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View Full Version : Interesting observation about rates of progression

ankyrin
12-27-2011, 01:58 PM
I was trying to figure out how the 1- build/upgrade time, 2- cost of upgrade and 3- income of the money buildings increase and noticed something very interesting:

- the cost of the upgrades increase exponentially (2^x)
- the income of the buildings increase quadratically (x^2)

I could not figure out exactly how the time increases, but it seems to follow some S-shaped curve.

Since you know the cost of the upgrades increase exponentially, ordinarily you would need 2 data points to figure out the whole curve. However, since the base of the exponent seems to be 5/3 for all money buildings, you only need to have 1 data point and you have the whole cost schedule (of course there is round off error). So if:

g(n) = cost to upgrade building to level n

then

g(n) = (5/3)*g(n-1)

Since the income raises quadratically, once you have 3 data points you can figure out the income schedule (recursively). I'll spare you the math and tell you that if:

f(n) = the income of a building at level n

then

f(n) = 3*(f(n-1)-f(n-2)) + f(n-3)

this holds for all buildings.

So you can verify that this works, I'll post the schedule for the armory:

level time income cost
1 0.05 400 1,500
2 0.25 800 2,500
3 0.75 1400 4,200
4 4 2200 7,000
5 12 3200 12,000
6 24 4400 19,500
7 30 5800 32,500

Hopefullly this will help some of you plan money building upgrades.

Tramp Stamp
12-27-2011, 02:36 PM

All money buildings are Type B in Modern War. Interesting note on times, which seem to follow a formula much more closely in Modern War. There is no pattern in Crime City. I assumed that values were assigned by lookup tables but apparently it's formulaic after all.

ankyrin
12-27-2011, 04:43 PM
I don't play Crime City, but that was a very interesting read. I think we are actually saying contradictory things. In that post, you say:

Growth Type A follows a Triangular numbers sequence (which is quadratic)

Growth Type B follows a Fibonacci sequence (which is essentially exponential where the base of the exponent is the golden ratio)

I agree that the Modern War buildings follow a Type B payout, however, the payouts do not grow exponentially. I'm claiming they grow quadratically and your Type B schedule is actually quadratic. In fact, I can give you a quadratic formula that fits your Type B table perfectly (except at level 10):

payout = 0.25*(level)^2 + 0.25*level + 0.5

I'm not sure how that Type A table grows, but it seems faster than quadratic and slower than exponential.

Tramp Stamp
12-27-2011, 05:10 PM
I don't think I ever claimed exponential growth. As for labeling the sequences, I just looked at the numbers and it was very obvious that someone had them in mind when designing the payout systems. Type A actually grows faster than a triangular sequence while Type B is nerfed compared to a true Fibonacci sequence.