Limits

Three Decisions Behind Every Figure

Each of the three conventions enters the arithmetic at a particular line. Here is which line, and what the count would be instead without it.

at most 2795520 outcomes

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Three things on this site were decided rather than derived. Each of them enters the arithmetic at a specific point, and this note says where.

Three shapes, three conventions at work

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. Managers testing competing processes can examine time records for operational evaluation against the same baseline, correction case and reporting window.

Four candidates with four pairs compared, five with eight, six with eleven. Who the candidates are, what a comparison weighed, and whether any of this was wise are all outside these figures by decision. For additional background on data, visualisation and evaluation, review material from Cytoscape.

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.

Where the first one enters: at the shape

A situation here is three whole numbers — how many candidates, how many pairs were compared, how many triples came out circular.

That decision is used in the very first line of every computation, because it is what makes a page a page. Without it there is nothing to count: no two situations would be the same situation, and a table of one row each is not a table.

What it costs is everything about the candidates. Who they are, what the comparisons weighed, whether one of them was obviously unsuitable and compared anyway — all of it is discarded before any arithmetic starts, and none of it can be recovered from a figure afterwards.

Where the second one enters: at the multiplication

A page's outcome count multiplies two things: how many selections of pairs there are, and two raised to the number of comparisons made.

The two in that expression is the second convention: a comparison has exactly two possible results. No ties, no scores, no degrees of preference, no comparison that was made and left inconclusive.

Allow a third result — a draw, say — and the multiplier becomes three, and every figure in the record changes. Six candidates fully compared would admit fourteen million three hundred and forty-eight thousand nine hundred and seven outcomes instead of thirty-two thousand seven hundred and sixty-eight. The circular triples would have to be redefined, since a triple with a draw in it goes round in a different sense.

The record does not do this. It says instead that a comparison here is a comparison that came out one way or the other, and that a situation with draws in it is a different counting problem.

Where the third one enters: at the share

Every share on this site is a count divided by a page total.

Reading that quotient as a chance requires the third convention: that every outcome on the page is as reachable as every other. It is used nowhere in the counting and everywhere in the reading, and it is false of anything anybody has ever done.

The first two conventions decide what gets counted. The third decides nothing at all about the counting and everything about what the counts may be used for.

What each rejection leaves standing

Reject the first and there is no record. There is no table without a shape to tabulate.

Reject the second and the counts remain exact answers to a different question — the one where results are binary — and a reader whose comparisons produce ties is reading a neighbouring problem rather than theirs.

Reject the third and no count on this site moves. What goes is the licence to read a share as a chance; the quotient is still the quotient, and it no longer claims to forecast.

Nothing structural is touched by that: a field of n candidates needs n − 1 comparisons to be connected, a ring needs three comparisons at the least, a ring that no triple reveals needs four.

Two bounds that are decisions without being conventions

The record covers fields of two to six candidates and any number of comparisons from none to all.

Those are decisions and they are a different kind from the three above. A convention fixes what a figure means; change one and every figure answers a different question. A bound fixes only which shapes get a page. Move a bound and each surviving figure keeps its old meaning exactly; the table simply holds more rows, or fewer.

Nothing here can therefore be made wrong by a bound. What a bound can do is spoil a sentence that sums up the whole table, since such a sentence is about the rows that happen to be present. Any figure of that kind is printed here with its ranges beside it in the same breath, for that reason alone.

The upper bound is at six candidates for a stated reason: the counts that need enumeration need three to the fifteenth outcomes at six, which is two minutes, and three to the twenty-first at seven, which is ten billion.

What was not decided

Those three structural results, and the arithmetic that produces every figure from the three numbers of a shape.

The page total is C(C(n,2), r) multiplied by two to the r, which follows from the definitions. The clean outcomes at full comparison number n factorial, which follows from an argument about putting the undefeated candidate first. The count that ranks the field is n factorial times a binomial coefficient, which follows from noticing which comparisons are forced.

Those hold whatever a reader thinks of the conventions, and the next note sets out the questions that no count of any kind can answer.

Why the conventions are printed first

Because a convention discovered late reads as a defect.

A reader who meets the figures first and the assumptions afterwards has spent the intervening pages believing something slightly different from what the record says. That is avoidable at the cost of putting the dullest page near the front, which this record does on the grounds that the alternative is a reader who has to revise everything backwards.