StarCraft army positioning and chokes

Lesson: 4 of 14
Watch: The lecture, 54 minutes, or a 53 minute cut
Practise: Homework 4, a checkpoint та a plan for your next game

This lesson is about the fights you lose with the bigger army. You will learn why arriving together wins by far more than the supply difference, how much of your army is really shooting, and what to hit when you do attack.

What you will learn

Why a bigger army wins by more than its size, with numbers: 20 against 15 is a crushing win, and two groups of 10 against 15 is a loss.
Why concentrated damage beats spread damage, and why that gap grows the longer a fight lasts.
What flux is: the share of your army that is actually firing. How chokes, lines and surrounds change it.
Why a broad line beats a column, and where the advantage comes from.
What to attack first: the things that do not shoot back.

Dead units do not shoot

Start with one fact. A unit you kill stops dealing damage for the rest of the fight. Every unit you remove early is damage you never take. That is why fights snowball, and it is the idea every other part of this lesson builds on.

The volley model

Picture two identical ranged armies. Both fire at once, in volleys. Side 1 has perfect micro: it focuses fire, so every volley kills units. Side 2 has the worst micro: it spreads its damage evenly, so nobody dies until everybody dies.

Focused against spread

N_2 M - \frac{M(M+1)}{2}\cdot\frac{N_1 D}{HP}

(1)

What the spreading side manages to deal over M volleys, counted in unit volleys.

Lesson 4, the volley model

Each volley, side 1 kills $N_1 D / HP$ of side 2's units and loses none. Those losses stack up as $1 + 2 + \dots + M$ , so the gap grows with the square of the number of volleys. A short fight barely separates good micro from bad. A long one separates them enormously.

Nobody is perfect and nobody is that bad, so the exact numbers do not matter. The growth rate does, and it tells you where to spend your attention:

Where your attention goes	How the advantage grows
Micro, focused against spread	$t^2$
Keeping your army together	$\cosh t$
Economy, with perfect macro	$e^t$

Exponential beats squared given time. Micro wins the fight in front of you. Macro wins the game. When you must choose where the next few seconds of attention go, a fight you can win with a-move is the place to spend less, and your production is the place to spend more.

The bigger army wins by more

Now make it continuous. Each side loses units at a rate set by how many of the other side are still alive. This is Lanchester's square law.

Two armies wearing each other down

\frac{dA}{dt} = -\alpha B, \qquad \frac{dB}{dt} = -\beta A

(2)

Alpha and beta are how fast one of his units removes yours, and one of yours removes his.

\beta A^2 - \alpha B^2 = \text{constant}

(3)

The square law. Army size counts squared, unit quality counts once.

Lesson 4, attrition

The second line is the one to remember. Army size enters squared. Double your numbers and your fighting strength goes up four times. Double your damage and it goes up twice.

Worked examples, identical units

Take marines against marines, same upgrades, open ground. With equal quality, the units left standing are the square root of the difference of the squares.

The fight	Left standing	What it means
10 against 7	about 7 of your 10	You lose 3 to kill 7
20 against 15	about 13 of your 20	You lose 7 to kill 15
Two groups of 10, one after the other, against 15	His 15 win with about 11 left, then beat your next 10 with 5 left	Same 20 units, and you lose everything

Read the last row twice. You had 20 and he had 15. Arrive together and you win with 13 alive. Arrive in two groups of 10 and he beats the first group with 11 left, then beats the second with 5 left. Splitting your army hands him the win. That is the most expensive mistake in this game, and the square law shows exactly how expensive.

It also works for you. Catch ten of his supply with fifty of yours and the fight ends before his units deal meaningful damage. Every isolated group of his is a free meal.

The derivation, in four lines

Differentiate the first attrition equation and substitute the second:

\frac{d^2A}{dt^2} = -\alpha\frac{dB}{dt} = \alpha\beta A

(4)

The solution that starts at A0 and loses alpha B0 per second at the start is:

A(t) = A_0 \cosh\!\left(\sqrt{\alpha\beta}\,t\right) - \sqrt{\tfrac{\alpha}{\beta}}\, B_0 \sinh\!\left(\sqrt{\alpha\beta}\,t\right)

(5)

Differentiate beta A squared minus alpha B squared and the terms cancel exactly, so it is constant through the whole fight. The numbers left standing in the table come from setting B to zero at the end.

The naive estimate A0 minus alpha B0 t pretends every one of his units fires for the whole fight. They do not, because yours are killing them. The difference is the damage that never landed, and it always favours the side that kills faster: dead units do not shoot.

Short fights and long fights

The model gives four pieces of advice, one per case:

His units kill yours much faster. Few of his die, so the fight is roughly linear. You are simply losing. Leave.
A short fight, one to three seconds. The snowball has no time to start, so the trade is roughly even. Short skirmishes are fair trades.
A long fight, twenty to forty seconds. The snowball dominates. You cannot predict a long fight by counting units, and a small early edge becomes a wipe.
You brought far more. The fight ends before the snowball matters, and you lose almost nothing.

That last case is the thesis of the lesson. Bring your whole army. A large army ends the fight fast, in the short regime, and barely loses anything.

Extending the model

Reinforcements and regeneration slot straight in:

Reinforcements and regeneration

\frac{dA}{dt} = -\alpha B + R(t) + rA

(6)

R(t) is reinforcements arriving. rA is regeneration of the units still alive.

Lesson 4, extending the model

Two practical lessons come out of it. First, compare rates before you worry about a term. Zerg regeneration is tiny next to the damage in a real fight, so it does not decide battles. It decides what you have for the next one. Second, when you cannot work out a unit's value from its stats, measure it. Dragoons are large, slow and hard to focus cleanly, so run the fight a dozen times in the Battle Lab and see.

Flux: how much of your army is shooting

Flux is the share of your army that can attack right now. Not what you own. Not the supply on the bar. The ones in range.

Flux

\Phi = \frac{\displaystyle\int D(u)\cdot d\vec{A}}{\displaystyle\sum D(u)}

(7)

The damage your army delivers from where it is actually firing, over the damage the whole army could deliver.

Lesson 4

One hundred percent means every unit is firing. A third is terrible. Two examples:

A ball of dragoons walking into a point. The back rows never fire until the front rank dies and they step up. You paid for them, and they are not in the fight.
Three zealots in a choke against a zergling flood. Only the few zerglings at the front can touch them. The rest mill about behind. A choke caps the attacker's flux, and that is the whole reason holding one works.

Flux is relative. If yours is 15 percent and his is 80, you lose, whoever has more supply. This is the fight where your bigger army walked in and evaporated.

Flux is also why the square law bends. In a choke, only the front of each army fights, so extra numbers behind it are worth nothing until the front dies. Pull a bigger enemy army into a narrow place and you turn his square law into a straight trade.

Broad line against single file

Should you walk in as a column or a broad line? Work it out. His army sits at a point with range $R$ . You advance into that range circle. When you have gone a depth $x$ into it, the width of your line inside the circle is a chord.

Why the line wins

(x - R)^2 + y^2 = R^2 ;\Longrightarrow; y = \sqrt{2Rx - x^2}

(8)

Half the width of your line that is in range, at depth x.

\left.\frac{d\phi}{dt}\right|_{\text{his}} = \lambda v, \qquad \left.\frac{d\phi}{dt}\right|_{\text{yours}} = \frac{R - x}{\sqrt{2Rx - x^2}}\,\lambda v

(9)

Lambda is how densely your line is packed, v is your speed. His flux grows steadily. Yours grows with the slope of the chord.

Lesson 4, the geometry

At first contact, $x$ is tiny, so the bottom of that fraction is tiny, and your flux grows far faster than his. Every step forward puts several more of your units in range while only a trickle of his come into range of yours. The rates only cross once you have closed about 29 percent of his range, at $x = R\left(1 - 1/\sqrt{2}\right)$ , and by then his front rank has been under fire the whole way in.

The practical version: arrive as a wide line or an arc, never as a column. A column puts one unit in range at a time, which is the choke, self inflicted.

The surround

The surround is flux taken to its limit: your whole arc fires at once while his ball can only answer from its edge. It only pays if the whole arc lands together. Misjudge the distance and one end of your arc engages before the other, and a ball that sees it coming can move to cut your army in half and kill each half on its own.

Our drawing, to scale, of the situation on the homework sheet. The cross marks where the arc is aimed, which is 10 squares to the right of where the Protoss army is standing. Every zergling on the arc is a different distance from the fight, and the question is what that costs.

The picture above is homework 4's surround problem: a zerg line 60 squares wide closing on a protoss ball 20 squares wide, with the distance misjudged by 10 squares. Work out how much earlier one side engages than the other, what that costs, and how far the protoss needs to move to cut the zerg in two. The Flux Lab lets you move the pieces and see the answer.

Attack what does not fight back

When you do attack, choose the target by one question: does it shoot back?

What you hit	What it costs him	Does it shoot back?
Workers and mining	The rate he gathers at	No
Production and supply	The rate he turns money into army	No
His army	Just those units	Yes

The first two cut a rate, for free. The third trades. So drops on the mineral line, run bys on production, and picking off isolated groups beat a head on fight with the main army, every time you can get them.

Sun Tzu's ratios make the same point: ten to one, attack; outnumbered, divide him or defend a choke; badly outnumbered, run. Every line is one rule. Fight only where he cannot hit you back in full. Running keeps your army and denies him damage. Defending uses terrain to cap his flux.

Common mistakes

Arriving in pieces. Reinforcements that trickle across the map one group at a time are the "two groups of 10 against 15" row. Gather, then move.
Walking up his ramp. You have just given him a choke. Pull him out onto open ground, or come in from a second angle.
Moving as a column. Units stream out of production in a line. Spread them before contact.
Counting supply instead of flux. Your 150 supply ball behind a choke is fighting as 40.
Chasing his army when his workers are exposed. Hit what does not shoot back.
Staying in a long fight you are losing. The snowball runs both ways. Leave early.

Watch the lecture

A 54 minute lecture: the volley model, attrition, flux and what to attack. The short cut drops the break.

Three models and the closing argument, without the break.

Captions are YouTube's automatic ones. Skipped in the short cut: 46:00 to 47:00 (The break).

The minutes worth your time: the growth rates at , the four regimes at , flux at , and "attack what does not fight back" at .

Practise

Homework 4 turns this lesson into four problems.

Supply at the cap. Zerg looks supply efficient in the midgame, yet at 200 against 200 it is much weaker, especially against a 150 supply terran ball of tanks and marines. Why? Start from flux: cheap melee units that must touch the enemy fight from the edge of the ball.
The surround. The problem pictured above. The second and third parts are the same question from two sides.
Positioning through the game. Where does your army stand early, mid and late, and what decides it? Early you cover places you cannot see. Later you cover things that cost you money.
Mobility. Not top speed: how much of the map your army threatens from where it stands. Speed, plus range, plus whether it can be somewhere else before the fight is over.

Do homework 4

Then run the fights yourself:

Battle Lab: 20 against 15 together, then 10 and 10 against 15. Count what is left.
Flux Lab: a line, a column, a choke and a surround, side by side.
Dead units do not shoot and Flux: chokes, arcs and the surround: the longer guides.

Checkpoint

Why does the bigger army lose fewer units than the size difference suggests?
You have 20 marines, he has 15. You send them as two groups of 10. What happens?
Your flux is 20 percent and his is 80. You have more supply. What happens?
You walk a line of ranged units into his army. When is your flux growing fastest?
His army is on your side of the map and his mineral line is open. What does this lesson say to hit?

Credits and sources

Built on Game Theory with Application to Starcraft, taught at UC Berkeley in spring 2009 by Alan Feng, Anthony Chen and Sherwin Mahbod, as its week 4. The teaching on this page is written in our own words. Not affiliated with UC Berkeley or anyone named here; if something should not be here, tell us.

All sources for the course