Zerg Hex Forum

Is zerg eco exponencial?

To answer this question i got into some weird math to get an answer. It's getting very technical, but I'll try my best to explain the concepts I'm using. I'm working currently on a zerg eco simulator on python and i can later confirm my conclusions.

Short answer is: No, it's not exponencial.

Btw I'm sorry for the low quality graphs, I dont have my full setup atm

Concessions:

I'll make a few assumptions so i can make a working mathematical model. This is far from a guide or a realistic situation, I'm only investigating if the eco is exponencial or not.
1. The player is a solo zerg
2. Starting supply isnt 54, it will be zero - wont really matter as I'm not looking for realistic numbers, just a working model on the long run.
3. Building count is not an integer variable. Again, I know you can't make 0.5 a spawner, but it doesn't matter if the investigation is to find out wheter the eco is exponencial or not.
4. Time will not be a factor. To analyze the behaviour of zerg eco growth, I'll consider the costs and benefits from each supply, so for this exercise it wont matter the time - I will cover it soon when i make a simulator.
5. The only buildings available are the slowling spawners and mineral extractors.

Metodology/The nerdy maths

1. Figuring out the ratio mathematically

When playing the game, players can follow the ratio by reading what's written on the extractor, but when making a mathematical that doesn't work. In a simulation it might, but we'll need to use a different strategy.

In economy we have a concept called PPF - (Production Possibility Frontier). It basically sets the limits of production you have, that in your case is how many spawners and extractors we can produce with the limited amount of 45 supply.

The PPF is then 4x + 15y <= 90n

That's a lot of letters and I'll explain each one of them:
x: Number of spawners. They cost 4 supply so the constant multiplying it is 4.
y: Number of extractors. They cost 15 supply.
n: Number of the supply. Each supply gives you 45 supply capacity, and when you're solo zerg the supply cost increases only every 2 supplies, so that's why its 90n and not 45n. So for now on, each time we see "n", we have to remember it means 2 supply worth.

This expression basically means that a combination of x spawners and y extractors can never use more than 90 supply for each supply you bought.

https://imgur.com/vJkRjt4

Now that we have set our limitations, we have to choose the best possible combination within our limitations. For that I'll use a utility curve. In a utility curve, all points have the same value, so in our case, in the utility curve all points represent the same amount of economy. The maximum value we can get from a supply is represented by the intersection between the utility curve and our PPF, because that's the highest utility avalible given the PPF restrictions.

https://imgur.com/6ZlbSh1

Our utility curve (economy) is given by:

U(x,y)=(54+x)*(12+y)

Again, x represents the number of spawners and y is the number of extractors. 54 and 12 are the starting values of spawners and extractors.

To figure out the intersection, the derivative of both functions must have the same value. Given that our utility curve is a tradeoff between 2 variables, we get 2 funcions back:

Spawner count law
x=(45n-18) / 4

Extractor count law
y=3n+ 6/5

This values are the amount of extractors and spawners for each supply to maximize the income.

To find out the proportion between spawners and extractors, all we need to do is devide x by z, which gives us:

15/4 - 15/(5n+2)

Now this is the first cool thing we can analyse. As you can see the "n", which represents supply is divinding numbers, meaning that the more time goes on, more valuable spawners become, meaning that you would need less extractors per supply as time goes on. If the supply went to infinity, the spawners/extractors ratio becomes stable at 15/4 = 3.75 (In each supply, for every extractor you build 3.75 spawners)

https://imgur.com/tz7Cwk1

Now that we have modeled our "ratio" depending on what supply we're on, we can calculate our income in a "n" supply. To figure it out, all we need to do is substitute our spawner and extractor count laws on our x and y values from our utility curve. That would mean that the player has followed the correct ratio perfectly. The outocme is:

Income per minute = 27/20 * (5n +22)^2

https://imgur.com/UJkM855

We just got to a second cool conclusion. The income per minute for each supply follows a quadratic formula. Do we have our answer then? No, it's never that simple xD. We haven't considered our costs. Our costs for each supply varies for 2 reasons: first the ratio keeps changing (spawners get more valuable over time) and second the supply costs will follow a 600+125n function.

2. Calculating supply cost variation

The supply cost variation is simple: 125 minerals for each supply bought.

The extractor and minerals one though, is hard, because the ratio keeps changing. As we know how many spawners and extractors we have on each supply (n), what we can do is to multiply the spawner count variation by spawner cost and the same for the extractors and then subtract the values from the last supply (n-1). The math looks like this:

Δc=(70∗(45n−18)/4+220*3n+6/5)−(70∗(45(n−1)−18)/4+220∗3(n−1)+6/5)+125

Δc=1572.5

Long story short, that means that the cost variation for each supply is costant, meaning that supply get more expensive linearly. How about the income, how is it's variation behaviour?

3. Calculating income variation for each supply

The process for this one is similar, as we have figured out the income per minute on each supply (n), all we have to do is subtract it from the last supply (n-1). The maths are:

ΔU=27/20∗(5n+22)^2−27/20∗(5(n−1)+22)^2
ΔU=27/4∗(10n+39)

Now if we pay attention to that, we will see that the variation of income for each supply varies linearly, which makes a lot of sense given that the income per minute on each supply is quadratic - it's variation would be the derivative.

4. Calculating effective economy variation for each supply

We have figured out how much the cost changes with suppy and how much income changes, to figure out the effective increase, we just need to devide ΔU/Δc (economy variation/cost variation). As cost variation is constant, the effective economy variation will also be linear, as it has the same degree as the income growth without considering cost,but it will be "slower" (the line will be less tilted):

ΔE=ΔU/Δc
ΔE=27n/629+1053/6290

https://imgur.com/9HdyK8p

Conclusion

We know that the economy effective variation with each supply is linear. Meaning that our variable "n" is powered by 1.

If we integrate the income per minute effective variation per supply, we will discover the income per minute per supply, and our "n" variable will be powered by 2. That would mean a quadratic funcition.

If we integrate that again, it would give us our minerals, which will be a function with "n" powered by 3.

Meaning that the income grows like a quadratic function and your accumulated mineral bank will grow cubically. That means that eco is not exponencial - it would be if stuff was powered by "n", not the other way around. Mathematically, exponencial are the fastest growing curve compared to all arithimatic ("n" powered by something) in the long run, but in short term (meaning early game), arithmatic funcions can be faster and then get slower than exponencial.

Problems/next steps:
As i have adressed, I haven't considered a lot of things that needed to be consider for these numbers to be comparable to a real game, but is only possible with a simulation I think. I'm currently working on a python script to simulate the economy and understand better whats going on.

You can see that after the fourth supply strikelings start being cheaper compared to zerglings, becuase they cost one less supply and that difference will only grow with time.

Good to know.  Thank you.

if this version on sc2? i didn't see it online when i searched.

Also, any word on the creator? Seriously open source the game, the balance needs lots of work.

Yes it is, and it is set to 1 i think. Both the gas and mineral income are per second, so the start is 12min/60gas and in seconds that would be 0.2min/1gas.

Version 1.1

Implemented the army expenses and army related upgrades. Still need to fix the energy and wall upgrades scaling up the cost.

Next update: I'll implement terran eco upgrades, such as Improved Farming and Gas Extraction.