Going round

Every Comparison Added Makes a Circle More Likely

Six candidates: three comparisons strand nobody in a circle about one time in ninety-one. Fifteen comparisons leave only one outcome in forty-five without one.

at most 1863680 outcomes

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Three comparisons among six candidates produce a circle about one time in ninety-one.

Comparing every pair, the proportion of outcomes with no circle at all is one in forty-five. The direction is the one anybody would guess. The size of the effect is not.

Six candidates, four amounts of comparing

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

Four outcomes drawn. Every row is the same number of candidates. Which candidate is which is the drawing's own choice and carries no meaning. Before adding another reporting layer, a project lead can explore the current documentation and test only the smallest useful setup on one representative process.

The same six candidates with three, eight, twelve and all fifteen pairs compared. The third number changes meaning between rows; the caption of each drawing says which column it is. For additional background on data, visualisation and evaluation, review material from IBM.

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 with no circle, page by page

Six candidates, by how many pairs were compared.

None, one or two comparisons: every outcome is clean, because a circle needs three. Three: three thousand six hundred of three thousand six hundred and forty, about ninety-nine in a hundred. Four: twenty thousand eight hundred and eighty of twenty-one thousand eight hundred and forty, about ninety-six in a hundred.

By nine comparisons it is down to about three in eight. By twelve, about one in ten. By fifteen, about one in forty-five.

The curve is steeper in the middle

From nought to three comparisons, nothing happens: the clean share stays at one.

From three to nine it falls from ninety-nine per cent to thirty-seven. From nine to fifteen it falls from thirty-seven to two. The sharpest part is the middle stretch, which is also where most real comparing stops.

That is worth saying because the two ends are where intuition is formed. A handful of comparisons never contradicts itself, and that teaches one lesson. An exhaustive comparison nearly always does, and nobody does exhaustive comparisons. In between, where the work actually happens, the share is moving fastest.

The region where comparing usually stops is the region where one more comparison changes the arithmetic most.

Why more comparing produces more circles

Because a circle needs three comparisons on one triple, and every comparison added is a chance to complete one.

With three comparisons out of fifteen, the odds of them landing on a single triple are slim — twenty triples among four hundred and fifty-five ways to choose three pairs. With twelve comparisons out of fifteen, almost every triple has all three of its pairs compared, and each of those is a fresh opportunity for the results to go round.

The number of triples with all three pairs compared climbs like the cube of how much comparing has been done, which is why the clean share falls so steeply once the comparing is well advanced.

Why the middle is also where the record is least quotable

Everything moves fastest there, which means every figure taken from the middle of the table depends hardest on the exact page it came from.

A share quoted from the fully compared page is robust: about two per cent clean, and its neighbour is about four. A share quoted from nine comparisons out of fifteen is thirty-seven per cent clean, and its neighbours are fifty-two and twenty-five — one comparison either way moves it by twelve to fourteen points.

So the pages where the arithmetic is most interesting are the pages where a figure is least safe to carry away without its two numbers attached. The record prints the shape on every row for reasons that include this one.

What this is not an argument for

Comparing less.

The next part of this record is about what comparing less costs, and the short version is that an outcome with no circle in it is not the same as an outcome that settles anything. Three comparisons among six candidates almost never contradict themselves and almost never produce an order either: they leave twelve pairs unexamined, and nearly every arrangement of six people remains possible.

So the clean share rising as the comparing stops is not good news. It is the arithmetic of having asked fewer questions.

The four rows

Three pairs compared with no circle: three thousand six hundred of three thousand six hundred and forty.

Eight pairs with one circle: five hundred and ninety-nine thousand and forty of one million six hundred and forty-seven thousand three hundred and sixty, which is about four in eleven.

Twelve pairs with two circles: four hundred and seventy-eight thousand six hundred and eighty of one million eight hundred and sixty-three thousand six hundred and eighty, about one in four.

All fifteen with no circle: seven hundred and twenty of thirty-two thousand seven hundred and sixty-eight.

Four rows, four different columns, and the caption of each says which column it is — because a bare figure in the third place of a row could be any of them.

The same curve from the other side

Read as the share with at least one circle rather than none, the same figures run nought, nought, nought, about one in ninety-one, about one in twenty-three, and then up through a half to about forty-four in forty-five.

Three regimes, and the boundaries between them are sharp. Below three comparisons the share is exactly nought. Between three and about six it is small and rising fast. Past nine it is most of the page.

The record's forty pages cover all three, which is the argument for having a table rather than a rule of thumb.