Scouting in StarCraft Brood War

Learner Competitor

The short version

What they have is what you saw plus what you did not see. Both halves are real, and the second one is arithmetic, not guesswork.
Scouting does two different jobs. It grows the part you saw, and it tells you which direction they spent the rest in. The second usually decides the game.
Information expires. From the second you look away, the unknown part grows at exactly the rate they mine.
The frontier of what they could have is a straight line, not an arc. Money adds up.
More options means uncertainty grows faster. Against a race with three answers, two minutes blind is much worse than twice one minute blind.

The situation

Your probe got in at five fifty. One hatchery, nine drones, gas taken, no spire.

It is now seven forty. You are deciding whether to expand, and you catch yourself thinking either "they were on one base so I am fine" or "they could have anything, I should play safe". Both of those are ways of not doing the arithmetic.

The arithmetic is not hard. It is roughly a thousand minerals, and a thousand minerals is a specific list of things, and about half of that list loses to you expanding right now.

See it

The gold line is the budget: every mix of the two things they could have bought since you looked. Slide time forward and watch it walk away from you.

Since you last looked 120 seconds What they mine 480 a minute

120 seconds since you looked, at 480 minerals a minute: 960 minerals they have spent on something. That is up to 38 more zerglings, or 19 more drones, or any mix on the line.

The straight edge is the budget line: every mix of the two that the money reaches. The faint curve behind it is the arc the lecture drew first, before a student pointed out that money adds up rather than adding in quadrature.

Two things to notice. The line, not the region: against a player who is not wasting money, the inside of the triangle is empty, because they are spending everything. And the straight edge, not the curve. Money adds up, so the frontier of what a thousand minerals reaches is a straight line. If you picture a curve, you will overestimate how much of two things they can have at once.

The one assumption everything comes from

What they really have is what you observed, plus a quantity you did not observe, pointed in a direction you have to infer.

Write it as a vector and the two jobs of scouting separate cleanly. Seeing more grows the observed part. Seeing anything at all constrains the direction. A probe that dies at the ramp without getting inside has still told you something: it has told you the direction is defence, because they spent on whatever killed it.

The reason this is worth writing down rather than just feeling is that players habitually collapse it to one term. They either believe the scout report as though it were still true, or they throw it away as though it told them nothing. It is a sum, and the second term has a size you can compute and a direction you can often guess.

Information expires at the rate they mine

The scout report is true at the instant of observation and never again.

From that instant, the unknown part grows by whatever they earn. That is the whole content of "information expires", and it is why a five fifty scout is not a report about seven forty. It is a report about five fifty, plus a hundred and ten seconds of money you did not watch them spend.

The practical shape of this: the value of a scout is not the information it gets. It is the information it gets, divided by how long that information has to last. A probe that sees a base at four minutes when the decision is at four thirty is worth far more than one that sees the same base at four minutes when the decision is at eight.

And the second look is worth more than the first, unit for unit, because the first look is nearly always spent learning things that were mostly forced anyway.

What you saw is a subtraction

The title is not a slogan. Scouting works by removing possibilities, and removal is what is useful about it.

Lesson 3 makes the bounds explicit, and they are the sharpest tool on this page. Before you scout at all, their army is not an unknown, it is a bounded unknown:

They cannot have more of anything than their money buys. Total resources divided by unit cost is a hard ceiling on every entry.
They cannot have what the tech tree forbids at this point on the clock. An ultralisk at six minutes is not unlikely, it is impossible, and that column is zero.
Everything else is weighted by what the map, the matchup and the last twenty games make likely.

Now scout. Seeing a spire does not add a spire to your model. It sets one probability to one and several others to zero, and the zeros are the valuable part. Seeing no spire at seven minutes against a player who took gas at three is a bigger piece of news than seeing one.

The working

Show the maths

The fundamental assumption. Their true state is the part you saw plus the part you did not:

R_{\text{true}} = R_{\text{obs}} + |R_u|\,\hat{u}

(1)

A magnitude you can compute, and a direction you have to infer.

The magnitude. The lecture first drew the set of possible states as a circular arc, treating the magnitude as a Euclidean length. A student objected on the spot that in a space whose axes are quantities of units, the size of a bundle is what it cost, and costs add:

|R_u| = c_1 n_1 + c_2 n_2 + \cdots + c_k n_k \quad \text{rather than} \quad \sqrt{c_1^2 + c_2^2 + \cdots}

(2)

An L1 norm, not an L2. The lecturer agreed at the board and redrew it.

So the frontier is the flat face of a budget set, not the surface of a ball. Everything they could have, given $R$ minerals of income you did not watch, is

\left\{\,n \ge 0 \;:\; \sum_i c_i n_i \le R \,\right\}

(3)

and against a player who is spending everything, only the boundary of that set is occupied. A region of guesses becomes a line segment you can reason along.

The growth. Observation is instantaneous. Afterwards,

R_u(t) = R_u(t_0) + \int_{t_0}^{t} f(\tau)\,d\tau

(4)

f is their income. Under a steady economy this is just money times elapsed time.

Why more options is worse than more time. The budget set above is a simplex, and in $k$ dimensions its volume is

V = \frac{R^{k}}{k!\;c_1 c_2 \cdots c_k}

(5)

Uncertainty grows as the k-th power of elapsed time, where k is how many things they could be buying.

With two plausible answers, going blind for twice as long makes you four times as unsure. With three, eight times. This is the formal version of a thing good players do by instinct: they scout hardest in the matchups and at the timings where the opponent has the most branches open, and they stop bothering once the branches have closed.

How often to look. If you scout every $T$ and their income is $M$ , the unknown pool sits at $M(t - t_{\text{last}})$ and averages

\bar{R}_u = \tfrac{1}{2} M T

(6)

Halve the interval, halve your average blindness. Not a quarter: this one is linear.

Which sets up the trade honestly. A scouting worker costs you its mining, and a scouting overlord or observer costs its build. Against that, halving $T$ halves the average size of the thing you are guessing about. The Scouting Lab is where you can put your own numbers into that trade.

Turning a report into a decision

The abstraction pays off in one specific way, and the lecture's worked example is the clearest version of it.

Draw two axes. Their economy on one, their army on the other. You have a decision to make that is discrete: expand now, or do not. If you expand, your own army for the next ninety seconds is whatever static defence and pulled workers you can manage.

Now put their frontier on the same picture. Every point along it is a mix, and for each mix you can say whether your expansion survives it. That splits the frontier into three pieces: the part where you can expand freely, the part where you can expand but must add defence, and the part where expanding loses immediately.

Your scout's job is exactly one sentence: which part of the line are they on. Not "what are they doing", which is unanswerable, but "which of these three regions", which is usually answerable from two or three facts. Gas timing. Worker count. How many production buildings. Whether the thing that killed your scout was cheap or expensive.

That is the difference between scouting as a chore and scouting as a decision.

Discrete and continuous, and why timing is a separate question

The last piece of lesson 7 is a distinction worth carrying into every other guide.

Some choices have no in between. Templar or robotics. Mutalisks or hydralisks or lurkers. Mech or bio. You cannot half commit, and neither can they, which means defending against a discrete choice means preparing for the whole of one branch. There is no intermediate state to meet in the middle.

Other choices are continuous, and for those the question is never whether but when. When to expand, when to attack, how far to push your anti-scouting units out, what ratio of the two units to build. A continuous choice made late delivers a fraction of what it would have delivered made early, and the fraction is set entirely by the timing.

Which is why timing is its own subject. A discrete choice gives you the whole effect or none of it. A continuous one gives you a share, and the share is the thing you are actually choosing.

What the lecture said

The lesson 7 lecture opens with the right question, at 34:30: why would you ever send a worker out of your own town, where it could be working. Everything above is an attempt to answer it with a number. The lesson page plays the lecture in teaching order.

The production cone and the resource vector are built from 8:00, the fundamental assumption is stated and then repeated at 12:00, and the correction to the norm happens across the next fifteen minutes: a student objects around 20:00, the lecturer agrees and redraws, and a second student names it as an L1 norm at 27:00. He thanks them twice and says he has been working with the wrong one.

That exchange is worth watching even if you skip the rest. It is a room doing mathematics together, and the answer it reaches is simply correct: drones plus two zerglings is a budget line.

Still true today

Checkpoint

You scouted at 5:50 and it is now 7:40. What has changed about what you know?
Why is the frontier of what they could have a straight line rather than an arc?
Which scout report is worth the most?

Sources for this page

Lesson 7 lecture, on YouTube, channel VioleTAK The video starts mid lecture: watch from 32:20 to the end, then from the start.
David Sirlin's write ups of weeks 7 and 8 He attended week 7 and published his account two days later
Lesson 3 lecture, on YouTube, channel VioleTAK The imperfect information section, 00:21 to 00:27, is where the bounds come from