Discussion about growth rates Linear/Exponential/Logarithmic

Discussion in 'Planetary Annihilation General Discussion' started by syox, March 7, 2013.

  1. syox

    syox Member

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    OK the link posted earlier get some thinking going on my side.

    I now want to encourage a result open discussion about income and economy growth rates and there impact on the gameplay.

    First lets look at the model current RTS use:
    The exponential growth rate
    This is achieved by a linear outcome of ressource generating facilitys.
    So in case of FA style games: if one mex has X income
    2 have 2*X
    3 have 3*X
    .
    .
    .
    N have N*X

    Now lets say ah Mex costs M of this ressource and the in time needed to get a new Mex the give mex has his X income.
    So Starting with one Mex you can have ((X/M) +1) Mexes after the first building round.
    (the steps/time needed to build new mexes is called a building round from start to finish)

    So now you have ((X/M)+1)*X income
    next round you can build:
    (((X/M)+1)*X)/M=((X/M)+1)*X/M new mexes witch will give you :
    (((X/M)+1)*X/M)*X +((X/M)+1)*X income
    this equals
    ((X/M)+1)*((X/M)+1)*X or X*(((X/M)+1)^2)
    More iterations will lead you to the conclusion of I iterations of building rounds lead to a
    Income of X*((((X/M)+1)^I)

    A exponential growth in income.

    Even if you not contribute all of your income to build new mexes, the fact that the growth rate is exponential still stays:
    If you just contribute 5% of your current income you will get 5% worth more income as in 1+(X*0.05/M) so still exponential.

    This will stay truth till there is no more Mex to gain.

    pros:
    + It is the best mapping of actuall reality to a game. Species/bacteria will grow exponential till they reach natural borders or other things that slow growing down.
    (though the player is limited in his capabilities so it may get overboard fast)
    + If you are in the lead, you stay in the lead and probably will get even more ahead
    + enables big/expensive weaponary (game enders/experimentals) at the end of the game.
    + losing many stuff may not be hurting if you can replace it all in no time
    + encourages: 'hurt your opponent early' or 'get vasts ammount of economy to be strong later' strategies
    + focus on economic play

    cons:
    -after a more or less lengthy starting phase the ammount of clicking will exponential grow as the economy (this will lead to a clickfest; some may see this as a + depends on preferences)
    -if you are behind you will probably get more behind.
    -more tactical gameplay (again some +; preferences)
    -probably not suited for jump in jump out multiplayer games

    The linear growth rate

    Easiest possibility to achiev this, fixed income whole game.
    Other possibilities:
    # Income limits (by linear income as X per Mex a cap of Z and if N*X>Z cut it back to Z; N Mexcount)
    # decreasing income per mex as in (Y^N; Y<1; N number of Mex)
    (maximum income here (with infinite number of mexes): 1/(1-Y) )
    # using upkeep systems (see WC3 for instance dont leed to linear there though)

    pros:
    + building phase starts fast gets slower over time
    + things to manage overall stay even, so no growth in things to manage therefore:
    best mapping of limit and constant awarenes and handle possibilities of the player to a game.
    + if you are behind five units and handle stuff from now as your opponent you will only be behind five units in the future.
    + force strategic play
    + good suited for jump in jump out multiplayer games
    + harder to lose games
    + focus an military play

    cons:
    - discourages rush and turtel play
    - discourages player to stay in game if no advancements are possible
    - harder to win (thus a win may be more fulfilling +)
    - not as good for expensive gameenders and experimentals
    - not as many sim citiy/expansion as in nonlinear economys
    - needs mapping of player capabilities to income or game may be to hard/easy all the time
    - games may tend to be longer but not as big in scale.

    Other growth rate models

    Other models are mainly between the two above, though even more or less income models are thinkable. But mostly the more income growth then exponential models are not playable anymore, they would need you to exponential increase your input to the game.

    Less then constant income models are thinkable (for instance, start with a amount of money, by things till money is out, fight! example: the fights in UFO/ X-COM or Total war series. As there is no possibility for reinforcement while the fight happens)

    pros:
    + finite game model, last unit die you lose
    + most strategic model

    cons:
    - may be most time/thinking intensive gametype (chess, X-COM, not as much suited for real time games)
    - most merciless gametype


    Possible models between Exponential growth and linear economy, might be logarithmical models, as seen in most strategic browser games. They have either decreasing income rates (but not as though the income is limited over all, for instance 1, 1/2, 1/3, 1/4,... sum still infinite) or have increasing spending needs (tec gets more expensiv with every level, more units cost more ressource)

    pros:
    + everytime there is the possibility to get more income/units, though encourage players to stay/advance
    + awareness/click capabilities not as fast increasing in demand as in exponential growth rate systems
    + probably best suited for long term play (though therefore being the most cruel gametype [i never forget my uncles face after he lost all his ships he worked 2 years for in ogame])

    cons:
    - long term play may produce most frustration (see above)
    - still increase in stuff to handle after all, so there will be a point where your capabilities 'to do all the stuff you want to may do' may not be enough anymore
    - endless growth, therfor never be finished with it may produce frustration also



    Disclaimer
    All pros and cons or systems and modes of action are made in allconscience of the author.
    This post claims in no way to be a treatise or in 100% precise.
    The pros and cons reflect the opinion of the author (me) and may therfor be appealing different for you.
  2. CrixOMix

    CrixOMix Member

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Personally I think economy GROWTH should be different types in two dimensions, creating exponential growth overall.

    Dimension one: Progressing down the "tech tree" or build order, or whatever. Early on, you can build basic energy generation. Then later in the game, you can build bigger generators, with this being somewhat exponential growth. As you have MORE energy to build MORE generators, etc.

    Dimension two: Having more and more AREA to build buildings in, so the rate at which you construct energy generation buildings is going up at a fairly linear rate. (Land acquisition, if anything, peaks mid-game and slows as the game progresses)

    Assuming you are progressing through both dimensions at the same time, resource growth is exponential.

    Personally I think the longer your economy goes, the more stable it should be, with larger storage values, longer build times. But then if you get a constructor army, you could drain it all quickly as always.

    And it's also important to have high-generation structures, otherwise late game, it's hard to pinpoint anything to kill to damage their economy. This is why the generating structures should be exponential in how much they generate, otherwise it wouldn't behoove someone to really try to take out that t3 generator.

    Another important note is that the higher tier resource structures MUST be expensive AND lengthy to build. They should take 5ish minutes minimum to pay for themselves, on top of taking a while to complete. As we will have games on the scale of hours, there needs to be a fairly large (larger as the game progresses) window where if you lose a building, it's an overall loss.

    Although I guess it wouldn't be too hard to do AOE damage on all the weaker generators either.

    Honestly I think a linear idea for resources is silly... While warcraft had linear growth for gold, it still had exponential growth in the sense that the richer you were, the more capability you had to get new goldmines. And forests were straight up exponential.

    The rich SHOULD get richer. That's the whole point of BEING rich. I watch a lot of Husky Starcraft 2 casts, and while the economies of that game and this one differ greatly, it's still very obvious that those who achieve strong economies early are able to secure strong economies later. If this wasn't the case, it would be much to easy for a player with a poor economy to smash a strong economist.

    It's all a balancing act, where you want economy to be an investment which is worth it, but also has risks. You should feel vulnerable building resource structures instead of battle units/defenses. Your enemy should scout so they can take advantage of when you build resource structures. It shouldn't be just another thing that you do when you're not busy fighting. "Oh I've got downtime, lets build 20 solar panels and some storage to keep my economy up". It should be big decisions which carry real weight.
  3. syox

    syox Member

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Well first of:
    Its not all about PA here, i imagined it as a more general discussion.

    I think i addressed that in Warcraft the upkeep didnt lead to a complete linear Economy. It was a Example that games use restrictions for economy.

    One issue i have with the exponential economy in general, is that things get overboard. If ther is no restriction and the game lasts long you wont be able to keep up with your economy. Thats where the games get silly. Or new growing(similar to the economy) input methods are needed.

    Also there should not be a system, where the player behind can easy beat the Leading player. But it should nevertheless be possible. While a exponential economy leads to one big decisively point in the game, a linear may lead to a series of minor decisively points. So being ahead doesnt let you stay automatically ahead, though its better to recover, harder to win but may be more fulfilling.

    After all most games have limits to economy or techlevels or bases or unit caps. So there is a final to achiev point wherefrom your only left option is to attack. Still a question i ask myself is:
    :?: Is exponential economy suited for the scale for PA? I cant imagine anything else, but can we as players keep up with the stuff happening then?
    :?: If i imagine PA with linear economy, does it work in the same scale as exponential?(I dont think so)

    Just some things i am currently thinking about in regards to PA.
    Now i am into PA again... ... damn.
  4. BulletMagnet

    BulletMagnet Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Syox, I think your maths was wrong; so I did it properly. Also, mine has less spelling mistakes. ;P


    So lets say an extractor costs M metal. Each extractor produces at a rate of X metal-per-second.

    Lets assume that, at the moment the first extractor is built, there is zero metal in the bank/storage. And lets further assume that there are no factories or engineers consuming metal.

    From this, the time taken to save up enough metal to build a second extractor is;
    Code:
    time = M / X
    Now we can build a second extractor, which we will do. Our income is now 2 * X metal-per-second. The time taken so save up enough metal for a third extractor is;
    Code:
    time = M / (2 * X)
    It takes half the time to save for third extractor, compared to the second extractor. After building the third extractor, the time taken for the fourth extractor is;
    Code:
    time = M / (3 * X)
    And so on and so forth. I'll save you a bit of boring maths and show you a pretty graph instead;

    [​IMG]

    This is the bog-standard 1/x curve. As you can see, once you get going you can afford new extractors very quickly.

    With some mathematical trickery, we can make a graph of how far the game has progressed per extractor.

    [​IMG]

    It's a little hard to interpret, but on the y-axis (up/down) is how long the game has been playing before each extractor is built. For example, the 10th extractor appears 2.93 lengths of time into the game. Note that the first extractor appears at one.

    The cleverererer members of the forum might be thinking to themselves; "That curve looks logarithmic. That's literally the opposite of exponential!" And you'd be 100% correct. It is a log-curve. But the kicker is; it's a log with time as the dependent variable. Time is the output of this, not the input.

    Complicated maths (not really):

    As I pointed out earlier, the time taken to save up for an extractor is;
    Code:
    time = M / (k * X)
    where k is the number of extractors you already have built.

    In maths-speak, the time taken for something is delta-t. The total length of time for a bunch of things is the integral of this.

    Complicated maths (okay, for reals this time):
    Code:
    delta-t = M / kX
    
    t = Integral { delta-t dX }
    
    t = Integral { M / kX dX }
    
    t = M/k * Integral { 1 / X dX }
    
    t = M/k Log(X)
    Hey, that's a log I talked about. I can use this formula to see how long it takes to get k number of extractors if they cost M metal, and each produce X per second.

    Now but when we juggle it around so that I can find how much X per second we get at t seconds into the game.
    Code:
    X = exp(tk/M)
    Well, damn. That is the exponential economy.

    Here's a graph of what it happens to look like;

    [​IMG]
  5. BulletMagnet

    BulletMagnet Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    crixomix, what you said made me cry.

    It has so many conceptual errors that it made me shed tears.

    This dimension is completely uneeded for exponential economy.
    No!

    Resource growth is exponential if you have your Dimenion 2 by itself. If you doubt me, have a read of my post above.

    I agree with this. But I'll add this; it should be made unstable by attacked by your opponents (ie. them blowing up your economy).

    NOOOOOOOO!

    Just no. Not ever.

    It's hard to pinpoint anything that will kill their economy in one go.

    There's nothing wrong with that. In fact, making it hard to kill in one go is something that you said you wanted in having the economy become more stable as time progresses.

    You can't have it both ways.

    Now, it's just as easy to kill little bits and pieces of their economy. In fact it's easier because they have lots of little bits.

    If you have fewer (but more important) economic structures, it becomes easier to defend. Being able to avoid confliect is important, but it shouldn't be easy.

    Honestly, I don't think you understand what the exponential economy is. Please go have a read of the other economy threads (oh yes, this forum has discussed them in depth before you know).

    Those rich that you speak of should do work to get richer. Not just sit on their thrown of cash and let it work for you. It's bad game design when the game plays for you.
  6. BulletMagnet

    BulletMagnet Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Guess what Syox's and my own posts assume to be true.

    Neither of us mentioned it, but we assumed it to be a fundamental truth of the game. It's two words;

    • Global.

      Economy.
      (highlight to see the answer)

    Yes, forumites, BulletMagnet is bringing up the local economy debate... again. I can hear your collective groans across the Intertubes, and it makes me cackle with laughter.

    With a global economy, you get crazy growth right up until you smack into the impassable barrier of; gobbling up all the metal deposits.

    But what happens when you can't funnel resources from one area into building more resource producing structures in another area of the map?

    I don't know. But I think it'd be damn fun to find out.
  7. syox

    syox Member

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Nah i think not. I used a discrete sequence approximation. Its easier understandable and close enough. :)
  8. syox

    syox Member

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Interessting you mention this. But even if not, logistics only work as a timedelay if they are not limited. If it is per planet economy without interchange it still is exponential. Then per planet and over all planets also, because of exponential growth in colonized planets.

    The problem still stays that the economy will grow to big to manage it all. Especialy on the scale PA is aiming.

    One option to handle this is scripts and automation.
    Or you could have a exponential amount of players joining the game o_O
    Or we accept this and let the fastest clicker win.
  9. dmii

    dmii Member

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Correction: You get crazy growth until you hit the barrier your enemy imposes on you.

    Whether the economy is global or not is irrelevant, what you actually assumed was, that there is enough Build Power to make use of all the incoming resources. Build power isn't global, so your giant intro of the global economy turned out to be a bubble which just popped.
  10. syox

    syox Member

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    I honestly dont see how local build power change the effects of a exponential economy.
  11. igncom1

    igncom1 Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    You can't funnel it all into one place at the same time.
  12. BulletMagnet

    BulletMagnet Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Both very good points.

    I thought about this. And thought about this some more. And had the brilliant idea of getting some triangular grid paper, and plotting it out step-by-step.

    I made this.

    Then I counted the dots up, and plotted them against time. It starts as an exponential, then degrades into polynomial growth because the first resource producing structures can't reach our to the edge any faster than dots on the edge can spread. While I'm producing more and more, it becomes impossible for me to spend that extra income on growing my income. I may as well be spending that on tanks.

    But, if I remove my self-imposed limit of only drawing in certain places, I can get a 2^k growth all day long.

    While it's not time-delay at all, you're right that localised economy can grow in exponential mode. But this only occurs in very idealised situations. Metal-makers/mass-fabs follow this curve, but deposits are likely to never follow.


    So which is better; 2^k, or e^k?

    Or, we increase the cost of metal extractors according to how many extractors we have.

    Extractor cost of
    Code:
    M = (2k - 1)m
    really does clobber the living crap out of exponential economy. That said, I happen to hate that idea because I don't think varying costs should used to balance things.
  13. bobucles

    bobucles Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Exponential growth is simply awful and every RTS worth a damn finds ways to mitigate it in some way. For starters:

    1) Metal is fundamentally limited by the number and quality of extraction points on the map. It takes time to approach new extraction points, time which is reduced by sending multiple engineers on multiple errands. No matter what happens, metal extraction has a PEAK which can not be surpassed with extractors alone.

    1A) Metal is further limited by enemy action. Extraction points require protection else they may be destroyed by enemy units. Destroyed points must then be fought over and re captured, further reducing the time they are generating metal for the game.

    2) Generators do not build generators. METAL builds generators. Similar to extractors, any energy cost for the generator merely determines how much time it takes for the generator to pay itself off. While technically, a player with infinite metal will have his production slowed down by energy, the pay off time is typically very fast(less than a minute per doubling).

    2a)If one were to create unique, separate energy economies, each new development would be very easy to build thanks to the pre existing metal supply and the fast payoff of generators. The cost in metal would remain very steep, however.

    3) Reclaim, reclaim, reclaim. Any beginning resources on the map are an integral part of the early game economy. Any wreckage created through combat is an integral part of the mid game economy. A huge metagame is going to exist around the capture, harvesting, and denial of open field resources. This aspect of the game is far too important to be ignored, even for the most basic of models.
  14. BulletMagnet

    BulletMagnet Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    If we accept that reclaiming typically happens to tanks (because dozens, if not hundreds, of tanks blow up for every extractor blown up) then reclaiming tanks has little effect on the exponential economy.

    What it does effect, however, is tank production. Using Sup1/FA as an example, reclaiming your own tank returned you 80% of the mass. You lost 20%, which you had to replace with newly extracted mass from your extractors.

    In effect, only one fifth of your economy was being spent on new tanks... the other four fifths were just cycling around in a giant circle of construction, destruction, and reclamation.

    Suppose that you and your opponent have the same number of extractors. Your metal income from them is the same. But you're reclaiming your tanks, and he's not reclaiming his.

    You can build tanks five times faster than he can. If you reclaim his tanks, you can build nine times faster.

    Sup2 nerfed wreckage. It was one of the things they did smashingly well.
  15. igncom1

    igncom1 Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    In what way did they nerf wreckage bullet? As I am curious.
  16. BulletMagnet

    BulletMagnet Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    If memory serves correctly, you only got 20% mass back.
  17. syox

    syox Member

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    A delay doesnt change the fact of an economy being exponential. Being forced to build generators either, it just adds to the cost of new mexes. Therefor both things just change the possible growth rate per time.

    What is essential is how many of your income you are able to reinvest into your economy. This mostly comes down to the behavior of you and your opponent.

    After all the economy forces you to:
    • get as much mexes as possible
      hinder your opponent from getting as much mexes as possible
      try to get a economical and or military edge over your opponent
    Well to be precise a economical edge now will translate into military edge later.
  18. igncom1

    igncom1 Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    @BulletMagnet

    Interesting, I can't say I really noticed it but then again in any SupCom game for me the little details about the reclaimed mass of a tank never really clicked in my mind.

    What's more interesting is the times that I have used reclaimable mass in SupCom2 to support a strategy, yet have always found the amount I am reclaiming to be more then suitable.

    What with cybran recycling towers and the strategy of using an army of engineers to zerg across the map, eating everything.
  19. bobucles

    bobucles Post Master General

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Perhaps not. But every element that plays into the starting resource grab IS. It takes time to get the first engineers out. It takes time to reach the first extractors. It takes time to move up and grab the next extractors in the queue. So the exponential growth is NOT unchecked. It is being slowed down every step of the way, by fabber limitations, map design, hostile activity, and the simple fact that extractor points have a definite MAX output. When those points are taken, economic growth is impossible without something new.

    On top of all this you have a SECOND resource grab happening in addition to the race for extractors. It is the hunt for rocks, trees, and neutral junk that litters a map. These factors may seem innocuous, but they're providing a starting surge that gets the base established, and allows for future growth. Every bit of extra metal grabbed in this way is automatically denied to everyone else. It's a lethal battle, which happens before the first tank battles even get started!
  20. CrixOMix

    CrixOMix Member

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    Re: Discussion about growth rates Linear/Exponential/Logarit

    Sigh. It was late and I didn't state myself well. Obviously you don't NEED bigger generators for exponential economy... I never said it was required, I was talking PA specific...
    And I disagree with you about land acquisition. It DOES slow down. Land acquisition is a linear function of a square area. A person can only do one thing at a time. And the average micro is only going to be able to deal with building one or two new bases at a time.
    About high generating structures, the idea isn't to be able to cripple an economy with one blow. My main point was that it should be WORTH your time to destroy the higher tier economy structures. It shouldn't cripple their economy. But it should be effective. I said damage their economy, not destroy it. The idea behind the larger structures isn't that they can do just as much as your individual 100 generators, but that they can do more for you than just building regular ones could, and that way the enemy is going to want to destroy it. I think it would be lame to have one generator and one metal extractor...
    First of all, I know what an exponential economy is. I'm a math major, I full well understand the difference between linear and exponential growth. I'm just saying I don't like the idea of linear growth. I think it's too slow for the scale of this game. I want my resource production growth (the growth of my rate of income) to be bigger between minutes 50 and 60 than in minutes 5 to 15...
    ... This is barely even related to what I said? I feel like you just extapolated everything I said to try and disagree with me... I never said they shouldn't have to work for it. I never said they should sit on a throne of cash and easywin... I never said the game should play for you... How did you get that out of what I said? I was simply saying a linear economy does not provide an important enough stress on the economy. I think an exponential economy provides the necessary drive people need to CARE about their economy. Also remember exponential doesn't mean it doubles every 2 minutes... People hear exponential and freak out, sometimes it's SLOW exponential growth, that's fine too...
    Overall I agree with a lot of what you said, I think you just misread my posts.

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