Fixing Influence - any ideas welcome

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Oberlus
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Re: Fixing Influence - any ideas welcome

#61 Post by Oberlus »

Ophiuchus wrote: Sat Aug 13, 2022 1:20 pm How about make a game rule setting for "number of empire-owned planets where you should stop consider growing" which might equal 100 according to your reasoning with current balance.

So if one doubles the number of systems one could double the new setting for a similar feeling of influence cost based on percentage of colonized universe.
If I get you right, the problem I see with that is that allowing bigger empires also makes influence more abundant during early game. Maybe that is not a problem, but it would change balance between content that needs IP (currently only policies) and content that doesn't need it.
With the percentage-of-galaxy-owned approach, early game (small empire) and late game (empire spanning most of the galaxy) has always the same balance/feeling regardless of any parameter (but still parametrizable to change ratio of influence-focused planets).

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Re: Fixing Influence - any ideas welcome

#62 Post by Ophiuchus »

I guess I need to reread your proposal.

I thought you meant scaling the upkeep formula by galaxy size.

My suggestion was to make that scaling a game rule setting instead.
So one could actually do that scaling by galaxy size, just have to do that manually.

Or in other words: I suggested putting a game rule value in your influence upkeep formula (whatever that may be) where you would put galaxy size
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Re: Fixing Influence - any ideas welcome

#63 Post by Oberlus »

Then I understood you well. If you place a scale factor to influence upkeep, when you play with that parameter, influence requirements depending on number of planets vary, but influence production from colonies stays the same. Factors for small galaxies (multiplying upkeep by bigger numbers) makes saving up IPs for mid game policies more difficult (needs more time or more planets set to influence) compared to factors for bigger galaxies. So, in order to be able to colonize huge galaxies you are forced to make influence negligible during early game.

Ophiuchus wrote: Sat Aug 13, 2022 2:05 pm I thought you meant scaling the upkeep formula by galaxy size.
Not exactly. It is similar to the pop growth equation we have, that makes growth slower when current value approaches target. Linear in this case.
When designing the equation (not the parameters to be configured by the playe), you set a desired upkeep value for empires with a single planet (0 or 1, but translates to "0% of colonies need to be influence-focused") and estimate the upkeep value that should cost a colony when whole galaxy is dominated by the empire to be balanced with what an average end-game empire (with well-developed techs and policies) can achieve. The balance parameter that the player could tweak would be "percentage of influence-focused colonies needed at end-game". If setting 80%, an average empire holding half the systems of a galaxy would require 40% of its planets set to influence, and would require 80% when dominating all the galaxy.

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Re: Fixing Influence - any ideas welcome

#64 Post by BlueAward »

Some more influence math for consideration. Not sure if I'm providing a new perspective to anybody but interesting for me at least and I'm sharing it

I talked about upkeep cost being limiting factor on empire's size. Assuming no outposts, I understand that the total upkeep cost is:

U = 0.4 * sqrt(n) * (n-1) - 3

0.4 sqrt(n) is upkeep cost per colony, but capital seems not to get that cost, hence times (n-1). Meanwhile, Imperial Palace gives +3 influence, so I've added that to the mix (as -3 to upkeep cost, though might have put as +3 to influence gain, same thing really - prefer it on the cost side though)

And now you want to pay that upkeep with x planets put on influence focus, with average influence output per planet being I

U = x * I

So you may ask, at what point x = n? In other words, what is the amount of colonies that means all of them have to be put on influence focus to pay for the upkeep? Adding any more would mean you can no longer pay the upkeep at all, so that's a pretty hard limit

If you plug in I = sqrt(30), it comes to... drumroll... 190 planets. Plugging in I = sqrt(36), or I=6, gives 227 planets:

U = 0.4 * sqrt(n) * (n-1) - 3 = 0.4 * sqrt(227) * 226 - 3 = 1362.01 influence upkeep cost

Meanwhile, 227*sqrt(36) = 1362 influence income (so ok, the roudning to 227 actually gives a wee bit not enough)

Things stop making sense before hitting that cap, though. There's a point where adding more planets causes having to switch other planets to influence, to pay for the upkeep, so fewer and fewer planets get production or science focus

Thinking about upkeep as function of number of planets n, U(n), you want the difference between U(n) and U(n+1) to be lower than I, otherwise the newly added planet can't even pay for itself i.e. forces other planets to switch to influence focus

So new planets are able to pay for themselves as long as

U(n+1) - U(n) <= I

Plugging in I = sqrt(30) and the formula for U(n) = 0.4 * sqrt(n) * (n-1) - 3 you get... drum roll.. 83 planets (~83.4989). Any new planet after that can't pay for itself in terms of influence, though other planets may pick up slack until the previously mentioned "harder" cap. For I = sqrt(36), you get 100 planets.

And of course any policy / species considerations change these numbers. For I = sqrt(30)*1.5, you get 187 planets instead of 83, how about that? Granted, for 83 planets and I=sqrt(30), you get 295.822 upkeep cost, and can pay for it barely not enough with 54 planets, so you'd need 55 on influence with some influence surplus, leaving you with only 28 useful planets (about 34% useful planets). For good influence species, the 187 planets cost 1014.40 influence and are covered by 124 planets set to influence, so you're left with 63 productive planets (again about 34%)

Hmmm were you designing things so roughly 2/3 of end game planets would need influence focus to pay for the most possible otherwise "productive" planets?

But note 63 productive planets are 225% of 28, not 150%. Then, it says nothing about dynamics of getting there to that theorized end state. So maybe still better to consider that for 83 planets, 150% species would need 37 planets on influence, not 55, so 46 productive planets, 164% of 28

The nature of this is discreet and considerations above don't describe the dynamics, so it may well make sense to stop even before the discussed caps, because more and more new planets need to be set to influence as opposed to anything else even before they stop being able to pay for themselves altogether, needing other planets to flip. So it's harder and harder to justify new planet even if it still could pay for itself and then some. You'd keep getting fewer and fewer production points extra, and even those would be even lower and lower percentage of what you already have, really diminishing the returns. But it's harder for me to put it into some equation


(EDIT: FWIW, if you set I = sqrt(30) + 4, as in artistic bonus, you get 249 planets instead of 83 on "soft cap", and 563, nearly 564, on hard cap, instead of 190! if you plug in U(n) = 0.4 * sqrt(n)*(n-1) - 3 - n, as in divine authority bonus, you get 116 planets on soft cap instead 83, and 265 on hard cap)

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Re: Fixing Influence - any ideas welcome

#65 Post by BlueAward »

Oberlus wrote: Sat Aug 13, 2022 12:34 pm I think a good solution for this is to make influence upkeep grow slower. 0.5*N^1.33 instead of 0.4*N^1.5. But Geoff expressed his preference for simple exponents (it was already a concession to use 1.5).
If you adjust exponent on the cost side but not the earning side, you lose certain... proportionality? Not clear if that was intention. Such formula gives hard cap of 1319 planets, and soft cap of 555 planets, with 139 set to production. If you put cubic root instead of square root on the earning side as well, like 30^(1/3), then it gives 101 soft cap at 27 productive planets, 245 hard cap.

This all could be designed with math if you know exactly what you want, and math ending up complicated for the player I don't think is a good argument. I can attest that it is confusing at first anyway, but all a player should learn is that upkeep per planet grows with amount of planets you have, so eventually you hit a cap where all planets would need to be set on influence. And that policies etc. help putting that cap away. No need for specifics I think.

Maybe I can offer again a perspective.

U(x) being upkeep for x planets, and I being average influence per planet, gives you y = U(x)/I of planets that need to be set to influence.

You can put current U(x) and I = sqrt(30) on a chart:
InfluencePlanetsVsTotal.png
InfluencePlanetsVsTotal.png (21.4 KiB) Viewed 727 times
I put f(x)=x there as well - you can see those intercept at about 190 i.e. the hard cap I've calculated earlier

Ultimately perhaps what is the most interesting thing is difference of those two functions, which is the number of "productive" planets

p(x) = x - y

p(x) = x - U(x)/I


for U(x) = 0.4 * sqrt(x) * (x-1) - 3 and I = sqrt(30) you have

p = x - (0.4 * sqrt(x) * (x-1) - 3) / sqrt(30)

and it charts as such:
UsablePlanets.png
UsablePlanets.png (9.28 KiB) Viewed 727 times
Where this hits 0 (for positive x), it's the hard cap - you get zero productive planets, all get set to influence (at 190 planets). Where this has maximum, is the soft cap - it's the maximum number of productive planets, 28 productive planets at 83 total planets

You can put those calculations into Wolfram Alpha and see:

https://www.wolframalpha.com/input?i=0+ ... rt%2830%29

and for maximum, too, which gives what I came to earlier, maximum of 28 productive planets at 83 total planets:

https://www.wolframalpha.com/input?i=ma ... 2830%29%29

You can tweak your function to fit predetermined points more or less. p(x) = 0 for hard cap, so you can "reverse engineer" some factors if you plug in the expected hard cap to p(x) and equate to zero

U(x) = a * x^b * (x-1) - 3

If I wanted to keep b as square root, but have hard cap at 400, then

400 - (a* 400^(1/2) * (400-1) - 3)/sqrt(30) = 0 => a ≈ 0.274923587721888

But then soft cap comes at

max(x - (0.274923587721888 * sqrt(x) * (x-1) - 3) / sqrt(30) )

I.e. 60 productive planets at 177 total planets

If I wanted to keep b as square root, but have soft cap at 100 planets, then need to get derivative to find maximum:

p = x - (a * sqrt(x) * (x-1) - 3) / sqrt(30) => p'(x) = 1 + (a*(1 - 3*x))/(2*sqrt(30)*sqrt(x))

p'(100) = 0 => 1 - (299*a)/(20*sqrt(30)) = 0

a = 20*sqrt(30)/299 ≈ 0.36637

And then hard cap comes at 226 planets, and for the soft cap of 100 planets you get 34 productive ones (really looks like with current formula what you get is 34% productive planets at soft cap), Though thanks to discreet nature of things, rather than going the differential route, you may go the earlier route I shown i.e. U(x+1)-U(x) <= I

Ultimately I think a different shape of p(x) would be preferable. Maybe something skewed right, so initially it takes longer to get to soft cap, but then you hit hard cap at roughly same time?
Maybe instead of exponential growth, you'd rather see logistic growth? Planet population should preferably grow logistically, which is more or less exponential at first then slows down / becomes more or less logarithmic. Don't know the actual formula used in FO for pop growth, but presumably along those lines, and seems like Oberlus wishes to have something like that for number of usable planets, where hard cap is the entire galaxy but you want to get there slowly after hitting some form of soft and hard-ish cap (I can envision cost that eventually means every planet on influence, and every new planet on influence, but never going negative so able to colonize entire galaxy)

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Re: Fixing Influence - any ideas welcome

#66 Post by Daybreak »

When population < maximum, population of next turn will grow by
population * 0.08 (or 0.12 with policy Population) * ( 1 + (1-population)/maximum )

However there is a proposal to mak it
population * 0.1 (or 0.15 with policy Population) * ( 1 + (1-population)/maximum )

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Re: Fixing Influence - any ideas welcome

#67 Post by Oberlus »

Krikkitone wrote: Mon Sep 23, 2019 7:50 pmMin + (Max-min)*(% of galaxy colonized)
Min could be 0 (currently that's what we have):

Max*(% of galaxy colonized)

Calculating Max requires more thought. It must be the average value for an end-game (all techs plus good combination of policies), relatively good (say Good Influence species), influence-focused planet of average size (medium size 3 * [good env. 3 + techs 5 + growth specials 3 + gaia 3 = 14] = 42, but lets take 36). Assuming Good Influence is +25%, that's 1.25*6 = 7.5 IP plus flat bonuses from policies. So Max could be 8 or 9 (would be a game rule).
Then add a parameter P in [0% .. 1000%] to allow the player configure the expected ratio of planets set to influence when galaxy is fully conquered:

P*Max*(% of galaxy colonized)

P=0% means no influence upkeep
P=50% means conquering the whole galaxy requires 50% of owned planets set to influence.
P=100% means conquering the whole galaxy requires 50% of owned planets set to influence.
P=200% means conquering half the galaxy requires 100% of owned planets set to influence (i.e. impossible to pay upkeep for whole galaxy).

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Re: Fixing Influence - any ideas welcome

#68 Post by BlueAward »

Daybreak wrote: Sun Aug 14, 2022 6:12 pm When population < maximum, population of next turn will grow by
population * 0.08 (or 0.12 with policy Population) * ( 1 + (1-population)/maximum )

However there is a proposal to mak it
population * 0.1 (or 0.15 with policy Population) * ( 1 + (1-population)/maximum )
thank you! Yeah to be honest I am not sure if that is exactly logistic growth but it's very close, and certainly makes more (practical) sense in context, like you're able to actually hit that darn limit rather than getting ever close to it

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Re: Fixing Influence - any ideas welcome

#69 Post by BlueAward »

Oberlus wrote: Sun Aug 14, 2022 6:53 pm
Krikkitone wrote: Mon Sep 23, 2019 7:50 pmMin + (Max-min)*(% of galaxy colonized)
Allright, so this new formula assumes end game empire should be able to make 8 IP per planet, so I can plug it into current formula instead of sqrt(30) for more apples to apples comparison of "end game" state

p = x - (0.4 * sqrt(x) * (x-1) - 3) / 8

That gives you hard cap of 402 planets, and soft cap at 178 planets, where you have 60 productive

The new formula for P = 100% and galaxy size 402, assuming end game empire does pull off the prescribed 8 IP per planet:

p = x - (8 * (x/402) * (x-1) - 3) / 8

Now this actually goes to 403 thanks to not counting capital (x-1) and the incosequential capital bonus of +3. Not clear to me if you want to include capital or not. Also assuming your empire can pull off MAX then this MAX pretty much cancels out in the end, though real world case you probably won't be pulling off that MAX so will get smaller cap)

Unsurprisingly soft cap is at about half the universe, about half of which would be productive (201 planets, 101 productive)

Current formula has 1.5 exponent rather than 2 in the proposal, so produces fewer productive planets overall when comparing at same hard cap. Another thought is that the current formula could be scaled to the size of universe, too, if you tweak the 0.4 to be something else,as I shown previously. I suppose the new approach has the benefit of easy parametrization, though.

Putting both on same chart:
UsablePlanets.png
UsablePlanets.png (13.9 KiB) Viewed 677 times
If you scale the current formula to 200 planets

200 - (a * sqrt(200) * (200-1) - 3) / 8 = 0
=> a≈0.56959

p = x - (0.56959 * sqrt(x) * (x-1) - 3) / 8
vs
p = x - (8 * (x/200) * (x-1) - 3) / 8

Chart looks like this:
UsablePlanets2.png
UsablePlanets2.png (15.26 KiB) Viewed 677 times
I mean the same thing just rescaled, which should not be surprising.

So, when comparing "apples to apples", the new formula gives you more productive planets than the current formula at any stage, and the ratio of productive planets predictably drops in linear fashion compared to how many planets you have, while old formula had non-linear progression of the ratio, and I think bit skewed left, too, so it soft caps sooner, too. Like 50% usable planets on soft cap in new formula vs 34% in old formula.

Old formula can be rescaled to a percentage of universe size, too, but new formula is more easily parametrized to apply the scaling, and has already baked in other considerations for parametrization

Guess depends on the ratio of usable planets you want to see, and if you think things should be skewed in some fashion - for example early game skewes things because you don't have those end game "influencers" considered for the formula, so perhaps you want a curve that is skewed right

I don't have enough experience to tell which is better but I don't see anything wrong with the new formula, and it's easier to parametrize, just keep in mind old formula could be scaled to universe size as well. But it is not, so that would require change as well, and overall I do like the idea of scaling to universe size

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Re: Fixing Influence - any ideas welcome

#70 Post by Oberlus »

BlueAward wrote: Sun Aug 14, 2022 11:51 pm Allright, so this new formula assumes end game empire should be able to make 8 IP per planet, so I can plug it into current formula instead of sqrt(30) for more apples to apples comparison of "end game" state

p = x - (0.4 * sqrt(x) * (x-1) - 3) / 8
The idea is to turn the upkeep equation for a colony from this
0.4*sqrt(x)
to this:
parameter * 8 * (% of galaxy colonized)

with parameter in [0, 10], default to 1, and % of galaxy colonized = #colonized planets / #planets in galaxy at game start

So the hard cap with default parameter would always be at the size of the galaxy set by the player at game start.

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Re: Fixing Influence - any ideas welcome

#71 Post by BlueAward »

Oberlus wrote: Sun Aug 14, 2022 11:59 pm
BlueAward wrote: Sun Aug 14, 2022 11:51 pm Allright, so this new formula assumes end game empire should be able to make 8 IP per planet, so I can plug it into current formula instead of sqrt(30) for more apples to apples comparison of "end game" state

p = x - (0.4 * sqrt(x) * (x-1) - 3) / 8
The idea is to turn the upkeep equation for a colony from this
0.4*sqrt(x)
to this:
parameter * 8 * (% of galaxy colonized)

with parameter in [0, 10], default to 1, and % of galaxy colonized = #colonized planets / #planets in galaxy at game start

So the hard cap with default parameter would always be at the size of the galaxy set by the player at game start.
Yes, I understand. Hence for the new way, number of productive planets is a function of

p = x - (parameter * 8 * (x/galaxy_size) * (x-1) - 3) / influence_per_planet

x/galaxy_size is the % of galaxy colonized, and I used "1" as parameter to have the hard cap at galaxy size, and also assumed the planets can pull of the "8" as influence_per_planet. I kept (x-1) part and -3 part from current game, which does not apply the cost per colony to capital and instead gives a flat bonus of 3.

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Re: Fixing Influence - any ideas welcome

#72 Post by BlueAward »

OK let me put things other way.

So the new upkeep per colony is

P * 8 * (x/galaxy_size)

Let's plug in P = 1 then upkeep is

8 * (x/galaxy_size)

Suppose you played a bit and wanted to mix up things and instead put it as

8 * sqrt(x/galaxy_size)

Maybe now you can see that

0.4 * sqrt(x)

Is just like the "new" way, except it has some baked-in, hardcoded galaxy_size, along with P and 8 hidden within the 0.4 factor. My point was that it all could be reclaimed as parameters instead of being just 0.4!

Where the "new" way is substantially different is that it is not using sqrt

EDIT: edited out some other thoughts for clarity

EDIT2: and I mean the new way actually does bring all the parametrization front end center, which is the main intention, and very nice indeed! That's why I like it!

EDIT3: ok final thought, I guess this means if you used P * 8 * sqrt(x/galaxy_size) instead, then it would be practically what we have now, except easily parametrized by galaxy size etc instead of having things hardcoded within "0.4" (my calc shows the "hidden" galaxy size in this 0.4 is 402 planets if you go with P=1, and sqrt... or rather 400, but extra comes from not paying for capital and it giving +3 influence instead... which is obvious if you go 8 * sqrt(x/galaxy_size) = 0.4*sqrt(x) => galaxy_size=400)

EDIT4: OK I lied,one more thought - it's not like I'm actually recommending going with sqrt in the new approach! I haven't considered the implications. If you went with sqrt, you could think of things currently scaled to galaxy size 400, so if you normally play on smaller galaxies then game would change quite a bit. Mission statement of going with % of galaxy conquered is clear, putting some exponents on it muddles things

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Re: Fixing Influence - any ideas welcome

#73 Post by Oberlus »

BlueAward wrote: Mon Aug 15, 2022 12:09 am Yes, I understand.
Gah! I'm sorry, I was reading too fast when I answered.

OK, so using Max*sqrt(colonies/gal_size) we would get a different curve, reaching soft cap sooner.
E.g. for 49% colonized galaxy, upkeep value would be 70% of max upkeep.

Thinking of balance during game, and the intended effect of making successful/lucky empires not so successful compared to the others, it seems better to have the soft cap sooner, so that alliances of two small empires have real chances of winning against a single big empire with twice their planets. <-- If this backfires (making alliances the only way to win) we could fall back to linear.

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Re: Fixing Influence - any ideas welcome

#74 Post by Daybreak »

Oberlus wrote: Mon Aug 15, 2022 7:54 am (making alliances the only way to win)
Won't it always be that way anyway.
Thinking of balance during game, and the intended effect of making successful/lucky empires not so successful compared to the others, it seems better to have the soft cap sooner, so that alliances of two small empires have real chances of winning against a single big empire with twice their planets. <-- If this backfires (making alliances the only way to win) we could fall back to linear.
Do we have to be careful here?
A lucky empire does not have to be large, but would have an advantage against a larger empire who have been unlucky, because you are hobbling them with higher IP cost. IE: The larger empire has to be larger to just compete against the lucky empire.

The two smaller empires theoretically may have better PP and RP (as they have lower IP costs), and in fact if a larger empire had spent more time colonising, then the smaller empires may have more fleet as well.

Maybe we are looking at this the wrong way. In this thread we are trying to play with the formula to fix this problem. In this thread viewtopic.php?f=6&t=12485 we are adjusting the influence percentages.

What if (Here we go again - I perceive a possible problem/solution, but may not explain it properly.

we base Colony costs on influence and planets, not just planets. We dictate a percentage, or a percentage range, so if -
* the number of planets required to be turned to IP vs other focus is lower than our dictated percentage range, then IP colony costs are higher.
* the number of planets required to be turned to IP vs other focus is higher than our dictated percentage range, then IP colony costs are lower.

To bring the number of planets focussed on IP up/down to achieve the dictated percentage range. We don't want it to be exact, as we may need to save IP for policies

Does that not counter if one empire is lucky of not lucky?

EDIT
I don't lnow how, but if that is then tied to how much of the galaxy you own, so the percentage range moves up making it more expensive in IP the more of the galaxy you own, and the percentage range moves down the less of the galaxy you own.

That means many small empires would still have a chance against a larger empire.
Last edited by Daybreak on Mon Aug 15, 2022 8:09 pm, edited 1 time in total.

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Re: Fixing Influence - any ideas welcome

#75 Post by Oberlus »

Daybreak, if you can come up with some number to back your arguments, I'll gladly answer to your concerns. Otherwise, I'm tired of that discussion.
_________________________

Edit:

A "lucky empire" (this comes from older threads) is one that has a nice combination of species, colonizable planets and specials around his starting world. Being lucky implies it can grow faster: get more/better planets and more population sooner -> get more PP/RP/IP sooner -> get more warships/troopers/colonizable planets -> get more PP/RP/IP sooner (etc.). This can never be bad for the empire by itself.

If a lucky empire chooses a bad strategy (like building up too much army without anything to do with it, or not building any army when there is a potential thread, or making bad choices of planets, techs and policies), that is up to the player and nothing that should be "fixed".

The motivation to hinder the growth of lucky empires is for gameplay experience: if a lucky (unlucky) start means you have already won (lost), you lose interest on the game.


A "successful empire" is one that is doing well compared to others. It can be a "lucky empire" that didn't do wrong or just an empire that is getting the most out of what the galaxy offered it at start, and doing it better than the neighbors.

The motivation to hinder the growth of a succesful empire is the same: to extend the game time that is thrilling.

_________________________

Forcing a fixed ratio of planets set to influence sounds dull and boring like hell. Let the players do a good or a bad job with what they get! Knowing that the upkeep will adapt to your needs, demanding less if you make bad choices of species, planets and policies, sounds just aweful.

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