Thresholds

Where The Staircase Turns Steep

Between four and seven of eight the intake falls from one hundred and sixty-three profiles to nine. Three steps, a factor of eighteen, nobody notices.

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A threshold of three on eight optional items admits two hundred and nineteen profiles in two hundred and fifty-six. A threshold of seven admits nine.

Four steps. A factor of more than twenty-four. Each step is one word changed in one sentence.

For an example beyond the counting model, teams can open this product resource and map available records to a specific capacity decision. For an independent technical reference, see SAP.

The steep part

Five thresholds drawn. The tread is flat because the threshold is a whole number: there is nothing between one item and the next.

Five treads from the wide end of the staircase to the narrow one. Each step is a single item on the rule and the fall between the first and the last is a factor of twenty-four.

Computed from the threshold. Each item is counted as a separate coin: this is the unit of measurement, not an estimate of any population.

The slope, step by step

Threshold	Admits	Change from previous
3 of 8	219/256	—
4 of 8	163/256	about a quarter lost
5 of 8	93/256	about two fifths lost
6 of 8	37/256	about three fifths lost
7 of 8	9/256	about three quarters lost

The steps get proportionally worse as they go. Raising the threshold from three to four costs a quarter of the intake; raising it from six to seven costs three quarters of what remains.

So the damage done by tightening a threshold accelerates, which is the opposite of the halving law, where every compulsory item costs the same proportion.

Why thresholds accelerate and compulsory items do not

A compulsory item is a coin: it removes half of whatever is left, always.

A threshold works on the distribution of how many items a profile meets, and that distribution is concentrated in the middle. Moving the threshold through the middle of the distribution removes large bands at a time; moving it out at the edges removes very little, because very little is there.

Between three and seven the threshold is cutting through the fat part. That is why four adjacent steps destroy twenty-four twenty-fifths of the intake, and why the two steps at either end of the staircase barely move anything.

The threshold is the only parameter in this record with accelerating cost. Everywhere else the arithmetic is flat, and here it is not, which is exactly why it is the parameter that gets set by feel.

Crossing the steep part without noticing

The language does not mark the crossing.

"Several of these", "most of these", "the majority of these", "nearly all of these" — four phrases that a writer would consider roughly interchangeable, landing on four different treads spanning a factor of twenty-four.

Nothing in the drafting process signals that a phrase has moved the filter by that much. The list looks the same, the items are unchanged, and the sentence reads slightly differently.

The eighteen, counted as profiles lost

The factor is easier to hold if it is read as what each step removes rather than as a ratio.

Going from four to five of eight removes seventy profiles — the whole band meeting exactly four items, which is the largest single band there is. Five to six removes fifty-six. Six to seven removes twenty-eight.

So the steepest single step is the first one, and it is steep because the distribution of how many items a profile meets is peaked in the middle. A threshold moving through the middle of a list walks straight through the fattest part of it.

Step	Profiles removed
4 → 5	70
5 → 6	56
6 → 7	28

Where the steep part lives in a document

Not in a paragraph anybody reads carefully.

It lives in a phrase: "most of the following", "several of these", "a strong majority of these". Each of those covers two or three adjacent treads, and the treads differ by factors of about two. The phrase is doing the work of a number and cannot carry it.

A reader who wants to know where they stand cannot resolve the phrase, and a second reader resolves it differently from the first. Both of those follow from the same fact: the language has about four words for this region and the region has four treads with an eighteenfold spread across them.

What this makes worth doing

Writing the threshold as a number rather than a phrase, every time.

"Four of the following eight" is unambiguous, computable, and identical for every reader. "Most of the following" is none of those things, and it sits in the steepest part of the staircase where the ambiguity costs the most.

A number also makes the rule arguable. Somebody can say that four is too many and propose three, and the difference between them can be stated: two hundred and nineteen against one hundred and sixty-three, a quarter of the intake. That is a conversation with a fact in it.

The asymmetry worth remembering

Because the cost accelerates, the cheapest useful loosening is always at the strict end.

A list stuck at seven of eight and admitting nothing can be moved to six for a nearly fourfold gain in intake, which is a larger improvement than any change to the items themselves could produce. A list at four of eight gains much less from moving to three, because it was never the binding constraint.

That gives a simple rule for a list that is admitting nobody: look at the threshold before looking at the items, and look at it first if it sits above the middle of the staircase.