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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:

cool graph

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.

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That's impossible to answer. If you would just ask about economy, then that can be easily calculated. The answer simply boils down to the fact that you should expand whenever it is safe and not expand when it's not safe. –  Let_Me_Be May 16 '11 at 18:31
    
How many workers can you transfer, how saturated was the base where they're coming from, how far is the transfer?' This is actually the main determining factor as far as I know... if you can't transfer any workers, it will take forever, if you transfer 20 workers from a base that previously had 40, it will be very fast. Race is also important, a larva locked zerg can get a bigger army the moment it finishes than they would have had otherwise. –  en1gmatic325 May 16 '11 at 18:32
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I think the problem is that there are perhaps too many factors in there. For example, your bulleted list enumerates a lot of factors that make the "green zone" vary so much. If there was a single equation that you would just plug all that into, that'd be one thing, but I'm under the impression it's not so simple as that. Is it possible to pare down the scope into something a bit more practical to handle as a question? –  Grace Note May 16 '11 at 18:45
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I have no idea even if there is answer, but people, take note. This is how you ask a question. => +1 –  DrFish May 16 '11 at 19:09
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Not a full answer, so I'll leave it as a comment. For the case of FE such as 15 CC or 15 nexus vs no expo, the key timing for the one base player will be to hit generally before the 7 minute mark, with the strongest timing generally being in the 5:30-7:00 range depending on builds used. Depending on tech paths of both players, as late as 8 minutes may be stronger, but generally by the 8 minute mark any advantage for the one basing player is gone. –  en1gmatic325 May 16 '11 at 19:13

2 Answers 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
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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...

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I think that someone should try it and say about this. –  Meta May 17 '11 at 4:57
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This might be a correct calculation for resources, but it does not address unit production. Because creating production facilities, upgrades and units costs time, thus even if you have all your money back in about 4 minutes you have fewer units and/or less tech compared to investing in an army from the start. I would add at least 30 to 60 seconds when it comes to comparing armies. –  ayckoster May 20 '12 at 0:10

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