Every pair

Every Pair Has Three States, Not Two

Compared with one result, compared with the other, or not compared at all. The third state is the one that gets dropped, and dropping it changes every figure here.

at most 240 outcomes

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A pair of candidates can be in one of three states.

Somebody compared them and the first came out ahead. Somebody compared them and the second came out ahead. Or nobody compared them.

The same four candidates, three 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

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. To test the same alternatives in live work, reviewers can check which options fit the decision and agree the decision, access rules and review date in advance.

Four candidates throughout. Three pairs compared, then four, then all six. The lines that are absent are as much a part of each row as the lines that are present. A broader reference for interpreting structured comparisons is available from UserTesting.

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.

Why the third state is not an absence of data

It is a state, and it is usually the commonest one.

Six candidates make fifteen pairs. Comparing all fifteen is work that nobody does without a reason, and most real comparing stops long before it. The pairs that were never put side by side are not a gap in the record; they are what the record is largely made of.

So this site counts them. A page here is a pool size and a number of comparisons actually made, and the pages where that number is small are not thin pages — they are the ordinary case, and they have their own arithmetic.

What it does to the count

Two states per pair would give two to the power of fifteen for six candidates: thirty-two thousand seven hundred and sixty-eight.

Three states give three to the power of fifteen: fourteen million three hundred and forty-eight thousand nine hundred and seven.

A factor of four hundred and thirty-seven, from admitting that a pair might simply not have been looked at.

The space of what could have happened is four hundred times larger than the space of what could have happened if everybody had been compared with everybody.

How the record splits it up

Not into one enormous total. Into pages.

A page fixes two things: how many candidates, and how many of the pairs were compared. Four candidates with three comparisons made is one page; four candidates with four comparisons is another. Each page has its own count of outcomes, and those counts add to the three-to-the-power total.

That split is what makes the figures readable. Fourteen million is a number nobody can do anything with. One hundred and sixty — the outcomes available when three of four candidates' six pairs are compared — is a number somebody can check.

Where the outcomes figure comes from

Two choices, multiplied.

First, which pairs get compared: choosing three of the six available pairs can be done twenty ways. Then, how each comparison came out: two ways each, so eight.

Twenty times eight is one hundred and sixty, which is the figure on the first row of the drawing.

The same arithmetic on the second row: fifteen ways to choose four of the six pairs, sixteen ways for four comparisons to come out, two hundred and forty. On the third row every pair is compared, so there is one way to choose them and sixty-four ways for six comparisons to land.

The three rows, read across

Four candidates in every row, and exactly one circular triple in every row.

With three comparisons: eight of the one hundred and sixty outcomes. With four: forty-eight of two hundred and forty. With all six: sixteen of sixty-four.

One in twenty, one in five, one in four. More comparing produces more contradiction, and it does so steeply — which is not an argument for comparing less, since comparing less also produces less of everything else. That trade has a part of this record to itself.

Where the three states come apart

The three states are not equally informative, and the record keeps them apart for that reason.

A pair compared one way and a pair compared the other way are mirror images: swap the two candidates and one becomes the other. Every count here that involves them is symmetric in that sense, which is why the outcome figure has a clean power of two in it.

A pair not compared is nothing like either. It cannot be swapped into anything, it does not contribute to any circular triple, and it is the thing that lets a long ring hide from a triple check. The third state is not the average of the other two; it is a different kind of fact.

What a page looks like at both ends

Six candidates with no comparisons made: one outcome.

Six candidates with all fifteen compared: thirty-two thousand seven hundred and sixty-eight outcomes, and only one way to choose which pairs, since every pair is in.

Both ends are narrow and the middle is wide, for the same reason a row of binomial coefficients is: the number of ways to choose which pairs runs out at both extremes. The widest page of all is ten comparisons out of fifteen, with over three million outcomes, and that page describes comparing two pairs in every three.

The convention being used

Every outcome is counted once, and no outcome is treated as more reachable than another.

That is false of anything anybody has ever done. Real comparing picks pairs for reasons, in an order that is never random, and stops when somebody decides it has gone on long enough. The arrangements this record counts include every outcome that could occur and weights them all the same.

The convention is declared here and on every page that leans on it. What it costs and what survives its rejection are the subject of the last part of this record.