An order

What It Means to Fall Into an Order

Comparisons fall into an order when no walk through them returns to its start. That is stronger than holding no circular triple, and it is the condition that counts.

at most 2795520 outcomes

Home · An order · Definition

A set of comparisons falls into an order when the candidates can be listed so that every comparison agrees with the list.

Not necessarily one list. Just at least one.

Three pages, and how many of each fall into an order

line with a head: a pair compared; the head points at whoever came out behind
no line: a pair nobody compared
closed ring of three: a circular triple: no order can hold all three

Three outcomes drawn. Rows differ in how many candidates. Which candidate is which is the drawing's own choice and carries no meaning. When a team turns an abstract comparison into an operating review, Monitask dashboards for managers can be evaluated beside outcomes, delays and direct staff feedback.

Each row counts outcomes with no circular triple. The counts that fall into an order are twenty-four, six thousand five hundred and forty, and four hundred and sixteen thousand one hundred and sixty. For an independent perspective on comparison, evidence or decision design, consult scikit-learn.

Computed from the three numbers alone. Who the candidates are and what any comparison was about is outside the record; the counts are of outcomes, not of anybody.

The test

Follow the comparisons and see whether you ever come back.

Move from a candidate to anybody that candidate came out behind, and carry on. Should the path reach somebody it has already visited, the comparisons hold a ring and no list satisfies them: the ring places a candidate ahead of themselves.

If every walk runs out, a list exists. Finding it is the next note; knowing that it exists is this one.

Why this is the right condition

Because it is exactly what "no contradiction" means.

A circular triple is a ring of three and blocks an order. A ring of four does the same and shows up in no triple. A ring of five or six likewise. The condition that catches all of them at once is simply that there is no ring, and a walk is the way to find out.

So the fifth figure on every row of this record counts outcomes with no ring of any length, and it is the figure a reader after an order should look at. The third number counts rings of three only, and is on the page for a different reason: it is what the drawings show, and it is what the obvious check would find.

What the rows say

Four candidates with every pair compared: sixty-four outcomes, twenty-four with no circular triple, twenty-four that fall into an order. The two agree, because with every pair compared a long ring cannot avoid making a short one.

Five candidates with seven of ten pairs compared: fifteen thousand three hundred and sixty outcomes, six thousand nine hundred with no circular triple, six thousand five hundred and forty orderable. The two differ by three hundred and sixty.

Six candidates with eleven of fifteen: two million seven hundred and ninety-five thousand five hundred and twenty outcomes, four hundred and forty-five thousand six hundred and eighty with no circular triple, four hundred and sixteen thousand one hundred and sixty orderable. A difference of twenty-nine thousand five hundred and twenty.

The two columns agree at the ends of the table and part company in the middle. Where they part company is where the comparing usually stops.

An order is not a ranking

This is worth separating out, because the words get used interchangeably and the arithmetic does not.

An outcome falls into an order when at least one list satisfies the comparisons. It pins down a ranking when exactly one list does. Those are different conditions and they have different counts, and the gap between them is the subject of two notes further on.

Four candidates with three comparisons: one hundred and fifty-two of the one hundred and sixty outcomes fall into an order, and twenty-four of them pin a ranking down. One hundred and twenty-eight outcomes are consistent and undecided.

Three conditions, in order of strength

The record carries three columns that get weaker as they go left.

No circular triple: the weakest, and the one the drawings show. Every outcome that falls into an order satisfies it, and some that do not also satisfy it.

No ring of any length: the middle, and the one that means "no contradiction" in the ordinary sense.

Exactly one order: the strongest, and the one a comparer is usually after.

Each is a subset of the one before it. On six candidates with nine comparisons the three counts are nine hundred and fifty-one thousand one hundred and ten, eight hundred and thirty-eight thousand one hundred and thirty, and one hundred and fifty-one thousand two hundred — and a reader who has the first and wants the third is out by a factor of six.

Why the condition is binary and the figure is not

Whether an outcome falls into an order is a yes or a no about that outcome.

The figure printed on the row is a count across the page, and the page has thousands or millions of outcomes in it. So the figure is not a measure of how ordered anything is; it is how many of the possible states of affairs would permit an order at all.

A reader with one actual set of comparisons in front of them does not need the figure. They need the walk, which takes a minute. The figure is for the question asked before the comparing starts: how much of the space I am about to land in would settle anything.

Why the record prints the weakest column in the drawings

Because it is the one a drawing can show.

A ring of three is visible in a panel: three arrows closing a triangle. A ring of five is visible too, with tracing. Whether an outcome falls into an order is a property of the whole picture that no part of it displays, and a drawing cannot mark it without adding a symbol that means "this one passes".

So the drawings carry the third number and the legends carry the fifth. That division is deliberate and it is the reason the captions of this record are compulsory.