Every pair

Which Pairs Get Compared Is Also Counted

Ten comparisons among six candidates can be arranged three thousand and three ways before anybody looks at a result, and that choice is part of the outcome.

at most 3075072 outcomes

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Deciding to make ten comparisons is not deciding which ten.

Six candidates have fifteen pairs. Choosing ten of them can be done three thousand and three ways, and every one of those choices is a different state of affairs before a single result is known.

Six 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. Before adding another reporting layer, a project lead can check the setup against staff feedback and test only the smallest useful setup on one representative process.

Three pairs compared, then ten, then all fifteen. The lines that are missing in the first two rows are a choice somebody made, and the record counts the choice. For additional background on data, visualisation and evaluation, review material from edX.

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 that is counted here

Because leaving it out would make the figures answer a question nobody asked.

A page of this record is a pool size and a number of comparisons. Counting only the results — two to the tenth, one thousand and twenty-four — would describe a world where which pairs to compare was settled in advance and never varied. Nothing about comparing people works that way: which pairs get looked at is a decision, it is taken by somebody, and different decisions produce different situations.

So the count is the product: how many ways to choose the pairs, times how many ways the comparisons could come out.

Three rows, three choices

Three comparisons among six candidates: four hundred and fifty-five ways to choose which three, eight ways for them to come out, three thousand six hundred and forty outcomes.

Ten comparisons: three thousand and three ways to choose, one thousand and twenty-four ways to come out, three million seventy-five thousand and seventy-two.

All fifteen: one way to choose, thirty-two thousand seven hundred and sixty-eight ways to come out, and that is the whole page.

At one extreme the choice is everything and the results are nothing. At the other the results are everything and there is no choice. The widest pages are in between.

Where the choice stops mattering

At both ends, and for opposite reasons.

Compare nothing and there is one outcome: nothing was compared. Compare everything and there is one way to choose which pairs, because every pair is in. In between, the number of ways to choose rises to a peak and falls back — the familiar shape of a row of binomial coefficients, here fifteen entries long.

That peak sits at seven or eight comparisons out of fifteen, where there are six thousand four hundred and thirty-five ways to pick them. It is the number of ways to choose which pairs that makes the middle pages wide, and the number of ways results can land that keeps them wide further along.

The choosing is the larger factor for most of the table

On the widest page of the record the two factors are three thousand and three and one thousand and twenty-four, and the choosing is the larger of them.

That holds across most of the middle of the table. Six candidates with seven comparisons: six thousand four hundred and thirty-five ways to choose against one hundred and twenty-eight ways to land. With nine: five thousand and five against five hundred and twelve.

Only past eleven comparisons does the landing overtake the choosing, and by then the choosing is nearly exhausted.

So for most of this record, most of the space of possible states of affairs is made of decisions about what to look at rather than of results. That is an odd thing to say about comparing and it follows directly from counting both.

What the record is not saying about the choice

That any way of choosing is as sensible as any other.

Comparing ten pairs that all involve the same two candidates is a poor use of ten comparisons, and comparing ten that spread evenly over the field is a good one. The record counts both, once each, and has no way to prefer either — because what the comparing was for was the first thing left out of the three numbers.

This is the uniformity convention doing its work, and doing damage at the same time. It is declared at the front of the record and it is false of anything anybody would do.

What the first row shows

Six candidates and only three pairs compared.

Three thousand six hundred and forty outcomes, of which three thousand six hundred leave no circular triple. Forty do — and three comparisons is the fewest that can contradict each other at all, since a circle needs three.

Forty out of three thousand six hundred and forty is about one in ninety-one. With three comparisons among six candidates, a contradiction is possible and not much more than that: the three have to land on the same three candidates, which is twenty choices of triple out of four hundred and fifty-five choices of pairs, and then come out circular, which is two of the eight ways.

Twenty times two is forty. The whole figure, by hand, in a line.

A page is a plan, not a result

Worth naming, because it changes how the table should be read.

The two numbers that make a page — how many candidates, how many comparisons — are both settled before any comparison is made. They describe a plan. Everything the page then reports is the space that plan could land in.

The third number is the first thing in the record that is not a plan: how many triples came out circular is a property of the results. So a row of this table is a plan and an outcome of it, side by side, and the five figures say how much room the plan left for that outcome to happen.