The Record Cannot Tell Anybody What to Compare

This record says that four comparisons cannot rank six candidates, that five can, and how many of the possible outcomes at each amount of comparing do what.

It does not say which pairs to compare, and this note is about why not.

Three plans with the same number of comparisons

• 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 review the present feature set and test only the smallest useful setup on one representative process.

Three shapes of one page. Seven comparisons among six candidates, in outcomes with none, one and two circular triples. Every plan to make seven comparisons lands somewhere on this page and nothing decides where. For additional background on data, visualisation and evaluation, review material from Wolfram MathWorld.

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.

What a method would need

The results of the comparisons already made.

Any sensible way of choosing the next pair uses what is known so far: compare the two that seem closest, or the two whose relative position is still open, or the candidate nobody has looked at yet against somebody in the middle. All of those read the results.

The record has no results in it. It has counts of outcomes, which is the whole space of what the results could be, weighted equally. A method that adapts to the results cannot be described in that language at all.

What a method would also need

What the comparing is for.

Ranking the whole field and finding the best candidate are different jobs with different cheapest methods, and the second is much cheaper. Neither job is in these three numbers, because the purpose of the exercise was the first thing dropped to make the counting possible.

A record that counted what the comparing was for would not be a record of arithmetic. It would be a record of somebody's judgement about a situation, which is the kind of page this one was written against.

What is available instead

Three structural facts, and they need no method.

A field of n candidates cannot be ranked by fewer than n − 1 comparisons, because each comparison connects at most one new candidate.

A contradiction needs at least three comparisons, and a contradiction invisible to a triple check needs at least four.

And when nothing is left uncompared, the arrangements that rank the field number n factorial, out of two raised to the pair count.

All three hold whatever method anybody uses, because they are conditions on what the comparisons can do rather than on how they were chosen.

The three rows

Seven comparisons among six candidates: eight hundred and twenty-three thousand six hundred and eighty outcomes.

Five hundred and forty-five thousand six hundred and forty have no circular triple. Two hundred and forty thousand one hundred and twenty have exactly one. Thirty-seven thousand and eighty have two.

Two plans to make seven comparisons, carried out by two people on the same field, land somewhere in that page. The record knows the sizes of the regions. It does not know, and cannot know, where either plan lands.

What it would take to say more

A fourth number, and the record would stop being checkable.

To price a plan rather than a quantity, the record would have to know which pairs were compared — not how many, but which. That is not one number; for a field of six it is one of thirty-two thousand seven hundred and sixty-eight possible sets of pairs, and the table would have a row for each.

A table of that size cannot be read, cannot be checked by hand, and cannot be summarised without throwing away exactly the information that made it larger. So the record stops at three numbers, and the fourth is named as missing rather than approximated.

The honest summary

What the figures here price is a decision about quantity: how many candidates, how many comparisons. Those two fix the size of every region exactly, and are commonly settled on grounds that never touch the counting at all.

What they cannot price is a decision about which comparisons, which is where most of the skill in comparing anything actually sits.

A record that pretended otherwise would be more useful and would be making it up.

What a reader can still do with it

Price the quantity decision, which is the one usually taken first.

How many candidates to compare, and how many comparisons to make. Those two numbers fix every figure in this record exactly, and they are normally settled for reasons — time, availability, custom — that have nothing to do with their consequences.

The record makes the consequences visible: the size of the space, the share that contradicts itself, the share that would settle the order, and the size of the blind spot in the obvious check. Four figures for two decisions, available before anybody is compared with anybody.

What a reader should not infer from the three rows

That the first of them is the plan to aim for.

Five hundred and forty-five thousand six hundred and forty outcomes with no circular triple is the largest of the three figures, and it is tempting to read the row as the good case and the others as failures. Nothing supports that reading. A plan does not choose a row; it produces an outcome, and the outcome falls where the comparisons fall.

Worse, the largest row is not even the one a comparer would want. Of those five hundred and forty-five thousand six hundred and forty, forty-eight thousand nine hundred and sixty hold a ring that no triple reveals — the single most dangerous region on the page, and the one that looks cleanest from inside.

Said plainly

This record prices how much, not which.

Every figure here follows from a field size and a number of comparisons. Nothing here follows from the choice of pairs, and the choice of pairs is where the work is.

Near this one

• What the Trade Comes To
  Fewer comparisons, fewer contradictions, less settled and a wider blind spot. Closing the part on comparing fe
• The Trade Nobody Prices Before Taking It
  Comparing fewer pairs makes contradiction rarer and leaves more undecided. Both halves have exact figures, and
• Below a Floor, Nothing Can Be Ranked
  Ranking six candidates takes five comparisons at the least, and the reason is not contradiction. It counts how
• Which Pairs Get Compared Beats How Many
  Five comparisons among six candidates rank the field in seven hundred and twenty of ninety-six thousand ways.