An order

What the Comparing Was For

Three columns, each a subset of the one before: no circular triple, no ring at all, and one order only. Closing the part on when the results settle something.

2562560 outcomes

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Six candidates, nine comparisons. Three figures about the same page, each a subset of the one before.

Nine hundred and fifty-one thousand one hundred and ten outcomes contain no circular triple. Eight hundred and thirty-eight thousand one hundred and thirty contain no ring of any length. One hundred and fifty-one thousand two hundred rank the field completely. A transparent review may include workforce time tracking for experiments when its purpose, access rules and limits are documented before any records are collected.

The three columns, on one page

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. A broader reference for interpreting structured comparisons is available from Codecademy.

Six candidates with nine pairs compared, in three of the page's five shapes. The page allows two million five hundred and sixty-two thousand five hundred and sixty outcomes; eight hundred and thirty-eight thousand one hundred and thirty fall into an order and one hundred and fifty-one thousand two hundred settle it.

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.

About a third, about a third, and one in seventeen

As shares of the page: thirty-seven per cent, thirty-three per cent, six per cent.

The first two are close enough to be mistaken for each other, and the gap between them is one hundred and twelve thousand nine hundred and eighty outcomes that pass the obvious check and settle nothing.

The third is a long way below both, and it is the one that corresponds to what the comparing was for.

What this part established

A set of comparisons falls into an order when no walk through them returns to its starting point. That is a stronger condition than having no circular triple and it is the one that matters.

Whenever an order exists, somebody came out behind nobody. Peeling that candidate off and repeating both proves it and finds the order; if the peel jams, the comparisons contain a ring.

The count of orderable outcomes is computed twice by routes with no step in common — walking the comparisons, and counting who could go first — and the two agree on all forty pages.

Falling into an order is not the same as settling it. The outcomes that settle the order number n factorial times a binomial coefficient: the order itself, plus a free choice over which of the remaining pairs were also compared. That column is the one symmetric column in the record, and it is symmetric for a reason rather than by accident.

Ranking a field of six takes at least five comparisons, and exactly seven hundred and twenty of the ninety-six thousand ways to make five comparisons do it — one for each order, which is about one outcome in a hundred and thirty-three.

And with every pair compared, falling into an order and settling it coincide, at n factorial outcomes out of two to the number of pairs: three quarters for three candidates, about one in forty-five for six.

The one claim worth carrying

Not contradicting yourself is easy; settling anything is not.

The easy half comes mostly from having asked too little to disagree with itself. The hard half needs the whole field covered, and covering a field is what comparisons are spent on.

A report that the comparisons agreed is therefore close to no information. A report that they produced an order is a much stronger claim, and the difference between the two is not visible from the inside of either.

The three columns on one more page

Six candidates with twelve pairs compared, for contrast with the nine-comparison page above.

One hundred and eighty-four thousand three hundred and fifty outcomes with no circular triple. One hundred and seventy-eight thousand two hundred and thirty with no ring at all. Eighty-six thousand four hundred that rank the field.

As shares of the page: about ten per cent, about nine and a half, about four and a half.

The first two have closed up — the blind spot that was one in eight at nine comparisons is one in thirty here — and the third has roughly halved. More comparing buys a more reliable check and fewer chances to rank the field, which is the trade the next part takes apart.

What comes next

The other half of the trade.

Comparing fewer pairs makes contradiction rarer and leaves more undecided, and both halves of that have exact figures for every page of this record. The next part prices the decision to stop early.

The figure this part is built on

n factorial, and how small it is.

Six candidates have seven hundred and twenty orders. That is the number of outcomes that rank the field when every pair is compared, the number when the fewest possible pairs are compared, and the multiplier in the formula for every amount of comparing in between.

Against a page of three million it is nothing, and against a page of ninety-six thousand it is nearly nothing. The whole of this part is the consequences of a factorial being small compared with the spaces it has to be found in.

The three columns at the other end of the record

Three candidates, all three pairs compared: eight outcomes.

Six of them have no circular triple, the same six fall into an order, and the same six rank the field. Three columns, one figure, and three factorial is six.

The remaining two outcomes are the two circular triples a field of three can produce — one going one way round, one the other. So the smallest fully compared field divides into six orders and two rings, with nothing in between and no blind spot to speak of. Every complication in this part of the record appears only once there are four candidates to put it in.

What is still missing after this part

How much a partial result is worth.

The record counts outcomes that rank the field and outcomes that do not, and says nothing about the enormous middle where some candidates are placed and others are not. That middle is where most real comparing ends up, and measuring it would take a count this record does not carry.