Comparing less

What the Trade Comes To

Fewer comparisons, fewer contradictions, less settled and a wider blind spot. Closing the part on comparing fewer pairs.

at most 32768 outcomes

Home · Comparing less · Analysis

Four comparisons among six candidates: twenty thousand eight hundred and eighty of twenty-one thousand eight hundred and forty outcomes contain no circular triple, and not one of them ranks the field.

All fifteen comparisons: seven hundred and twenty of thirty-two thousand seven hundred and sixty-eight contain no circular triple, and every one of those ranks the field.

Those are the two ends of the trade.

The trade at its two ends

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

Two outcomes drawn. Every row is the same number of candidates. Which candidate is which is the drawing's own choice and carries no meaning. For repeated pairwise evaluations, Monitask time tracking for structured reviews can add project and time context without turning one activity signal into a verdict about quality.

Four comparisons and all fifteen, on the same six candidates. The first page is almost free of circular triples and ranks nothing; the second ranks the field in seven hundred and twenty of its outcomes and leaves a circle in all the rest. Readers examining the figures against practical research methods can also see Brilliant.

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 this part established

Comparing fewer pairs makes a contradiction rarer, by a lot. For six candidates the share of outcomes with no circular triple runs from one at two comparisons down to about one in forty-five at fifteen.

It also leaves more undecided, and below a floor it decides nothing at all. A field of n candidates needs n − 1 comparisons simply to be connected, and fifteen of this record's forty pages sit beneath that line, with nothing on them able to rank anything — for a structural reason rather than an arithmetical accident.

Reaching the floor is necessary and nearly worthless: five comparisons rank six candidates in seven hundred and twenty of ninety-six thousand ways, about one in a hundred and thirty-three. The number of comparisons is the thing people plan; the arrangement is the thing that decides, and the record cannot see it.

The blind spot in the obvious check is worst in the middle. At nine comparisons out of fifteen, one in eight of the outcomes that pass a triple check still contains a ring.

And the pages themselves rise and fall: the widest page for six candidates is ten comparisons rather than fifteen, because selections of pairs become scarce before results do.

The one claim worth carrying

What stopping early purchases is not a tidier answer but a tidier silence.

Comparisons that contradict nothing may be complete and correct, or may simply never have asked enough to fall out with themselves. From the inside the two are indistinguishable. The figures tell them apart: at five comparisons, a consistent outcome ranks the field about one time in a hundred and seventeen; at fifteen, every consistent outcome ranks it.

The two ends, once more, as figures

Four comparisons among six candidates: twenty-one thousand eight hundred and forty outcomes, twenty thousand eight hundred and eighty with no circular triple, twenty thousand seven hundred and ninety that fall into an order, none that rank the field.

Fifteen comparisons: thirty-two thousand seven hundred and sixty-eight outcomes, seven hundred and twenty with no circular triple, seven hundred and twenty that fall into an order, seven hundred and twenty that rank it.

The first page is ninety-six per cent clean and settles nothing. The second is two per cent clean and settles everything it does not contradict. Between them lie fourteen pages and the whole of the decision.

The same trade on a smaller field

Five candidates, and the shape of it does not change.

Three comparisons out of ten: nine hundred and sixty outcomes, nine hundred and forty with no circular triple, and none that rank the field, since the floor for five candidates is four. Ten comparisons: one thousand and twenty-four outcomes, one hundred and twenty clean, and every clean one ranks the field.

Ninety-eight per cent clean and nothing settled, against twelve per cent clean and everything settled. The percentages differ from the six-candidate pair, the direction does not, and that is the point of printing both: the trade belongs to the arithmetic rather than to any one field size.

What the record still will not say

Whether to stop.

That depends on what the comparing is for, how much a wrong order costs, and how much time there is — none of which is countable from a pool size and a number of comparisons. The record prices the arithmetic half of the decision and is silent on the rest, which is the arrangement it has had from the first page.

What comes next

The conventions all of this rests on, stated rather than assumed, and the questions no figure here can answer.

Three things were decided rather than computed, and each can be rejected on its own. The last part says what survives each rejection, which is less than the record and more than nothing.

Where this part leaves a reader

With a table and two floors.

The table says, for a field of six, what every amount of comparing produces. The first floor says a contradiction needs three comparisons. The second says a ranking needs five.

Between the floors and the ceiling the figures move in four different directions at once, and the record's contribution is that they are figures rather than impressions. What to do with them is not a question a count can answer, and the last part of this record is about the boundary of what a count can answer at all.

One figure from this part

One in a hundred and thirty-three.

That is how many of the ways to make the fewest possible comparisons on a field of six actually rank it. The floor is necessary, it is nowhere near sufficient, and the gap between those two statements is the whole of this part.