Do early workers really have an exponential effect?

Question

Commentators such as HD sometimes mention that a worker advantage in the early game is of extreme importance since it has an "exponential effect" (here's an example video which mentions that at the start).

While I agree workers have a large impact in the early game, it seems to me that

1. Worker production rate is usually capped by building/larva limits rather than mineral balance, so losing an early worker or two doesn't necessarily mean the next workers will be produced slower.
2. I always thought that in the late game it doesn't really matter if one player have a couple of workers more, when the workers already count in the dozens.

So it seems to me the impact of having an early worker disadvantage doesn't have an increasing effect - just losing a more-or-less fixed amount of money for the lost mining time. If anything, it has the most effect on the early game, where every single unit matters.

Could somehow explain what is the true impact of early worker disadvantage, and why do people say it's exponential?

Answer 1

The effect is mostly that you earn less resources, which means that you can not expand at the same time as the enemy and you can only produce workers at a slower pace unless you accept falling behind in unit numbers.

You can recover from an early worker disadvantage by either killing a lot of his units and then expanding and getting a lot of workers while he rebuilds his army (especially true for zerg) or by killing enemy workers with drops while you have a similar sized army.

If you fail to do either then you will slowly fall behind because you always have less resources and less units than the enemy while the enemy economy is increasing faster than yours, just because he can afford to spend more resources on it instead of on unit production.

Simplified Math:
1 Worker gathers 0.7 minerals per second (according to this Teamliquid Thread).
This means a single worker gathers the resources required for 1 Marine every 70 seconds.
So being one worker behind doesn't have an effect immedatly, but after those 70 seconds you will be one marine behind unless you skip a worker and instead produce an additional marine, which would put you another worker behind, resulting in being behind a marine every 35 seconds unless you skip another worker. Continue until you are dead.

Answer 2

So let's go through this line of thinking for a second:

• Assume there is time requirement for workers
• Assume you can build an infinite number of workers simultaneously

Story Time

1. Player 1 builds 1 worker, it works for 70 seconds and builds a second worker.
2. Both workers work together for 35 seconds and build a third worker
3. The three need only 24 seconds to build a fourth
4. The four need 18 seconds
5. Five need 14 seconds

After 161 seconds she has 6 workers and can turn out a Marine every 12 seconds.

1. Player 2 does the same, 2 workers at 70
2. 3 workers at 105
3. But instead of a fourth worker, he builds a Marine, which takes 24 seconds
4. It takes him another 24 seconds to build his fourth worker
5. fifth worker comes at 171

After 171 seconds he has 5 workers, 1 marine and can turn out a Marine every 14 seconds.

Assuming neither player builds more workers, after 6 Marines, Player 1 is going to catch up to Player 2. He'll actually get his 8th before Player 2 does.

This illustrates the simplest principle: Money is worth more now than later. Anyone with a basic understanding of Economics knows this principle.

So your first point was: Worker production rate is usually capped by building/larva limits. If I take my trivial example and add build time in, I think we can both agree it will delay the advantage of building that fourth worker earlier, but it will show in the long run. Instead, let's talk about what happens when we cross the 17 second threshold (and I'm ignoring Zerg here for the moment, as their threshold is 6 as I'll show later). At timings less than 17 seconds, we accumulate more money than we can reasonably spend. Let's look at that example again (this time assuming 17 second cooldown between building workers):

Player 1: 2 @ 70, 3 @ 105, 4 @ 129, 5 @ 147, 6 @ 164, Marine @ 173

Even though there was overlap between 5th and 6th workers, that Marine still pumped out at 173, same as before. If both players continue as in the above example, Player 1 will still over take Player 2, by the 8th Marine.

Let's reexamine what these 8 Marines mean. Since our worker production is capped by 6, these 8 Marines constitute surplus (another economic concept). If we then use this surplus to build a second Command Center (a value of 8 Marines), once again our worker production become uncapped. If we view our ability to expand, as a function of our surplus, and our production rate is a function of our expansions, then we quickly see that our production rate is a function of our surplus, and thus the number of workers we have. What's more we can even model Army size as overhead, and we immediately begin to see the basics of Economic theory. This is why we often refer to worker growth as exponential. Even though the curve isn't smooth (or a very high exponent) on a long enough time line, worker production becomes exponential.

So while having 65 vs 60 workers late game may not seem like a lot, if those 5 workers were at a crucial time, that means the worker with 65 not only can support a larger overhead, but also reached that number faster, and thus has a larger inventory (read: army/tech).

The Zerg Problem

Zerg are slightly unique in worker production because a Hatchery + Queen (500 mineral investment) can produce 4*9+3*8 Larva every 360 seconds (45*8=40*9) or a larva every 6 seconds. As a trade off they must choose between using this for Drones or Army. As a Zerg player often will commit an entire batch of Larva to one or the other, a delay in Drones has an even more pronounced effect as it may constitute a set back of 4+ workers