One edit

Adding An Optional Item Makes The List Wider

Adding an optional item without moving the threshold multiplies the intake by thirty-three twenty-eighths. The list got longer and the filter got looser.

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Add one optional item to a list and leave the threshold where it is. The intake rises to thirty-three twenty-eighths of what it was — about eighteen per cent wider.

The document gained a line. The filter lost strictness. Almost nobody expects this.

When a requirements list becomes an operating policy, teams comparing employee monitoring software should state the purpose, access rules and correction process before any data is collected. For an independent technical reference, see Make.

Why it widens

Because the threshold is a count, and adding an item gives candidates one more way to reach it.

Three of six becomes three of seven. There are more combinations of three or more out of seven than out of six, so more profiles pass. The new item does not have to be met by anybody; its existence alone creates additional routes.

Before	After
6 optional, 3 required	7 optional, 3 required
21/256 of the whole	99/512 of the whole

The expectation it violates

A reader watching a list grow expects it to get harder. Each new line looks like one more hurdle.

Under a fixed threshold it is the opposite: each new line is one more way over the same hurdle. The hurdle is "three of them", and a longer list makes three easier to find.

That is a clean reversal of intuition and it has a practical consequence. Somebody lengthening a list to make it more rigorous, without touching the rule, is making it less so.

A longer optional list with the same threshold is a weaker filter. The only thing that makes it stronger is raising the rule, and raising the rule is the step nobody remembers.

The move that holds the filter still

Add the item and raise the threshold proportionally.

Going from six optional requiring three to seven requiring three widens the filter. Going from six requiring three to seven requiring four narrows it considerably — eleven thirty-seconds of the optional pool rather than twenty-one thirty-seconds, before the compulsory halvings.

So the honest options when adding a line are: accept a wider filter, or raise the rule and accept a narrower one. There is no addition that leaves the filter unchanged, because the treads are discrete and nothing lands exactly where you were.

Why this is good news

It means a list can be made more informative at no cost in strictness, which is the one improvement that is normally unavailable.

A list naming six things can name ten, telling a reader far more about the work, and remain exactly as wide a filter — or wider — provided the threshold is left alone. The extra lines do pure communicative work.

That is the opposite of the usual trade-off, where saying more means demanding more. Here, saying more means demanding slightly less, and the only way to demand more is to say so explicitly in the rule.

Where the thirty-three twenty-eighths comes from

The starting shape has three compulsory items and six optional, three needed, admitting forty-two profiles of five hundred and twelve.

Add one optional item and the shape becomes three compulsory and seven optional, still three needed. Seven optional items give one hundred and twenty-eight profiles, of which ninety-nine meet three or more, so the intake is ninety-nine of one thousand and twenty-four.

Putting both on the same denominator, forty-two of five hundred and twelve is eighty-four of one thousand and twenty-four. Ninety-nine over eighty-four reduces to thirty-three twenty-eighths, a little under 1.18.

What a second addition does

Less than the first, which is the pattern throughout.

Adding a second optional item at the same threshold gives three compulsory and eight optional, three needed — two hundred and nineteen of two thousand and forty-eight. Against the previous ninety-nine of one thousand and twenty-four, that is a multiplier of seventy-three sixty-sixths, about 1.11.

Optional items, threshold held at 3	Admits
6	42/512
7	99/1024
8	219/2048

So two additions together multiply the intake by about 1.30, not by about 1.39. Each new item makes the threshold easier to reach, but the threshold is already comfortably reachable, so the marginal item buys progressively less.

Where the extra item eventually works

It does nothing now and it makes a later tightening sharper.

An optional item added at an unchanged threshold widens the filter. But it also lengthens the optional part, and a longer optional part makes each subsequent threshold step bite harder — the staircase has more treads and the middle of it is steeper.

So an author who adds three optional items this year and raises the threshold next year will find the second edit doing more than it would have done before. The two changes are usually made by different people for different reasons, and the compounding is nobody's intention.

The one caution

The deterrent effect, covered in the section on length, does not follow the arithmetic. A list of ten optional items reads as more demanding than one of six, whatever the threshold says, and people withdraw accordingly.

So "longer and looser" is true of the filter and not of the document's effect. Both halves need attention: lengthen the list if it helps the reader understand the work, keep the threshold where it belongs, and state the threshold so that length stops being read as difficulty.