = 3000.

From 1 to 8 provinces countrysize multipliers are obtained as (tech cost at n provinces)/(tech cost at 1 province). Results are exact +/- 1d. [ie (3630-3025)/3025=0.2 exactly]. After 8 provinces tech costs increase by flat 75d (again, +/- 1d) which is exactly 2.5% of 3000 (base cost).

As one can see from the last column, religion techspeed penalty is flat 60d or 2% of base cost.

All other modifiers (innovativeness, tech group speed, inflation etc) are applied multiplicatively on top of this table.

Thus, for practical purposes it is safe to assume that each province past 8 costs 2.5% of base cost.
[/Comments]

Some unexpected observations:
1. As you can imagine, no other country had land tech 18 in 1419. Nevertheless, 17 tech levels ahead of competition Tuscany still receives a neighbour bonus to land tech:
3d at 1 province
2d at 2-8 provinces
1d after 8 provinces.

I have no idea what is going on there. Either there is something wrong in the neighbour bonus calculations, or Tuscany provides a neighbour bonus to herself.

2. Carmagnola with 5/5 army actually managed to defeat my 25/5 rather badly despite 3 CRT levels tech difference

cheers,
-- mueller

[satan]

Very interesting/helpful, thanks!

More or less meets what I expected.

[Bocaj]

Great stuff Mueller! *cheers*

Isn't it interesting that after getting 12 provinces, the tech amount only increases just over twice it's original value at one province. That neighbour bonus is weird...

[pcasey]

Originally posted by mueller
Here are results:

Size	TS=0	Mult.	TS=-1	TS penalty
1	3025	1.000	3085	60
2	3630	1.200	3690	60
3	4235	1.400	4295	60
4	4537	1.500	4598	61
5	4840	1.600	4900	60
6	5445	1.800	5505	60
7	5748	1.900	5808	60
8	6050	2.000	6111	61
9	6126	2.025	6186	60
10	6201	2.050	6261	60
11	6277	2.075	6338	61
12	6352	2.100	6412	60

Interesting. You know, it could just plain have a base cost of 3025 All the number work out cleanly if you do if you assume evens rounding on 0.5s.

What it looks like, for large empires, is that:

Where n = Province Count
And N > 8

CostTech(n) = BaseCost * (2 + (n-8) * 0.025))

Example:

CostTech(10) = 3025 * (2 + (10-8) * 0.025))
= 3025 * (2 + 0.05)
= 6201


Note that this simplies to the easier to computer, but harder to read:

CostTech(N) = BaseCost * (1.8 + 0.025N)

Which lets us calculate the incremental cost increase for going from n provinces to n+1 provinces.

Delta(N+1,N) = 0.025B

Thus, adding provinces leads to a linear increase in tech costs e.g. going from province 10->11 adds the same *absolute* tech cost as going form province 100->101.

[mueller]

Yup, that's what it is. 75d per province cost increase after 8 provinces is a clean 2.5% of 3000d but some weird 2.47933% or 3/121 of 3025d so I was not entirely happy about this 25d discrepancy. But it is not terribly important for practical purposes.

And, to be fair, I have not run it all the way to 101 provinces but a linear trend is fairly obvious.

[JohnMK]

Very good work, it's nice to have something like this to go by.

[Wetboy]

Thank you very much. I will study this post till I know it by heart.

[Peter Ebbesen]

Originally posted by mueller

Some unexpected observations:
1. As you can imagine, no other country had land tech 18 in 1419. Nevertheless, 17 tech levels ahead of competition Tuscany still receives a neighbour bonus to land tech:
3d at 1 province
2d at 2-8 provinces
1d after 8 provinces.

I have no idea what is going on there. Either there is something wrong in the neighbour bonus calculations, or Tuscany provides a neighbour bonus to herself.

This is the proliferation bonus, which is independent of techlevels and only depends on the number of nations in your techgroup:

(This is old information, but I believe it is still mostly valid)

Originally posted by AndrewT on behalf of Johan a long time ago. Proliferation bonus highlighted by me
Neighbour bonus is just not how far behind you are, as it also takes into account how many nations there are in your tech group and how big you are.

First its the difference between your tech and the highest tech.

Example: Lets say 3 levels behind

Multiply this by 8.

Example: 3*8 = 24

If the original difference was above 2, then you get the square off the difference added.

Example: 24 + 3*3 = 33

Then 1 is added for every 8 countries in our tech groups.

Example: Lets say Latin has 40 countries, 33+5 = 38

Divided by your size modifier.

Example: We have 6 cities, 38/0.9 = 42

Divided by 4.

Example: 42/4 = 10.5

Multiplied by the NMR

[SJG]

Originally posted by pcasey
Where N = Province Count
And N > 8

<SNIP>

CostTech(N) = BaseCost * (1.8 + 0.025N)

Using the same definitions and restrictions

=> TechMultiplier(N) = 1.8 + 0.025N

=> TechMultiplier(N)/N = 1.8/N + 0.025

=> The cost of tech per province is an exponential decrease towards 0.025

Assuming the average income per province does not fall at a worse rate (I think that's fairly safe)

=> More provinces give faster tech

[Ironfoundersson]

So this means the technology calculations is not like this (from the current FAQ)

Actual value = (Base Value BV that depends on the researched technology)
* (1 + M * (cities count modifier) - M * (techspeed of state religion)/10 + (No contact modifier) )
* (date modifier)
* (tech group multiplier)
* (DP multiplier, from -35% to +35%, i.e. x0.65 to x1.35)
* (Inflation level, from x1.00 for 0% inflation to ...)

But like this (for nations with N>7)

Actual value = (BV-2*BV*(techspeed of state religion)/100)
* (1.8+0.025*N) <=size modifier
* (date modifier)
* (tech group multiplier)
* (DP multiplier)
* (Inflation level)

Where the no contact modifier should come in somewhere and according to ws2_32 should be something like this

w = max(whiteman, 5)
c = max(foreign capitals known, 19)
isolation penalty = f(c) = f[(19-c)/19]
whiteman penalty = f(w) = f[(5-w)/5]
No contact modifier = f([(5-w)/5 + 0.28*(19-c)/19])

[Hardu]

Basically the size penalty to tech costs is linear. Thank you. While the notion itself is strange I really do not see why people should get upset about it.

[mueller]

Originally posted by Peter Ebbesen
This is the proliferation bonus, which is independent of techlevels and only depends on the number of nations in your techgroup:

... which confirms the argument I made in this thread that orthodox-tech nations are somewhat screwed up due to low number of countries in their tech group.

[mueller]

Originally posted by SJG
Using the same definitions and restrictions

=> TechMultiplier(N) = 1.8 + 0.025N
=> TechMultiplier(N)/N = 1.8/N + 0.025

=> The cost of tech per province is an exponential decrease towards 0.025

Hyperbolic, to be precise.

Assuming the average income per province does not fall at a worse rate (I think that's fairly safe)

=> More provinces give faster tech

Tech cost = f(n), linearly increasing in # of provinces, n
Tech investment = g(n, t) = M + NB(n) + Trade(t) + Tax(n), where
M -> monarch skill
NB(n) -> neighbour bonus, a declining function of n
Trade(t) -> trade income, generally an increasing function of time, t
Tax(n) -> tax income, an increasing function of n

Techspeed(n, t)=f(n(t))/g(n,t) -> quite complicated function of n and t.

I would not jump to conclusion that "more provinces give faster tech".

Let's use an example I've made in Tech groups should be spread apart again thread:

Navarra, 2 provinces, early game, ignore trade altogether:
infra 3 tech cost = 3630 (from above table)
infra 3 investment = 2d (budget) + 7d (monarch) + 6d (neighbour bonus) = 15d
researsch time = 3630/15 = 242 months

Provinces that are close to Navarra are worth between 10d/year and 14d/year (exceptions are Catalonia, 22d, and of course Toledo with its gold mine). Let's make an assumption that each province that Navarra can conceivably capture in the early game is worth 12d/year=1d/month.

Say, Navarra expands to 10 provinces:
infra 3 tech cost = 6200 (from above table)
infra 3 investment = 10d (budget) + 7d (monarch) + 1d (neighbour bonus) = 18d
research time = 6200/18 = 344 months

18 provinces
Infra 3 tech costs = 6800
infra 3 investment = 18d(budget) + 7d (monarch) + 0d (NB) = 25d
research time = 6800/25 = 272 monts, still slower than at 2 provinces.

Notice that:
(1) trade income will make results only worse, since trade income does not depend on countrysize in any way;
(2) Navarra does not have to research infra tech only. It will necessarily lag behind in all other techs;
(3) Tech-slowdown due to expansion is much worse for non-latin, narrowminded, decentralised nations
(4) I completely ignored (negative) effects of increased stability costs and badboy value, and the cost of capturing and holding aforementioned 18 provinces.

The only equalizer between small and large nations are manufactories that large nations can build more of. But with vastly increased manufactories costs in 1.07betas it is not all that easy to build a boatload of refineries.

Now, getting back to the "steppes are worthless" debate that motivated this thread, Sibir provinces are worth a lot less than Cantabria and Gerona, contribute virtually nothing to census, and cost 100d in stability. It is very hard to argue that conquering Sibir is a smart move for Russia. Same is true for England/Eire, Ottomans/Moldova and all other quoted examples.

cheers,
-- mueller

[Peter Ebbesen]

Originally posted by mueller
Tech cost = f(n), linearly increasing in # of provinces, n
Tech investment = g(n, t) = M + NB(n) + Trade(t) + Tax(n), where
M -> monarch skill
NB(n) -> neighbour bonus, a declining function of n
Trade(t) -> trade income, generally an increasing function of time, t
Tax(n) -> tax income, an increasing function of n

Techspeed(n, t)=f(n(t))/g(n,t) -> quite complicated function of n and t.

You left out Production(n, i_p), and TradeTax(n, t_p), where i_p is the infrastructure percentage and t_p the trade percentage in your tech investment equation. They are not inconsequential.

[mueller]

Originally posted by Peter Ebbesen
You left out Production(n, i_p), and TradeTax(n, t_p), where i_p is the infrastructure percentage and t_p the trade percentage in your tech investment equation. They are not inconsequential.

You are right PE. They were incorporated into Tax(n) [I have used total province income in the above example] but it really should be Tax(n, t) [since both i_p and t_p are perhaps best modeled as functions of time].

[Edit] On the second thought, even more complex than this:

Total_tax=f(n, i_p, t_p)
Trade=h(t, t_p)

and both i_p and t_p are complicated functions of t and n (since if you believe above argument higher n makes it harder to reach next i_p and t_p levels).

[SJG]

While it's all very well saying that all these factors contribute to income it is not getting us anywhere nearer answering the question of whether increasing the number of provinces you own increases or decreases research speed.

To do any meaningful analysis we need to try and get an equation that contains as few other factors as possible. One can certainly make a few reasonable assumption that greatly simplify the equation.

For a start time, the monarch's skills, neighbour bonus, production efficiency and trade efficiency are not related to the size of the country in any way. There are indeed a complex set of interdependencies that exist between almost all facets of the game but if you are going to consider all these factors then it is just as relevant to include a plethora of other factors. Continue along this path and soon you will have to look at the situation on a case by case basis, knowing the exact situation.

[Peter Ebbesen]

Originally posted by mueller

and both i_p and t_p are complicated functions of t and n (since if you believe above argument higher n makes it harder to reach next i_p and t_p levels).

i_p and t_p, depending also on dp-settings and the number of goods mnf. and refineries built, cannot adequately be described as functions of t and n, even though they often tend to rise with time.

(Counterexample: The Mughal Empire I took over in 1653 in the Tsunami game had lower t_p and i_p than it had when the game started in 1492. I blame the previous players)

You cannot easily use the argument advanced above, that higher n makes it harder to reach next i_p and t_p levels since 1) it was a conclusion based on the calculation that should now be changed to include them as a factor, and including the conclusion as a premise was all days a bad idea, and 2) higher n also means a higher overall income, and thus the ability to construct more manufactories (up to a certain limit, where manufactories no longer are profitable)

[Freiherr vStein]

Your Navarra example continued: Suppose Navarra expands to 50 provinces, but the average income per province drops to 0.7(instead of the 1 ducats per province you were willing to asume). Navarras techcost size modifier would be 3.05, total tech costs 9226.25 or something like that. The research time would be just 220 months. With 1 ducats income per province it would be 161 months.

I think it is save to asume that average research time as function of province numbers usually increases form 1-8 provinces and decreases after that. This should be true in all but the most special cases. I.e. The netherlands might be better of techwise without further expanding (after 8 provinces), but most other nations wouldn't. Pagans with zero trade income and a crappy monarch would have faster tech with 8 provinces than with one, but most nations would have faster tech as one province miniors. Of course expanding into extremely poor regions might hurt tech even after the 8th province.

[mueller]

Originally posted by Peter Ebbesen
You cannot easily use the argument advanced above, that higher n makes it harder to reach next i_p and t_p levels since 1) it was a conclusion based on the calculation that should now be changed to include them as a factor, and including the conclusion as a premise was all days a bad idea, and 2) higher n also means a higher overall income, and thus the ability to construct more manufactories (up to a certain limit, where manufactories no longer are profitable)

(1) I did include both production income and trade taxes as a factor in above calculations, I just did not put it as a part of the formula. Numbers were taken from looking at a map in a game I am trying to play in between posting here

The year is 1425 and provinces are worth
Cantabria - 14, 9+1.9+3 (taxes/production/trade taxes)
Asturias - 14, 8+3.8+3
Gascogne - 13, 8.7+2.1+3
Poitou - 10, 5.2+2.4+3,2
Roussillon - 14, 6.3+3.8+4.1
Gerona - 12, 5.2+2.9+4.1
Leon - 11, 6+2.1+3.2
Estramadura - 12, 7+2.1+3.2

and Catalonia is an exception, as I said: 22d (14+4.6+3.8)

But Navarra will also get some malus due to nationalist revolts etc (they are luvky to have both iberian and french - would be much worse if they only had basque), so 12d/year or 1d/month are ballpark-right.

I said "both i_p and t_p are complicated functions of t and n (since if you believe above argument higher n makes it harder to reach next i_p and t_p levels)". Why would this innocent statement warrant "including the conclusion as a premise was all days a bad idea"?

(2) I do not disagree, to quote earlier post: "The only equalizer between small and large nations are manufactories that large nations can build more of. But with vastly increased manufactories costs in 1.07betas it is not all that easy to build a boatload of refineries"

Now, you seem (please excuse me if I misinterpret your argument) to suggest that I rigged the numbers in favor of Navarra. The opposite is true. If Navarra has any trade income at all, it will need a hell of a lot more provinces to offset tech cost increase. I only counted one tech branch, infrastructure. Add other branches and things will get worse. Navarra is a latin-tech country with relatively rich same-culture, same-religion provinces nearby. Ottomans/Poland/Russia etc. do not have such a luxury. Etc.

[mueller]

Originally posted by Freiherr vStein
Your Navarra example continued: Suppose Navarra expands to 50 provinces, but the average income per province drops to 0.7(instead of the 1 ducats per province you were willing to asume). Navarras techcost size modifier would be 3.05, total tech costs 9226.25 or something like that. The research time would be just 220 months. With 1 ducats income per province it would be 161 months.

I think it is save to asume that average research time as function of province numbers usually increases form 1-8 provinces and decreases after that. This should be true in all but the most special cases. I.e. The netherlands might be better of techwise without further expanding (after 8 provinces), but most other nations wouldn't. Pagans with zero trade income and a crappy monarch would have faster tech with 8 provinces than with one, but most nations would have faster tech as one province miniors. Of course expanding into extremely poor regions might hurt tech even after the 8th province.

This is right - more or less
(i) Remember I ignored trade altogether. As time goes by, trade income usually dwarfs other kinds of income which makes expansion worth relatively less
(ii) But as time goes by, production/trade efficiency increases (and DP settings will be improved) which increases the value of captured provinces and makes expansion worth more
(iii) I completely ignored stabilty costs, which make expansion worse
(iv) But I also completely ignored manpower, which makes expansion better
(iii) 50 provinces is more than Navarra's BadBoy allowance

What these numbers seem to indicate is that you get best results as either
(a) small one province minor
(b) huge 50+ provinces colonial empire.

But even for colonial empires it makes little sense to capture/colonize poor provinces. Russia/the Steppes being the prime example.

I would not advocate sitting around and doing nothing as Albania. But for Tuscany/Mecklenburg (rich provinces, great monarchs) it is a good strategy to stay small, get stellar infra/trade tech and then explode in all directions using newfound firearms tech against technologically backward Aragon, France, Austria, and suchlike.