Going round

When Three Comparisons Close a Circle

A ahead of B, B ahead of C, C ahead of A. Nothing is wrong with any one comparison, and together they cannot be put in order at all.

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Three candidates, all three pairs compared.

The first came out ahead of the second. The second came out ahead of the third. The third came out ahead of the first.

Every comparison was made carefully. No two of them disagree. And there is no way to put the three candidates in order.

One circle, and one field without one

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. Teams comparing routes, outcomes and corrections can review this option in context, then check whether the resulting evidence improves the decision in practice.

Three candidates with all three pairs compared, twice: once with the results closing into a circle, once without. The third row is a larger field with exactly one circle in it. A separate technical or research perspective relevant to the comparison can be found at Tableau.

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.

What a circular triple is

Three candidates whose three comparisons close into a ring.

That is the whole definition. It needs all three pairs to have been compared, since a ring with a gap in it is not a ring, and it needs the directions to go round rather than lead somewhere.

Of the eight ways three comparisons can come out, six lead somewhere and two go round. Six orderings, two circles.

Nothing is wrong with any part of it

This is the part that is hard to hold on to.

Take the three comparisons one at a time and each is a perfectly ordinary result. Take any two and they agree: the first says the first candidate is ahead of the second, the second says the second is ahead of the third, and from those two a reader would predict the third comparison. The third comparison then comes out the other way.

So a circular triple is not an error. It is a thing that comparisons do, and it does not announce itself in any single comparison or in any pair of them.

A contradiction among three comparisons is invisible in any two of them. There is nothing to notice until all three are on the table at once.

How the record counts it

Each shape here carries a third number: how many triples came out circular.

Three candidates have one triple, so that number is nought or one. Four candidates have four triples, five have ten, six have twenty — the number of triples is C(n, 3) and it grows like the cube of the pool.

So the question is never whether a circle is possible but how many of the available triples closed. Six candidates fully compared have twenty triples, and the record's largest shape there has eight of them circular at once.

The three rows

Three candidates with all three pairs compared, no circle: six outcomes of the eight.

The same page with one circle: two of the eight. Those two are the clockwise ring and the anticlockwise one.

Five candidates with five of their ten pairs compared, one circle: one thousand five hundred and sixty outcomes of eight thousand and sixty-four, which is about one in five.

Why the figure is worth having

Because the usual way of thinking about this treats a circle as a sign that something went wrong.

It can be. It can also be exactly what an honest set of comparisons produces when the candidates differ in several ways that do not line up — one is stronger on this, another on that, and the third beats the first on something neither of the others weighs.

The record takes no view on which it was, because it does not know what the comparing was about. What it can say is how many of the possible outcomes contain a circle, exactly, for any pool from two to six candidates and any amount of comparing.

Why three and not two

A pair of comparisons can disagree in ordinary language and cannot disagree here.

Two comparisons either concern four different candidates, in which case they say nothing about each other, or they share one, in which case they place three candidates in a line. Neither can be contradicted without a third comparison, because there is nothing for a third opinion to be at odds with.

So two is the number of comparisons that can be surprising and three is the number that can be impossible. The distinction is exact and it is the floor of this whole part of the record.

What a circle is not

It is not a tie, and it is not a near-miss.

A tie would be a comparison with no result, which this record does not have: a comparison here came out one way or the other. A near-miss would be a comparison that was close, which the record cannot see either.

A circular triple is three decisive results that cannot all be true of any order. Nothing about it is soft, and that is why it can be counted exactly rather than estimated.

Reading the drawing

A line with a head is a comparison; the head points at whoever came out behind.

A circular triple shows as three lines forming a closed ring with all three heads going the same way round. It is the one figure in these drawings that can be spotted without reading any number, and on a crowded row it is worth looking for before reading anything else.