Philosophical: Long-term prospects

[UncleJJ]

It is 7. The maximum cost of the "number of cities" component of city maintenance is capped at 7 for Emperor in BtS (but I'm still playing with the 3.13 patch).

I'm surprised that Ozyar says the player's civic upkeep costs are the same on Diety and Emperor. Is the only difference between difficulties the discounts that the AI gets on its costs and upgrades and its increased bonusses for production and research etc? Are there no further penalties for the player at higher difficulties? Does anyone have a link to a discussion of the differences between difficulties? I used to have one but can't find it

 

[Maxit]

@Maxit Google asymptotics

I'm not sure how you think asymptotic analysis applies, unless it's the diminishing return of GPPs with increasing N.

 

[Iranon]

Re the GPP formula (current, not cumulative): It's a bundle of linear formulae that approaches a quadratic as a whole.

10th GP costs 1000
20th GP costs 3000
30th GP costs 6000
...

For every multiple of 10, the GP cost 5x^2+50x, quadratic.
However, in between the cutoff points the progression is only linear.

This means the total GPP required for x GPs is a cubic function seen as a whole, behaving as a quadratic within each 10-GP-interval.

Since the time period between 10 GP is a significant portion of the game yet we're likely to produce considerably more than 10 in many games, we can't ignore either aspect.

***

Also, I would disagree with the statement that you can never forgo the benefits of your trait as a philosophical leader. When I first started the game I never ran specialists (I thought growth was more important; which it often is), and when I went up in level I would/could not build any wonders. In those cases I would never generate a GP, so being philosophical would just be a waste. (Though I now think my tactics could have been better)

Based on your current grasp of the game... can you really regard this as anything but extremely poor play?
Let's assume you didn't pop a GP yet. Before you are way into the Renaissance, working even 2 mature Financial towns for 34 turns only gives you 340. Working 2 farms to support a scientist instead gives you 102 and 1 Great Scientist, which is much, much better. This is without PHI, without excess food from resources, unrealistically assuming you will have mature towns further boosted by a trait you might not have and discounting Representation which you might.

If you get less than 3 or more than 5 additional Great People out of PHI, you almost certainly played badly.
The former implies neglecting specialists to an extent that would be a mistake with or without PHI. The latter implies a strong focus on raw GPP without using other GPP multipliers, not even getting a decent GP farm and building the National Epic there. Maybe excusable in some weird games (oddities happen), but usually not,

It's possible to squeeze a little extra out of PHI by running a few specialists across the empire before you have a decent GP farm, thus getting a decent head start... but that's about the only thing that requires active leveraging.

 

[Celebithil]

Asymptotics are relevant as I was almost explicitely talking about them when I claimed the number of GPP required was quadratic. Besides, the title of the thread contains "Long term prospects".

Speaking of asymptotics you can easily calculate the quotient of the number of GP produced by a philosophical and a non-philosophical leader, given they have the same single GP farm in the limit as the game goes forever. In fact (using quadratic asymptotics, and assuming the NE is built in both GP farms), this converges to (3/2)^(1/3) (the factor of the extra amount of GPP to the power 1/3 as the amount of GPP necessary to get the first n GP goes as a cubic (in asymptotics).

Similarly one can calculate the effects of adding secondary GP farms with the same amount of base GPP. The results are (when a single GP farm for a non-Phi non-Pacisfist leader is scaled to 100, and all results are rounded to integers) for 1 through 5 GP farms in each case:

non-phi, non-pac: 100, 114, 126, 136, 144
phi or pac : 114, 136, 152, 165, 177
phi and pac : 126, 152, 171, 187, 200

So the effect of being philosophical is certainly underwhelming (1/7 extra GPs for 1 GP farm), and extra GP farms are better for a philosophical leader (instead of giving an extra 14 for the second GPfarm it becomes an extra 22).
Still a quarter of a single GP farm is not extremely good.

Note that this table disregards the effects of cut-off points and GPP being wasted in cities not making GPs. Also it disregards the fact that a PHI leader will get his GPs earlier. Therefore I would presume the 114 value of one GP farm under a PHI leader is underestimated, while the (same) 114 for 2 GP farms for a regular leader is an overestimate. (This again shows that in general it really is a bad idea to keep several GP farms, at least in the long run).

 

[Bleys]

One thing I havent seen discussed is how well the faster GPeople helps get to the techs that offer free GP to the first to discover. We all know how well it can speed up the road to Lib, but what about the other non-religion "Bonus" techs? Whether you bulb/trade or settle them, I think there is a lot to be said for getting to techs that give free GP's faster. Economics, Physics, even Communism, etc, can all be reached quicker, even on higher levels. This is kind of an unmeasurable intangible.

 

[Celebithil]

Some more thinking along the lines of "quotient with the one GPfarm case, but now with a more precise formula for the number of GPP needed to obtain n GPs has led me to change my values of different scenarios. The formula I used for F(n), the total number of GPP needed to get n GPs is
F(n)=5 x^3/3 + 55 x^2/2 + 325 x/3 + 165/4 + G( (n/10)-[n/10]),
where the G-part is a 10-periodic error term, not exceeding 81 1/4 (to be precise the errors are (41 1/4, 78 3/4,81 1/4, 58 3/4, 21 1/4, -21 1/4,-58 3/4, -81 1/4, -78 3/4, -41 1/4), the constant in the asymptotics obviously chosen to make this list symmetric under multiplying by minus one and reversing the order.

Subsequently I estimated the number of GPs acquired up til a certain point in a situation with k GPfarms as f^(-1)( GPP/ turn * #turns) - # GP farms/2 (where the last correction is added to simulate that you always have some remaining GPP in your cities, on average to the amount of half a GP per city.

Considering the relevant length of the game to be the area in which you have between 1,000 and 10,000 base GPP in your GPP farm (the lower bound also avoids some problems with singularities obtained in the calculations) and just taking as the value the combination of the maximum and minimum amount of the quotient of number of GPs obtained in a scenario and the base scenario (of one GPfarm for a normal leader), we get the following scores (1 GP farm, 2 GP farms, ... , 5 GP farms in each scenario). Note that in most cases the maximum value is attained at 1000, and the minimum at 10,000.

nonPHI, non PAC: 100, 116-117, 128-129, 138-139, 146-147
PHI or PAC:119-126, 144-157, 162-178, 177-194, 189-206
PHI and PAC: 134-148, 165-187, 187-214, 205-235, 225-259

Note that the values are all considerably higher (increasing the value of extra GP farms and the PHI trait). Also there are quite big changes in time of these values, but the general idea is clear.

Also the decrease in time suggests that the relative benefits are highest in the earlier game, which corresponds to the cubic term in F(n) taking over the quadratic term only after 16 GPs (which is the order of size we should consider changing the growth in number of GPP necessary for n GPs to become cubic instead of quadratic).

 

[Maxit]

Interesting analysis. The treatment of GPP vs. N with a polynomial function seems proper.

I just completed GOTM 35 (Elizabeth/Monarch/Standard Map/Normal Speed) with a cultural win. In that game, I think Phi was of more value than Org would have been, since the strategy used strongly depends on generating numerous Great Artists (32% of the required culture points came from GAs). So Phi was valuable "long-term" in this case, since the win is faster due to the quicker GA generation. At game end it was 80 GPP/t vs. 60 GPP/t without Phi in my GP farm with 6 artists/NE/Caste System. I finished with 13 GPs total, so it saved me 25-35 turns more or less (SWAG). Full analysis depends on effect of timing of NE build and GP Farm full capacity, but the more specialists, the better the benefit.

In general, it seems that evaluating the long term value of Phi boils down to the long term impact of the more/sooner GP. I think it is clear from this thread that there will be a different evaluation of the trait for each game strategy/situation.

 

[UncleJJ]

Some more thinking along the lines of "quotient with the one GPfarm case, but now with a more precise formula for the number of GPP needed to obtain n GPs has led me to change my values of different scenarios. The formula I used for F(n), the total number of GPP needed to get n GPs is
F(n)=5 x^3/3 + 55 x^2/2 + 325 x/3 + 165/4 + G( (n/10)-[n/10]),
where the G-part is a 10-periodic error term, not exceeding 81 1/4 (to be precise the errors are (41 1/4, 78 3/4,81 1/4, 58 3/4, 21 1/4, -21 1/4,-58 3/4, -81 1/4, -78 3/4, -41 1/4), the constant in the asymptotics obviously chosen to make this list symmetric under multiplying by minus one and reversing the order.

Subsequently I estimated the number of GPs acquired up til a certain point in a situation with k GPfarms as f^(-1)( GPP/ turn * #turns) - # GP farms/2 (where the last correction is added to simulate that you always have some remaining GPP in your cities, on average to the amount of half a GP per city.

Considering the relevant length of the game to be the area in which you have between 1,000 and 10,000 base GPP in your GPP farm (the lower bound also avoids some problems with singularities obtained in the calculations) and just taking as the value the combination of the maximum and minimum amount of the quotient of number of GPs obtained in a scenario and the base scenario (of one GPfarm for a normal leader), we get the following scores (1 GP farm, 2 GP farms, ... , 5 GP farms in each scenario). Note that in most cases the maximum value is attained at 1000, and the minimum at 10,000.

nonPHI, non PAC: 100, 116-117, 128-129, 138-139, 146-147
PHI or PAC:119-126, 144-157, 162-178, 177-194, 189-206
PHI and PAC: 134-148, 165-187, 187-214, 205-235, 225-259

Note that the values are all considerably higher (increasing the value of extra GP farms and the PHI trait). Also there are quite big changes in time of these values, but the general idea is clear.

Also the decrease in time suggests that the relative benefits are highest in the earlier game, which corresponds to the cubic term in F(n) taking over the quadratic term only after 16 GPs (which is the order of size we should consider changing the growth in number of GPP necessary for n GPs to become cubic instead of quadratic).

I've tried to understand what you're saying here and then decide how relevant it is to an actual game. You seem to be comparing the added benefit of adding several GP farms all with the same GPP output (a very unlikely occurence). But these GP farms all spring into existence suddenly and the number of GPs produced before you start the counter is zero.

The way the GPs are produced in most of my games is very different from that. It doesn't matter really whether I'm Philosophical or not, the same pattern is followed although it's much easier and quicker with the trait. My first 2 or 3 GPs are usually produced either by an early wonder like Stonehenge or Great Wall, or by running 2 scientists from a library. So there is an early phase to GP production which is very different from what you have assumed.

Then later in the middle game I might build a capital with multiple wonders and add the NE or have a high food GP farm that runs Caste System until Liberalism and Chemistry have been lightbulbed. Pacifism may or may not be used in this phase. The first golden age might be used to switch civics and boost GPP production temporarilly. This middle phase can be considered to start when the Great Library and / or NE are built and ends when Biology, Mercantilism and /or SoL is generating widespread GPPs.

Then there is a late game phase when there can be multiple captured cities with wonders and good food. This is when there is a chance of multiple high output GP farms. The National Park national wonder might be a significant contributor. If corporations are used, particularly one of the food ones, then GPP production can really blossom.

So it seems to me that the profile of GPP rate of production is very different from your assumptions. It starts small and grow rapidly through a middle phase and then can explode in the late game due to technologies, captured cities and corporations. This profile is totally different from the assumption of a fixed number of GP farms (one with the NE perhaps) all starting with considerable GPP production and all starting with zero GPs produced.

Will these significant differences significantly affect the results? How can you justify your assumptions? or reconcile them to a real game?

 

[KJIOYH]

You cant really compare PHI to other traits using some sane mathematical formulas because: a) PHI benefits different strategies differently b) In the end, you got to compare leaders (traits+UU+UB) during certain conditions (food, research, MP or SP, difficulty, map type etc etc) and strategies (SE, CE etc.) c) Ive no idea how you plan to give a mathematical number for quick creation of first few GP.

I personally think that Philosophical trait benefits most in certain situations, SE and cultural victory mostly. I definately wouldnt take Phi if i went for CE and i dont know . .. .. .. . about cultural victory.
I think everyone will agree that SE's efficiency and gambits are based mainly on GP and Pyramids+GL (one cant influence food allocation without reloading). +50% research from academy in capital early game is ~50% of the countries research, espercially with GL. Having academy 2x faster because of PHI is a nice bonus because its like getting a free tech. GL+academy leads to greater bonus. GL+academy+quick GS leads to quick Oxford. From my experience in SE the capital generates ~70% or more of the countries research > Academy+Oxford gives horsehockeyloads of research aka benefits SE more than CE. After Oxford PHI can RIP as much as i care.
Second issue with SE and GPP is it's dependance on Pyramids and GL. Assuming MP or higher difficulties, If theres no stone available really close (because of maintanance etc.) which is rare, the only way to build these wonders faster than other determined people is with engineer and PHI is cruical here.

Overall: 1) SE w/o PHI = SE w/o Pyramids+GL = absurd
2) PHI main benefit - 2x speed of generation of the first GP and maybe next few. It is deffinately more beneficial than some GP late game which might never come and when you generate god knows which GP due to pond corruption.
3) PHI opens possibility of Oracle>forge>GE>pyramids+GL gambit for SE which is quite important. W/o PHI it will take way too long.
4) IMHO PHI is only good for SE. SE depends on food which you see when the game has started. SE is hardly if at all better than CE. Conclusion: If you take PHI you are gambling a lot that you will have food and will be able to build Pyramids and GL and start efficient SE, getting allmost nothing in return.
5) CE with PHI w/o leveraging ends with those 2 bonus GP in the end. I think its better to rush courthouses with some other trait than trying to leverage PHI with GL+academy+rush for oxford/liberalism combo.

my perosnal overall conclusion: Its bettter to forget about whatever+PHI if efficiency is of most improtance, less you got some evil plan involving early mainly first GP.

 

[ParadigmShifter]

Meh, you don't need the mids for an SE but it helps. Food is king.

GLib is great anyway, phi makes it better. It's a dud for cultural games though.

 

[Unconquered Sun]

Overall: 1) SE w/o PHI = SE w/o Pyramids+GL = absurd

False.

3) PHI opens possibility of Oracle>forge>GE>pyramids+GL gambit for SE which is quite important. W/o PHI it will take way too long.

It's not important at all.

4) IMHO PHI is only good for SE.

PHI is a great sidekick trait for CE.

SE is hardly if at all better than CE.

CE shortcomings are in fact considerable.

 

[Shadowcrow]

I guess it would be like the difference between a graph and the area under a graph.

But if you were to graph the number of great people you have, sur ein the end you only have 4 more, but what you want to look at is the area (represented better if the graph resents the bonus given by GPs) between the PHI curve and the Non-PHI Curve. That essentially gives you the accumulated benefit.

 

[KJIOYH]

You can ofcourse go for SE with one city in the mountains. This is prolly wrong topic so ill STFU.

 

[Iranon]

@ Shadowcrow:

I prefer working with the number of GP I'm ahead at any given time. It's fairly easy to convert 'GP ahead' into 'specialistturns ahead' if you have the complete information over the game if that's what you're interested in... and in some cases you only care about volume anyway (e.g comparing output per turn to other economic traits, assuming you want to lightbulb but researching the prerequisites for juicy stuff takes longer then generating the GP).

***

I finally went and made a table for how much you benefit from additional cities... I found the results quite interesting.

Assumptions: The main GP farm has twice the base GPP of anything else and also benefits from the National Epic. Period length is the time it takes a support cities to generate 100 base GPP, to make it easy to compare different city sizes without having the nubers become too unwieldy - we'll simply change our period length to the appropriate amount of turns (roughly 11 for 3 specialists).

Each group of 3 colums gives the amount of total GP spawned with the number on top denoting the amount of additional specialist cities. First one without further GPP bonuses, the second with PHI or Pacifism, the third with both.

		0				3				6				10	
	2	3	3		2	3	3		2	3	3		2	3	3
	3	4	5		3	4	6		3	4	6		3	4	6
	4	5	6		4	6	8		4	6	9		4	6	9
	5	6	7		5	8	9		5	8	11		5	8	11
	5	7	8		5	9	10		5	10	12		5	10	12
	6	8	9		6	10	11		6	11	13		6	11	13
	7	8	10		7	10	12		7	12	13		7	12	15
	7	9	10		8	11	13		8	13	14		8	13	16
	8	9	11		9	11	14		9	14	15		9	14	18
	8	10	11		10	12	15		10	14	16		10	15	19
	8	10	12		10	13	15		10	14	17		10	16	19
	9	11	12		11	14	16		11	14	18		11	17	20
	9	11	13		12	14	16		12	15	19		12	18	21
	9	12	13		12	15	17		12	16	20		12	19	22
	10	12	14		12	15	18		12	17	21		12	19	22
	10	12	14		13	16	18		13	17	21		13	19	23
	10	13	15		13	16	19		13	18	22		13	19	24
	11	13	15		14	17	19		14	18	22		14	20	24
	11	13	15		14	17	20		14	19	22		14	21	25
	11	14	16		14	17	20		15	19	23		15	21	26

This confirms much of what has been claimed in this thread so far. For moderate amount of spare cities the long-term benefits are unspectacular, but it can be a noticable early boost (for 3 and 6 additional cities, we're further ahead of a non-PHI leader after period 6 as we are at period 20).

It also comfirms that additional cities become more beneficial for PHI leaders, as GP generation is less dominating by the NE city. In this example - 3 additional cities are almost as good as 6, and any more do nothing at all. With either PHI or Pacifism or both, we get a credible amount of additional GP.


***

Now another set, without a dedicated GP farm at all (maybe we don't have suitable land and/or we are planning to get a ridiculous National Park city).

	3				6				10	
1	2	3		1	2	3		1	2	3
2	3	4		2	4	6		2	4	6
3	4	6		3	6	8		3	6	9
3	5	7		4	7	9		4	8	10
4	6	8		5	8	10		5	10	11
4	7	9		6	9	11		6	10	12
5	7	9		6	10	12		7	11	13
5	8	10		7	10	13		8	12	14
6	9	11		7	10	14		9	12	15
6	9	11		8	11	14		10	13	16
6	10	12		8	12	15		10	14	17
7	10	13		9	12	16		10	14	18
7	10	13		9	13	16		11	15	19
7	11	13		10	14	17		11	16	20
8	11	14		10	14	17		12	16	20
8	12	14		10	15	18		12	17	21
8	12	14		11	15	18		12	18	21
9	12	15		11	16	19		12	18	22
9	13	16		11	16	19		13	19	22
9	13	16		12	16	19		13	20	23

Note how a large amount of moderate GP farms can compensate for the lack of a decent GP farm, but only if you are PHI or use Pacifism.