Thresholds

Six Optional Items, One Compulsory One

Six optional items requiring three admits twenty-one profiles in thirty-two. One compulsory item admits sixteen. The shorter list is the stricter one.

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One compulsory item admits half of all profiles: sixteen in thirty-two.

Six optional items requiring three of them admit twenty-one in thirty-two.

The list six times longer is the more generous filter, and by a margin of about a third.

A short strict list against a long loose one

filled: a requirement that must be met
open: one of the optional ones
area of the inner square: the share admitted
a stroke instead: too small for this panel to draw

Four shapes drawn.

One compulsory item is a narrower filter than six optional ones requiring half. It takes a threshold of four before the six-item list becomes the stricter of the two.

Computed from the shape of the list. The shape is a stated convention; the count of admitted profiles is arithmetic on subsets.

Why this is counterintuitive

Because length reads as difficulty. A reader encountering six requirements and a reader encountering one will judge the first to be the harder proposition, and in this case they are wrong.

The strongest case for remote team management analytics is a narrow workflow question, a short pilot and a review that combines records with participant feedback. For an independent technical reference, see Oracle.

The reason is that the six-item list has slack built into its rule. A candidate can miss any three of the six and still pass; there are twenty-one distinct ways to succeed. The one-item list has no slack at all — meet it or do not — and there is exactly one way to succeed out of two.

Slack is what the threshold buys, and slack is what the eye does not see when it counts lines.

Where the crossover is

Raising the threshold on the six-item list eventually makes it stricter than the single compulsory item.

List	Admits
1 compulsory	1/2
6 optional, 3 required	21/32
6 optional, 4 required	11/32
2 compulsory	1/4

At a threshold of three the six-item list is wider. At four it is narrower — eleven in thirty-two against sixteen — and it has crossed the single compulsory item without crossing two of them.

So the crossover sits between the third and fourth tread. A six-item list with a middling rule is somewhere between one compulsory item and two, which is a much weaker filter than its length suggests.

The length of a list is a measure of how much was written down. It is not a measure of how much is required, and the two can point in opposite directions.

What this does to comparing lists

It makes line-counting useless as a comparison, and that is worth being blunt about, because line-counting is what everybody does.

Faced with two lists — one of three items, one of nine — almost any reader will treat the second as more demanding. Whether it is depends entirely on the rules attached, and under plausible rules it can easily be the looser of the two.

A nine-item list requiring four is a wide filter. A three-item conjunction is one profile in eight. The nine-item list looks three times as hard and is several times easier.

The honest way to read a long list

Find the rule. If it is stated, the record gives the figure. If it is not stated, the list is one of the shapes with an unstated threshold, and the page on a threshold of none covers what that actually means.

If the rule cannot be found and cannot be asked for, the safest reading is the conjunction — because that is how a list with no stated rule is treated by at least some of the people applying it, and it is the narrow end of the range of possibilities.

Where the crossover actually falls

Against a single compulsory item, which admits sixteen profiles in thirty-two, the six-item list crosses over between a threshold of three and a threshold of four.

Three of six admits forty-two profiles in sixty-four. One compulsory item admits thirty-two in sixty-four. So the six-item list is still the wider of the two.

Four of six admits twenty-two in sixty-four, which is narrower than the single item. The crossover is therefore somewhere between three and four, and since thresholds are whole numbers it is not a point that any rule can sit on.

Rule	Admits (out of 64)
3 of 6 optional	42
1 compulsory item	32
4 of 6 optional	22

Two lists a reader cannot rank by eye

Put the two documents side by side and nothing visible separates them.

One page carries six lines and the word "desirable". The other carries one line and the word "required". The six-line page looks like the demanding one — it is longer, it asks about more things, and it takes longer to read.

Under a threshold of three it is the more generous of the two by a wide margin. Under a threshold of four it is narrower. The deciding element is a single number that may not appear on either page, and no amount of reading the lines will recover it.

The use of this for writing

If the intention is a wide filter that nevertheless communicates what the work involves, a long optional list with a low threshold does it well: six items at three admits two profiles in three while naming six things.

If the intention is a narrow filter, a short compulsory list does it with far fewer words and no ambiguity: two compulsory items admit one in four.

The failure mode is using the first structure while intending the second, which produces a document that names a great deal and demands very little, and then surprises everybody when it admits almost anyone.