How long does it take for an expansion to become an advantage?

When you expand, you sacrifice the "strength of your army" (i.e. tech + army units) for a while. What is this duration? To illustrate what I mean, here's a makeshift graph of 1-base production versus a 1-base into expansion production:

My question is: How long is that green line? How can you measure it / calculate it?

The paragraph-style description of the graph is: While you save for the expo, your army is not getting stronger. When you build the expo, you can continue to produce on 1-base, but you will be behind the cost of the expo. Once the expo is up, your income will gradually increase, allowing production to accelerate. At some point the acceleration will elevate you past the point you would have been if you stayed on one base.

I have a very hard time sensing this moment in the game. It's easy to feel when income increases, but hard to know when the expo actually led to a stronger army relative to if you hadn't expanded.

There are a lot of factors that influence the answer of course, but they can be ignored as follows:

• What race are you? For example if you're terran, you can benefit just by building an orbital command and summoning MULEs. Or just double-producing SCVs. Assume building right at the expo location as this is more general. If you're protoss, assume that you build a pylon at the expo, but that you needed it anyway so it's not really a cost of expanding.
• Will you need gas geysers? Ignore gas, because the geysers cost minerals but return gas, and cannot be compared. It would be a separate graph entirely.
• How many workers can you transfer, how saturated was the base where they're coming from, how far is the transfer? Assume transferring a practical handful of 5-8 workers. If you are transferring more, then there's not much question: you apparently NEED to expand regardless of the return, because you're over-saturated.
• How many resources are remaining in your other bases? similar to the last item, if you are desperate to expand, then it's not really necessary to calculate when it will be advantageous
• Unit producing structures you must create in order to . This could actually be considered as part of the cost to expand, however it can also be ignored because the goal here is only to find point 4 (on the graph). Beyond that is not so important, because you know you're benefitting.

So the idea is to get a general sense of "+/-30 seconds, how long is this duration?"

I think this would be helpful making in-game decisions, calculating riskiness of expanding (balancing need versus risk of being attacked during the green line), for developing new builds and considering FE, or making in-game adaptations. Or just plain curiosity.

Solution 1:

CC/Hatchery/Nexus is building 100 seconds. After that each 25 seconds you will build 1 more SCV/drone/probe (don't consider chrono boost for now: you won't be able to use it from new nexus right after start). If I not wrong each harvester on non-saturated base give 47 minerals per minute...

I will do math for CommandCenter:

400+X*50 (expenses to build new CC + workers there) <= (Y/2+X)47(t-100)/60 (income of new base) t=100+x*25

X - amount of SCVs produced by new CC;

Y - amount of SCVs brought from 1st CC to the expansion (I've split Y by 2 because those SCVs could bring some minerals on the main base, but smaller amount due to main base saturation. So I assume that on new base they will work twice more efficient).

Math could be complicated, because X SCVs don't start work immediately. To compensate that I will add a factor 0.5 to the X:

400+X*50 <= (Y/2+X/2)47(t-100)/60

also, lets assume Y=6 (we move 6 workers from main to expo):

400+50x <= (3+x/2)*19.58*x

9.79x^2+8.74x-400>=0

x1=-6.79, x2=6

x=x2=6, t = 100+25*6 = 250 => 4 minutes 10 seconds.

So, if you bring 6 SCVs from main and keep constantly building SCVs on new base in 4 minutes, 10 seconds you will build 6 more new SCVs and will compensate ALL expenses.

Is the math is correct you could extrapolate that to case when you build OC, Hatchery, use Chrono boost, etc...

Though that is a theory it give you some understanding of the process. If you want to know the time, when opponent is biggest investment (time when he invested a lot CC + 1st SCV + 2nd SCV, but didn't receive feedback at all), you could take derivation of suggested formula...

Solution 2:

According to manual test on teamliquid it takes for about 3:00 - 3:30 for expansion to payback.

But think about this:

• additional CC gives you additional supply and mana regeneration for scans / mule (larvae for Zerg, etc.)
• all calculations should be put in context of strategy - sometimes player expands not because of minerals but gas. In this case you'll get benefit of expansion very quickly