Limits

There Is No Comparer in This Record

Every figure counts outcomes that could exist. None of them describes anybody producing one, and the gap between those two things is where the record is weakest.

at most 2562560 outcomes

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The record counts outcomes. Nobody in it produces one.

That reads as a quibble and it is the widest gap between these figures and any use somebody might put them to, which is why it has a page.

Three shapes, no comparer in any of them

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. For repeated pairwise evaluations, workforce analytics for distributed teams can add project and time context without turning one activity signal into a verdict about quality.

The counts describe a space of possible outcomes. Nothing in them describes anybody moving through it. Readers examining the figures against practical research methods can also see Observable.

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.

How comparing actually proceeds

Somebody has a field and a reason to sort it.

The pairs they look at are the ones in front of them, or the ones that seem closest, or the ones they know something about. Each result is seen before the next pair is chosen. The work stops when the answer looks clear or the time runs out, which is nearly always before every pair has been examined.

What comes out is one of the outcomes counted here, arrived at along a route with nothing random in it.

Why that unsettles every share

A share reports what fraction of a page an outcome occupies. Nobody is drawing from the page.

Take six candidates and nine comparisons: nine hundred and seventy-one thousand four hundred of the two million five hundred and sixty-two thousand five hundred and sixty outcomes hold exactly one circular triple, which is roughly three in eight.

Whether any given person produces such a result three times in eight depends wholly on how they go about it. Somebody with a firm view of the field will produce far fewer circles. Somebody weighing several things that do not align will produce more.

The fraction measures the room. Which corner of it a particular exercise ends in is a fact about whoever ran the exercise, and nobody here has met them.

Which way the error leans

Unusually for this record, that can be said.

Earlier results travel with the comparer into the later comparisons. A candidate who has twice come out ahead is met the third time by somebody who half expects it, and whether the expectation is well founded or not, its effect runs the same way: toward the uncontradicted corner of the page.

So the shares printed in this part are almost certainly higher than any real rate of contradiction. How much higher is unknown and varies with whoever is doing the comparing — two unknowns too many for a rate, and not enough to put the direction in doubt.

Saying exactly what is absent

Comparing done one pair at a time is a sequence of decisions: which pair to take up, and how it came out.

Every outcome counted here sits at the end of at least one such sequence, and most sit at the end of very many. How many is not the same from one outcome to the next.

That inequality is the entire distance between this record and anything anybody does. Destinations are counted here and weighted alike; a person walks routes, and the destinations with more routes into them arrive more often.

Counting routes is a different problem whose answer changes with every description of how a person compares, which is why no attempt is made. There is not one route count; there is one for each model of a comparer, and every model is invented.

Why a page rather than a line in the conventions

Because these figures read as though somebody were in them.

A share of one in eight sounds like a report of what happens, and the sentence that most naturally carries it has already taken the step before the writer notices. A convention printed at the front of a record does not stop that sentence being written. A page can, by naming the step.

So this one exists to be set against the record's own figures — the page to point at when one of them shows up in a claim about what will happen. No figure can do that job, because what has gone wrong is not arithmetic.

What remains with nobody in the picture

Four results, and they are worth more for being few.

A field of n candidates needs n − 1 comparisons before any outcome can rank it, so nobody ranks six candidates on four comparisons. A ring takes three comparisons at the least, and a ring that hides from a triple check takes four. Compare every pair and the arrangements that rank the field number exactly n factorial.

None of those turns on how anybody compares. They describe the space, and they are what should leave this page with a reader.

The one thing a comparer can take from the space

A bound on what any arrangement of comparisons could have achieved.

A reader who knows only how many candidates there were and how many pairs were looked at knows the floor, knows whether a ring was even possible, and knows what fraction of the space would have ranked the field. None of that describes their own case. All of it rules out claims about their own case.

Five comparisons on six candidates cannot have ranked them if one candidate was never compared with anybody. Two comparisons cannot have contradicted themselves. Those are statements about the space that apply to every point in it, which makes them the only kind the record can lend anybody.

The shortest honest summary

The figures describe a space. A comparer occupies a point in it.

Everything the record says concerns the first, everything a reader wants concerns the second, and the bridge between them is a convention the record states and does not believe.

Better counting would not close that gap. It is the shape of what counting does, and saying so is what keeps the counts usable.