An order

Pinning the Order Down

An outcome can be free of contradiction and still leave the field unranked. The count of outcomes that settle it completely has a formula, and it is small.

at most 3075072 outcomes

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Four candidates, three comparisons. One hundred and fifty-two of the one hundred and sixty outcomes contain no ring at all.

Twenty-four of them say what the order is.

Three pages, and how many outcomes settle the order

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.

Each row counts outcomes with no circular triple. The counts that settle the order completely are twenty-four, two thousand four hundred and one hundred and eighty-one thousand four hundred and forty. Managers testing competing processes can examine privacy-aware team monitoring against the same baseline, correction case and reporting window.

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 additional background on data, visualisation and evaluation, review material from Kaggle.

The difference

An outcome falls into an order when at least one list satisfies the comparisons. It settles the order when exactly one does.

Three comparisons among four candidates can be consistent in many ways that leave the field open. Compare the first with the second, the first with the third and the first with the fourth, and the first candidate's position is known while the other three are in no order at all. Six lists satisfy that outcome.

To settle the order, the comparisons have to form a chain running through every candidate.

The formula

Choose the order: n factorial ways.

Any outcome that settles that order must contain the comparison between each pair of neighbours in it — n − 1 comparisons, all forced. The remaining comparisons are free: each of the other pairs may or may not have been compared, and if it was, its result is forced by the order.

So the number of outcomes with exactly r comparisons that settle a given order is the number of ways to choose r − (n − 1) of the remaining C(n, 2) − (n − 1) pairs. Multiplied by n factorial for the choice of order.

Four candidates with three comparisons: twenty-four times one, which is twenty-four. With four comparisons: twenty-four times three, seventy-two. With all six: twenty-four times one again, twenty-four.

Checked at the end where it is easy

With every pair compared, an outcome settles the order exactly when it falls into an order, since nothing is left undecided.

The formula gives n factorial times one, and the record's orderable count for a fully compared page is n factorial. They agree for every pool from two to six: two, six, twenty-four, one hundred and twenty, seven hundred and twenty.

A formula that reproduces the table at the end where the table is easy is doing its job. One that disagreed there would be wrong everywhere else as well.

The three rows

Four candidates with three comparisons: one hundred and sixty outcomes, one hundred and fifty-two orderable, twenty-four that settle the order. Fifteen per cent of the outcomes, and about one in six of the orderable ones.

Five with seven: fifteen thousand three hundred and sixty outcomes, six thousand five hundred and forty orderable, two thousand four hundred that settle it.

Six with ten: three million seventy-five thousand and seventy-two outcomes, six hundred and ninety-one thousand and twenty orderable, one hundred and eighty-one thousand four hundred and forty that settle it.

About one orderable outcome in six, then one in three, then one in four. The proportion is not monotone, because the two quantities behind it peak at different amounts of comparing.

The forced comparisons and the free ones

The formula divides every outcome that settles an order into two parts, and the division is worth seeing directly.

Six candidates in a settled order have five neighbouring pairs: first with second, second with third, and so on. Every one of those five must have been compared, because if any neighbouring pair were left out, the two candidates either side of the gap could swap and the comparisons would not notice.

The other ten pairs are free. Each may have been compared or not, and if it was, its result is forced by the order — a candidate earlier in the list came out ahead.

So an outcome that settles an order is: the order, plus a choice of which of the ten free pairs were also looked at. Seven hundred and twenty times a binomial coefficient over ten.

Where it peaks

The count of order-settling outcomes rises and falls with the amount of comparing, like everything else here.

For six candidates it runs seven hundred and twenty with five comparisons, seven thousand two hundred with six, thirty-two thousand four hundred with seven, eighty-six thousand four hundred with eight, one hundred and fifty-one thousand two hundred with nine, one hundred and eighty-one thousand four hundred and forty with ten, and then back down by the same figures in reverse.

That symmetry is not a coincidence: the formula is n factorial times a binomial coefficient, and binomial coefficients read the same forwards and backwards. It is the one symmetric column in this record, and it is symmetric for a reason that can be stated in a sentence.

The smallest and largest of the column

Seven hundred and twenty at both ends, for a field of six — five comparisons and fifteen comparisons give the same count.

At five, the comparisons must be exactly the chain and there is nothing else to choose. At fifteen, every pair is compared and again there is nothing to choose: the order determines every result.

Between them the free pairs make the difference, peaking at ten comparisons with one hundred and eighty-one thousand four hundred and forty. The two ends are tight for opposite reasons, which is the same shape the page sizes have and arrived at independently.