Scouting vs. Random analysis

Scenario: You send one of your first 6 SCVs to scout

Let us assume that for the first 10~ scvs, your mineral income per minute per scv can be represented as this:

g(n) = k*n

where k is an arbitrary value of the average income each scv makes per second. depending on the type of speed you play SC on, it is around 2/3.

Of course, as soon you add in more than 10 scvs, this function needs a piecewise-representation, as the return diminishes the more scvs you have. Furthermore, this function assumes there is a linear relationship of the amount of minerals recieved for the first 10 scvs, which does create a margin of error, but it is not that bad.

Thus, we say that:



Now, when you scout, you are effectively down by one scv, let us define the income of the opponent as



If we assume that both players continuously pump scvs for the first 10 supply, we can say that the number of SCVs with respect to time (in minutes) can be modelled by the function:



That is to say, number of SCVs for the first 100 seconds of the game can be calculated by floor(time / 20).
Because it takes 20 seconds to build an scv. I put a restriction of 100 seconds because in 100 seconds 5 scvs are produced, which brings our total count of SCVs to 11. However, one will be surely sent to build a structure by then, which fulfills our restriction of a maximum of 10 scvs mining the minerals at a time.

which we can substitute:




Now that we have the rate of income at time t of each player, we can integrate and find the total amount of money each player has gotten (from the new scvs), with respect to time

Thus now we can approximate the accumlated minerals gathered (by the new scvs) for the first 100 seconds can be modelled like this for each player.





We can find the advantage of Player 1's total minerals over Player 2's total minerals if player two starts off with 1 less scv (scout) by




If we plot this graph, we can see how big of an advantage, that your opponent has over you, if you sent one of your first 6 SCVs out right away. For these plots we will assume K = 4/6 minerals per second

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Thus if player two sends one of his initial six scvs to scout, by the 100 second mark, player one will have a 200/3 mineral advantage over player one.

This indeed makes sense, seeing as if the opponent has an extra scv over me for 100 seconds, and the scv gains an average of 4/6 minerals a second, he should have 4/6 * 100 minerals over me. This case is proven true.

But now we want to know, what happens if I send the 7th scv instead of the 6th? Or the 8th? Or the 9th?

Then we can model MY income by:



Where x is the time for which I sent my scouting SCV out. If it my 7th SCV, this is 20 seconds into the game. 8th SCV, 40 seconds in. Etc.

This makes sense because my income and my opponents income are exactly the same (note the f(x) part) up until i send my scout out (at time x). Then my mineral income diverges from my opponents, and I have the g(t) function which implies i have 1 less scv thereafter.

This creates the relationship of my total gathered minerals versus the total gathered minerals of my opponent. obviously, the later i send my scout out, the closer my mineral income is to his:

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Now for the real stuff; I want to know 100 seconds in, how much each of these options costs me, so I can draw some sort of application.

My opponents final income will always be the best case scenario, lets say f(100) in this case. I want to know how much greater his income will be over me depending on when I send my SCV scout out.

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There you have it; If i send one of my first 6 SCV scouts i will be at a defecit of 200/3 minerals by the 100 second mark. And so on...

Lets say we don't scout until after 10;

80 seconds in you have made 4 scvs, which brings you to a total of 10. You start with 50 minerals. 4 Scvs also cost 50 minerals each. By the time your 10th scv is out, you have gained f(80) minerals.

Thus, how much minerals you have when your 10th SCV comes out is:

f(80) + 50 - 50*4 = 350/3 = 117


This means, under optimal conditions, as soon as your 10th SCV pops out, you should have 117 minerals ready to spend on the first supply depot. I can spare 17 minerals without losing any efficiency.

So if i decide to scout at 6, when my 10th scv comes out, I will have this many minerals:

b(0,80) + 50 - 4*50 = 190/3 or 64 minerals.
Thus, I am behind by 36 minerals to making my supply depot.

Currently, I have 9 SCVs mining. If my rate of income is g(t) = [(2/3)*t]*9, and g(t) = 36, then t must be 6 seconds.

I will have to wait 6 extra seconds to make my supply depot if i scout on 6.

Similarily, we can continue this for every individual variable for which SCV scout was sent.