An order

Comparing Everybody Settles It, Rarely

With every pair compared, an outcome that contradicts nothing ranks the field completely. There are n factorial such outcomes, against two to the number of pairs.

at most 32768 outcomes

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Compare every pair and the ambiguity disappears: an outcome with no ring ranks the field completely, with nothing left open.

There are n factorial such outcomes, and that is the good news and the bad news in one figure.

Every pair compared, four sizes of field

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

Four outcomes drawn. Rows differ in how many candidates. Which candidate is which is the drawing's own choice and carries no meaning. A team translating this comparison into a limited operational pilot can review the available time context before deciding which time and project records are genuinely necessary.

Three, four, five and six candidates with nothing left uncompared. The figures on the right are the outcomes that contain no circular triple, which on these pages are exactly the orders. For an independent perspective on comparison, evidence or decision design, consult DataCamp.

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 nothing is left open

Every pair has a result, so for any two candidates the comparisons say which is ahead.

A consistent set of such results is a total order by definition: every pair is decided, and no walk comes back. So on a fully compared page, falling into an order and settling the order are the same condition, and the record's two columns agree.

They agree nowhere else except on the pages where no circle is possible at all.

The figures

Three candidates: eight outcomes, six of them orders. Four: sixty-four, twenty-four. Five: one thousand and twenty-four, one hundred and twenty. Six: thirty-two thousand seven hundred and sixty-eight, seven hundred and twenty.

Three quarters, three eighths, about one in nine, about one in forty-five.

The collapse, stated once

The orders grow like a factorial. The outcomes grow like two raised to the pair count, and the pair count itself grows like the square of the field.

A factorial has no chance against that. Each candidate added multiplies the orders by the new field size and the outcomes by two to the power of the new comparisons, and the second multiplier overtakes the first at four candidates and never looks back.

Exhaustive comparing removes every ambiguity and leaves the clean outcome as one possibility in forty-five. Thoroughness buys decisiveness and pays for it in how much could go wrong.

What this does not mean

That a thorough comparison will probably contradict itself.

The figure counts possible outcomes under the convention that each is as reachable as any other, which describes nothing anybody does. A comparer with a consistent view of a field produces consistent results far more often than the arithmetic alone would give.

What the figure does is price the claim. Somebody reporting a clean ranking of six candidates from fifteen comparisons is reporting an outcome that is one in forty-five by arithmetic alone. Whatever produced it was doing something other than deciding each comparison on its own.

That is not a doubt about the ranking. It is a statement about how much work the consistency is doing, and it is available to anybody who can multiply.

The same statement in reverse

Read the other way, the collapse says something about agreement rather than about contradiction.

A clean ranking of six candidates from fifteen comparisons is one outcome in forty-five. If the comparisons were independent — each decided without reference to the others — that is what it would be worth.

They are not independent. A comparer carries a view of the field from one comparison to the next, and that is precisely why clean rankings are common in practice and rare in this arithmetic.

So the figure measures how much structure the comparing imposed. A clean ranking is evidence that the comparisons were not independent, which is usually the intended state of affairs and occasionally the problem.

The four rows against the rest of the record

These four pages are where the record's two order columns coincide and its third number tells the whole truth.

Everywhere else — thirty-six of the forty pages — they come apart. On the middle pages the gap is at its widest: outcomes with no circular triple that cannot be ordered, and outcomes that can be ordered without the order being determined.

So the fully compared page is the easy case and the unrepresentative one. It is also the only case most people picture when they think about comparing a field, which is why it is worth drawing four times and saying plainly that the rest of the record does not look like it.

Where the two conditions part company

Everywhere except the pages drawn here and the pages where no ring can form.

Twenty-four of this record's forty pages have falling-into-an-order and having-no-circular-triple as the same figure. On the other sixteen they differ, and on one page they differ by one hundred and twelve thousand nine hundred and eighty outcomes.

A reader who meets the fully compared page first — which is the page everybody pictures — will carry away the impression that the two are the same thing. They are the same thing exactly here, and the rest of the record is the correction.

Why the collapse is worth a page of its own

Because it is the one page in the record where a shortcut is safe, and shortcuts travel.

With every pair compared, checking the triples is a complete check: no ring of four or more can exist, since every pair that such a ring would need to leave out has in fact been compared. So a comparer who has looked at everything may check the triples alone and be finished.

On a small field the saving is real: four candidates have four triples, and checking four things is quicker than tracing arrows. It is also exactly the habit that goes wrong one comparison earlier. At fourteen of fifteen pairs the triples are no longer sufficient, and nothing about the work feels different.

The figure to remember

One in forty-five.

That is the share of outcomes that rank six candidates when every pair has been compared, under the record's convention. It is the cleanest single statement of how much agreement a clean ranking represents, and it takes a factorial and a power of two to get.