Chokes, arcs and surrounds in StarCraft

Learner Competitor Advanced

The short version

Flux is the share of your army that can actually fire at this moment.
It is set by shape and by terrain, not by how many units you own.
What decides a fight is your flux against theirs. Fifteen percent against eighty percent loses even when you have more units.
A choke is a flux cap you can impose on somebody. That is the entire reason three units can hold a ramp against twenty.
Flux is also the dial between the two laws in T2. High flux, square law, the big army wins enormously. Low flux, linear law, the big army wins by subtraction.

The situation

You have twenty four dragoons. They have sixteen. You a-move into their position at the top of a ramp and you lose, badly, and the replay shows your back two thirds standing still in a column behind the units that were dying.

You paid for twenty four. You fought with eight. That is the whole story, and no amount of unit counting will ever explain it, because unit counting is exactly the thing that is wrong.

See it

The gold ring is the distance at which your weapon reaches them. Only units standing on that ring are firing. Widen the wrap and more of them fit on it; narrow it to the width of a ramp and most of your army is a queue.

How far you wrap 90 degrees Your range 4 body widths

Twenty four units wrapped 90 degrees around them: 12 weapons in range, a flux of 50 percent.

The gold ring is the distance at which your weapon reaches them. Only units standing on it can fire, and only as many as fit. A choke is the same picture with the wrap forced down to a few degrees.

Notice the second slider. Longer range does not only mean you shoot sooner. It means the ring you are allowed to stand on is bigger, so more of your army fits on it at once. Range buys flux, which is a second reason range upgrades are worth more than their damage numbers suggest.

What flux actually is

Flux is the number of your units that can attack right now, as a fraction of the number you own. One hundred percent means every weapon you paid for is being used. A third is, in the lecturer's word, terrible.

Two pictures make it concrete, and both are on the board in the lesson 4 lecture.

A ball of dragoons converging on a point. The ones at the back never fire at all. They are not out of position in any way you could point at on the minimap. They are simply behind other dragoons, and they stay behind them until the front rank dies and they move up into the gap. You paid for them and they are not in the battle.

Three zealots holding a choke against a horde of zerglings. Most of the zerglings are milling around behind, doing nothing, because only a few of them can reach. The choke is a cap on the attacker's flux, and holding one works for exactly that reason and no other.

Only relative flux matters

Here is the part to keep. Your flux alone tells you nothing. A fight is decided by the ratio.

If your flux is fifteen percent and theirs is eighty, you lose, whatever the supply counter says. Twenty four dragoons at fifteen percent are fighting with three and a half dragoons. Sixteen at eighty percent are fighting with thirteen. The fight you think you are having is twenty four against sixteen. The fight you are actually having is three and a half against thirteen, and you lose it so badly that you conclude the units must be bad.

This also explains a thing that looks like magic when a stronger player does it to you. They do not have a better army. They arranged for the moment of contact to happen where their shape was good and yours was not, which is a decision made thirty seconds before the fight, on the map, not in the fight.

The arc beats the column, and here is by how much

The class polled the room first and found, unsurprisingly, that people do walk into fights in a single file line. Then it proved on the board that they should not.

The geometry is simple once you see it. Your enemy's weapons reach a circle around them. Your units can only shoot from inside that circle, so the question is how many of yours are inside it at the moment the shooting starts. A broad line crossing the edge of the circle cuts a chord, and a chord is wide the instant it exists. A column crossing the same edge enters one unit at a time.

So at the moment of contact, which is the only moment that matters, the broad formation has many more weapons live, and it is not close.

The working

Show the maths

The definition, as it was written on the board: flux is the damage your units can deliver from the area they are actually firing from, over the damage your whole army could deliver in principle.

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

(1)

A normalised integral. One hundred percent is every weapon firing.

The chord. Put the enemy at the centre of a circle of radius $R$ , their weapon range. Advance your line into it. When your leading edge is a depth $x$ past the circle's near boundary, the half-width of your line that is inside the circle is

y = \sqrt{2Rx - x^2}

(2)

The board's own expression. It is the chord half-width of a circle of radius R at depth x.

Two formations, same penetration. Let your units stand at a linear density of $\lambda$ per unit of width. A broad line has, in range,

n_{\text{line}}(x) = 2\lambda\sqrt{2Rx - x^2}

(3)

and a single file, entering one behind another, has

n_{\text{file}}(x) = \lambda x

(4)

The ratio of the two is $2\sqrt{2Rx-x^2}\,/\,x$ . At a tenth of the way into their range it is about nine. At the full range it is two. The two are equal only at $x = 1.6R$ , which is to say four fifths of the way through the enemy's entire position, and you will never get there, because the front of your line has been under fire for the whole walk. The broad formation is better for every penetration that actually happens.

The rate version, which is what the board did. Differentiate:

\frac{dn_{\text{line}}}{dx} = \frac{2\lambda\,(R - x)}{\sqrt{2Rx - x^2}}, \qquad \frac{dn_{\text{file}}}{dx} = \lambda

(5)

Ours, re-derived. The board carried the same expression for the line, without the factor of two.

The line's rate is unbounded at first contact and falls as you push in. It drops below the column's only at $x = R\left(1 - 1/\sqrt{5}\right) \approx 0.55R$ . So there is a theoretical crossover, and the lecture was careful to say it is theoretical: you do not reach it, because the enemy's front rank is dying the whole way in and the fight is decided long before your rear rank arrives.

Where this meets T2. The square law assumed every unit on each side can shoot every unit on the other. That assumption is exactly $\Phi = 1$ . Write flux into the attrition model and it becomes a ceiling on the loss term,

\frac{dA}{dt} = -\min\!\left(\alpha B,\; \alpha \Phi_B N_B\right)

(6)

The cap a choke imposes: losses stop depending on how many they own.

and when the cap binds, losses become constant rather than proportional and the square law degenerates into the linear law. The two laws are one model at two settings of one dial.

Raising yours and lowering theirs

Flux is not a thing you know, it is a thing you do. Six ways, in rough order of how much they are worth.

Choose where contact happens. This is worth more than everything else on the list combined. If you are the larger army, fight in the open where your numbers can spread. If you are the smaller one, fight at a ramp or in a treeline where they cannot. Almost every fight you lose while ahead was a fight you took on their ground.

Arrive on a broad front. An army that walks in a column arrives in a column. If you want an arc at contact, you have to be wide before contact, which means splitting into two or three control groups on the approach and closing them at the end, not at the start.

Stop, then engage. Units that are still walking when the fight starts spend the first seconds resolving their own pathing instead of shooting. Holding position a body's length short of range and then moving forward together is worth several units, free.

Put the slow things behind and the ranged things at the edge. Flux is per weapon, not per unit. A ranged unit standing at maximum range occupies a slot on a bigger circle, and bigger circles have more slots.

Make your own chokes. Building placement is the cheapest flux cap in the game. A wall that forces an army through a two unit gap has bought you a permanent eighty percent discount on everything that comes through it, for the price of two buildings you were going to build anyway.

Break theirs. If they are holding a choke, the answer is almost never to walk into it. It is to make them come out: take the ground behind them, threaten something they have to answer, or arrive somewhere the choke does not cover. A choke defends one line. It does not defend a map.

The homework

The lesson 4 homework is the surround, and it is a geometry problem rather than a strategy one: an arc of attackers sixty long, closing on a body twenty wide, with the attacker's estimate of the distance wrong by ten. Work out what fraction of the arc is actually in range, and what the error costs.

The reason it is a good problem is that it is the fight you have every time you try to surround something. Get the radius right and the whole arc fires. Get it wrong by a little and the wings are out of range while the centre takes all the damage, which is worse than not surrounding at all. Try the full problem on the homework page, then check your answer in the Flux Lab.

What the lecture said

Flux is defined at 36:40, with the normalised integral going on the board at 39:40. The choke example, three zealots against a zergling horde, sits just before it, and the broad line derivation runs from 42:10 to the break.

Flux first comes up in lesson 3 at 42:00, during a clip of dragoon control, as the answer to a question the class had already run into: how much of your army can attack him at once. The lecturer put a name on something the room had already noticed, which is the correct order to do that in.

A common mistake. Players learn "arc good, column bad" and then form a beautiful arc thirty pixels too far away. An arc only counts if its units are in range. Walk the arc in until the front edge touches their range, then stop. The wings should fire in the same second as the centre.

Still true today

Still true

Flux is the most durable idea the course produced

It holds completely, and it has aged better than the vocabulary the wider community settled on. Players say concave, which names one shape. Flux names the quantity that shape is trying to change, which means it also covers chokes, ramps, unit size, range upgrades, building placement and the reason a column dies to a line.

You do not have to take it on trust. The Flux Lab lets you drag the same army into a column, a line and an arc against the same choke and watch the share of weapons in range move, and the Battle Lab runs the fight in the engine afterwards so you can check that the shape mattered as much as the geometry says.

One honest caveat about the model above: it treats units as points on a ring with a fixed range and no collision. Real Brood War units have bodies, push each other, and path badly, all of which make a column worse than the arithmetic says rather than better. The model understates its own case.

Checkpoint

Your flux is 20 percent, theirs is 80. You have twice as many units. Who wins?
Why does a range upgrade raise your flux as well as your damage?
You are the bigger army and they are sitting behind a ramp. What is the mistake?

Sources for this page

Lesson 4 lecture, on YouTube, channel VioleTAK Flux is defined at 00:35 and the broad line derivation runs to 00:46
Homework 4 (archived copy) The surround problem. The capture is a scan and yields no text
David Sirlin's notes on lesson 4