Going round

Three Comparisons Are the Fewest That Disagree

With two comparisons no contradiction is possible, whatever the pool. With three it is, and the record counts fourteen pages where the question cannot arise.

at most 3640 outcomes

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One comparison cannot contradict anything. Two cannot either.

Three can, and that is the floor.

Two comparisons, then three

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. Every row is the same number of candidates. Which candidate is which is the drawing's own choice and carries no meaning.

Six candidates in every row. With one or two pairs compared every outcome is free of circles; with three, forty of the three thousand six hundred and forty are not.

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. For repeated pairwise evaluations, project hours and decision evidence can add project and time context without turning one activity signal into a verdict about quality.

Why two is safe

A contradiction among comparisons means a ring, and the shortest ring needs three links. Readers examining the figures against practical research methods can also see SciPy.

Two comparisons can share a candidate or not. If they do not, they concern four different people and say nothing about each other. If they do, they say that one candidate is ahead of a second and the second ahead of a third — which is a statement nobody could contradict without a third comparison.

So every page of this record where fewer than three pairs were compared has exactly one column: no circular triples, every outcome.

Fourteen pages of the forty

The pages with nought, one or two comparisons, for every pool from two to six candidates.

Six candidates with no comparisons: one outcome, and nothing is known. With one: thirty outcomes, since there are fifteen pairs to pick and two ways for the comparison to land. With two: four hundred and twenty.

On all three of those pages the share of outcomes with no circular triple is one, and what that means here is worth saying out loud: contradiction is impossible rather than inevitable. Elsewhere a share of one carries the opposite sense, and only the shape beside it says which.

What the third comparison opens

Three comparisons among six candidates can be chosen four hundred and fifty-five ways and can come out eight ways each, for three thousand six hundred and forty outcomes.

Forty of those contain a circle. For that to happen the three comparisons must land on the same three candidates — twenty of the four hundred and fifty-five choices do — and then come out circular, which is two of the eight ways.

Twenty times two is forty. The entire figure derived in a line, and it is the first non-trivial count in the record that a reader can produce without any table at all.

About one in ninety-one. Three comparisons among six candidates can contradict each other, and the arithmetic says how rarely, exactly.

What the floor is really a statement about

Shape, not luck.

Three comparisons can contradict each other only when all three fall on the same triple. That is a condition on which pairs were compared, and it can be checked before any result is known: a comparer who has compared the first candidate with the second, the first with the third, and the first with the fourth cannot possibly contradict themselves, whatever the three results are.

So the floor divides into two parts. Fewer than three comparisons: contradiction impossible for arithmetic reasons. Three or more: contradiction possible only if the comparisons form a closed shape, which is a property of the plan rather than of the answers.

That second half carries further than the record's table does, and it is the useful form of the result.

The floor does not move with the pool

Three comparisons are the fewest that can disagree whether the pool is three candidates or six.

What changes is how likely the three are to land on one triple. In a pool of three there is only one triple and three pairs, so three comparisons must be it: two of the eight outcomes are circular, which is one in four. In a pool of six the same three comparisons have four hundred and fifty-five places to fall and only twenty of them are triples.

One in four against one in ninety-one, for the same number of comparisons. The pool size does not change the floor; it changes how much room there is above it.

Where it stops being so easy to count

At four comparisons.

Six candidates with four pairs compared: twenty-one thousand eight hundred and forty outcomes, of which nine hundred and sixty carry a circle. That is about one in twenty-three, against one in ninety-one with three comparisons — a fourfold jump from one extra comparison.

At five pairs compared the page runs eighty-five thousand seven hundred and sixteen outcomes with no circle, ten thousand two hundred with one, and one hundred and eighty with two. Two circles at once has become possible, which it was not a moment before, and it takes six comparisons before two circles can be drawn without sharing a candidate.

Past that the counts stop being derivable in a line and start coming from enumeration. What stays derivable is the floor: below three comparisons a contradiction cannot occur, and at three it occurs in twenty times two of the ways available.

The floor in one sentence

A contradiction needs a closed shape, and the smallest closed shape among pairs has three sides.

Everything in this part follows from that. Below three comparisons no closed shape exists. At three, one does if all three land on the same triple. Above three, closed shapes multiply faster than comparisons do, because every new comparison can close several at once.