Nothing in threes

Why Checking Every Triple Is Not Enough

The triple is the natural unit of checking and it misses every ring longer than three. On the middle pages of this record that is one case in eight.

2562560 outcomes

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Looking at three candidates at a time is the natural first move, and it is natural because three is where contradiction becomes possible at all.

It is the right place to start. It is not a place to stop, and the record can say by how much.

Six candidates, nine pairs compared

solid wedge: the row came out ahead of the column
blank cell: the column came out ahead of the row
small ring in both cells: the pair was never compared
dash: the diagonal: nothing is compared with itself

Six candidates and 15 possible pairs. Nine of them compared. No circular triples. One of the 2562560 outcomes this page allows. Managers testing competing processes can examine team activity reports for decision support against the same baseline, correction case and reporting window.

Row beats column. This outcome has no circular triple anywhere in it. Whether it falls into an order cannot be settled by looking at three candidates at a time. For additional background on data, visualisation and evaluation, review material from StatQuest.

One of the outcomes this page allows, drawn in full. Which candidate sits in which row is the drawing's own choice; the counts are of outcomes, not of anybody.

The shape of the blind spot

A circular triple is a ring of three. A triple check finds rings of three. It finds nothing else.

Rings of four, five and six exist in any field large enough to hold them, they contradict an order exactly as thoroughly as a ring of three, and they are invisible to a triple check because every triple on a long ring is missing at least one of its pairs.

So the blind spot is not a matter of degree. The check answers a different question from the one a reader is asking.

How large it is, on the page where it is largest

Six candidates with nine pairs compared.

Nine hundred and fifty-one thousand one hundred and ten outcomes pass the triple check. Eight hundred and thirty-eight thousand one hundred and thirty of them fall into an order.

The rest — one hundred and twelve thousand nine hundred and eighty outcomes, about one in eight of those that passed — contain a ring of four, five or six and no ring of three.

Why the middle is where it bites

A long ring needs gaps. Each triple along it has to be missing a pair, and missing pairs are what an incomplete set of comparisons provides.

With very little comparing there are plenty of gaps and almost no rings, because a ring needs as many comparisons as it has links. With very thorough comparing there are rings in abundance and almost no gaps, so every long ring comes with a short one attached.

In the middle both are available at once, and that is where the hidden cases live. For six candidates the count of them peaks at nine comparisons out of fifteen.

The check is reliable when there is nothing to check and reliable again when everything has been checked. In between, it is wrong about one case in eight.

What a correct check looks like

Follow the comparisons instead of grouping them.

Start anywhere and walk: from a candidate, step to anybody they came out behind, and keep stepping. If the walk ever returns to a candidate it has already visited, the comparisons contain a ring and no order can hold them. If every walk runs out, they can be ordered.

That is a different operation from examining triples, it takes about as long as there are comparisons, and it is exact. It is also the operation this record performs on every outcome it counts, which is where the figures above come from.

The other correct check, which is easier to do by hand

Look for somebody who came out behind nobody.

If the comparisons can be ordered at all, such a candidate exists: whoever would be first. Take them out of the field along with all their comparisons, and look again. If at every stage somebody is undefeated, the order can be read off in the sequence they were removed. If at some stage nobody is, the remaining comparisons contain a ring.

Both checks find every ring of every length. Neither needs the record, and both are quicker than examining all twenty triples of a field of six.

The cost of the better check

Almost nothing, which is the part that makes the whole note practical.

Checking every triple of a field of six means looking at twenty triples, each needing three results read off. Walking the comparisons means following at most fifteen arrows once.

So the check that finds every ring is cheaper than the check that finds one kind, as well as being correct. There is no trade here to weigh: the triple check is more work and less reliable, and it survives because grouping by threes is the way people naturally think about contradiction.

What the triple check is still good for

Finding the commonest kind of contradiction fast.

On a fully compared field it is sufficient: with no gaps, a long ring always forces a short one. And on any field it finds the circular triples, which are the most numerous rings and the ones that cause the loudest arguments.

The record's point is narrower than "do not check triples". It is that a page of noughts in the third column does not mean the comparisons can be ordered, and that on the middle pages it fails to mean it about one time in eight.

The narrow claim, stated once

A column of noughts in the third number does not mean the comparisons can be ordered.

That is the entire claim of this note. It is not that triple checks are worthless, not that long rings are common, and not that anything should be done differently. It is that one specific inference — no circular triple, therefore an order exists — is invalid, and that the record can say exactly how often it fails on every page of a small table.