Comparing less

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 many candidates one comparison can connect.

at most 823680 outcomes

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Six candidates cannot be ranked by four comparisons.

Not rarely. Never, in any of the twenty-one thousand eight hundred and forty outcomes that four comparisons among six candidates allow.

Below the floor, at it, and above it

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. If several people repeat this comparison, managers may see how the platform approaches this while interpreting every signal beside completed work and participant feedback.

Six candidates with three, five and seven pairs compared. No outcome in the first row ranks the field; seven hundred and twenty do in the second; thirty-two thousand four hundred in the third. Readers examining the figures against practical research methods can also see Farnam Street.

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.

The argument

A comparison connects two candidates. The first connects two; each one after it connects at most one new candidate to what is already connected.

So r comparisons connect at most r + 1 candidates. Four comparisons reach five. The sixth candidate is left with no path to the others, and if there is no path between two candidates then nothing in the comparisons says which of them is ahead — so at least two orders satisfy the outcome and none is settled.

That is the whole proof, it has no arithmetic in it beyond counting to one, and it holds for any field of any size.

The floor, page by page

Two candidates need one comparison. Three need two. Four need three. Five need four. Six need five.

In this record, fifteen of the forty pages sit below their floor. On every one of them the count of outcomes that rank the field is nought — not small, nought — and that is the one figure in this record that is zero for a structural reason rather than because the arrangement happens to be impossible.

What those pages are not

Empty, or uninteresting.

Six candidates with three comparisons allows three thousand six hundred and forty outcomes, and three thousand six hundred of them contain no circular triple. They are consistent, they say something — about three candidates at a time — and they cannot rank the field.

That combination is the ordinary state of a half-finished comparison, and it is worth having a name for: consistent, informative, and short of the floor.

A set of comparisons below the floor cannot be wrong about the order. It also cannot be right about it, because it does not have one.

The floor is about connection, not about contradiction

Worth separating, because the two limits in this record sound similar and are not.

The three-comparison floor is about contradiction: fewer than three comparisons cannot disagree with each other. It is a statement about rings.

The n − 1 floor is about connection: fewer than n − 1 comparisons cannot link the whole field. It is a statement about reach, and it holds whatever the results are.

A set of comparisons can be above one floor and below the other. Three comparisons on a field of six are above the contradiction floor and well below the connection floor: they can tangle and they cannot rank.

Reaching the floor is not enough either

Five comparisons among six candidates can rank the field and usually do not.

Of the ninety-six thousand and ninety-six outcomes with five comparisons, seven hundred and twenty rank the field — the chains, one for each order. Eighty-three thousand nine hundred and fifty-two are consistent, so the vast majority are consistent and short.

The floor is a necessary condition and a long way from a sufficient one. Reaching it means the comparisons could rank the field; being in the right arrangement means they do.

The three rows

Three comparisons: below the floor, nothing ranks the field, and three thousand six hundred of the three thousand six hundred and forty outcomes have no circular triple.

Five: at the floor, seven hundred and twenty rank the field out of ninety-six thousand and ninety-six.

Seven: above it, thirty-two thousand four hundred rank the field out of eight hundred and twenty-three thousand six hundred and eighty, which is about one in twenty-five.

The jump from five comparisons to seven multiplies the ranking outcomes by forty-five and the page by eight and a half. That is the shape of the trade when it is going the other way.

Why this floor is the more useful of the two

Because it can be checked against a plan rather than against results.

The contradiction floor says what the results could do. The connection floor says what the plan can do: count the comparisons intended, compare with the field size less one, and if the first is smaller the plan cannot finish whatever happens.

That makes it the one figure in this record a comparer can use before starting, and the only one that does not need the three numbers of a shape at all.

The floor on the record's largest field

Six candidates, five comparisons, and the record holds five pages below it.

Those five pages — no comparisons through four — carry twenty-five thousand nine hundred and thirty-one outcomes between them, every one of which ranks nothing. That is a small share of the record and a large share of what half-done comparing looks like.

The floor in one line

A field of n candidates needs n − 1 comparisons before any outcome can rank it, because each comparison connects at most one new candidate to the rest.

No results are involved, no convention is used, and the argument holds for fields far larger than this record covers.