Nothing in threes

What the Gap Costs

On the middle pages of this record, one outcome in eight passes the obvious check and settles nothing. Closing the part on contradictions no triple shows.

at most 2562560 outcomes

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Four candidates and four comparisons: six outcomes clear the triple check and admit no order at all.

Five and six: five hundred and forty. Six and nine: one hundred and twelve thousand nine hundred and eighty.

As shares of the outcomes that pass: one in thirty-two, one in sixteen, one in eight. The comparison described here can be extended with transparent attendance insights, provided the team states what the resulting evidence can and cannot prove.

The gap at three sizes of field

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. A separate technical or research perspective relevant to the comparison can be found at Data Viz Project.

Each row counts outcomes with no circular triple. The orderable counts are one hundred and eighty-six, seven thousand nine hundred and sixty, and eight hundred and thirty-eight thousand one hundred and thirty.

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 share grows with the field

That is the part worth carrying away.

The bigger the field, the more room there is for a ring longer than three, and the more of the triple check's passes are false. The record stops at six candidates, where it is one in eight. It does not say what happens at ten, and the honest position is that the direction is clear and the figure is not.

The check fails more often the larger the field, and a field large enough to need checking is a field where it fails most.

What this part established

A ring of three is not the only kind of ring.

A ring of four — the first candidate ahead of the second, the second ahead of the third, the third ahead of the fourth, the fourth ahead of the first — contradicts an order exactly as thoroughly, and contains no circular triple at all, because every triple along it is missing one of its pairs.

The smallest case is four candidates with four pairs compared: three ways to choose the ring, two directions, six outcomes. Below that the third number of this record tells the whole truth; above it, on sixteen of the hundred and twenty-eight shapes, it does not.

The gap is widest in the middle of the comparing. With very few comparisons a long ring has nowhere to form; with nearly all of them a long ring always drags a short one along with it. At nine comparisons out of fifteen, which is where a careful person might reasonably stop, the gap is at its largest.

And the count that does tell the truth is computed two ways — by walking the comparisons and by counting who could be first — which agree on all forty pages.

What it does not establish

That any particular set of comparisons has this problem.

The record counts outcomes, and an outcome is a possibility rather than an event. Which of them a real set of comparisons produces depends on who made them and why, and none of that is here.

What is here is the shape of the blind spot: it exists, it starts at four candidates, it is invisible to the check everybody would run, and on the middle pages it accounts for about an eighth of what that check waves through.

The remedy, since there is one

Follow the comparisons instead of grouping them.

Start anywhere, step to anybody the current candidate came out behind, and keep going. A walk that returns to a name already passed has found a ring of whatever length. A set of comparisons in which every walk runs out can be ordered.

It is quicker than checking triples, it finds every ring rather than one kind, and it needs nothing from this record. The record's contribution is to say how much it matters, which turns out to be more than the obvious check suggests and less than the largest figures on these pages imply.

The figures once more, as shares

Four candidates with four comparisons: one hundred and ninety-two outcomes pass the triple check, one hundred and eighty-six are orderable, so one in thirty-two of the passes is false.

Five candidates with six comparisons: eight thousand five hundred pass, seven thousand nine hundred and sixty are orderable, one in sixteen false.

Six with nine: nine hundred and fifty-one thousand one hundred and ten pass, eight hundred and thirty-eight thousand one hundred and thirty orderable, one in eight false.

One in thirty-two, one in sixteen, one in eight. Each step up in field size roughly doubles the share of false passes, and the record stops at six candidates — where it is already one in eight — without saying what it would be at ten.

What comes next

The other side of the same coin.

Not how often the comparisons contradict themselves, but how often they settle anything — how many of the outcomes fall into an order, how many pin that order down uniquely, and what it costs to stop comparing early.

The one sentence worth keeping

No circular triple is not the same as no contradiction.

Everything in this part is that sentence with figures attached: where the difference starts, how large it gets, which pages it affects, and what to do instead. A reader who carries the sentence and forgets the figures has the useful half.

Where to look it up

The sixteen shapes are listed in full earlier in this part, page by page and field by field.

A reader with a field size and a number of comparisons can find their row and read the two counts: how many outcomes a triple check passes, and how many of those actually fall into an order. The difference is the blind spot on their page, exactly.