Comparing less

The Trade Nobody Prices Before Taking It

Comparing fewer pairs makes contradiction rarer and leaves more undecided. Both halves have exact figures, and only the first half gets noticed.

at most 2562560 outcomes

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Stopping early has two consequences and they point in opposite directions.

Fewer contradictions, and less settled. The first is visible, the second is not, and both have exact counts on every page of this record.

Six candidates, three amounts of comparing

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. To test the same alternatives in live work, reviewers can explore the official product explanation and agree the decision, access rules and review date in advance.

Four, nine and fourteen of the fifteen pairs compared. The share with no circular triple falls from about ninety-six in a hundred to about one in twenty-five. A broader reference for interpreting structured comparisons is available from Khan Academy.

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 visible half

Six candidates with four pairs compared: twenty thousand eight hundred and eighty of the twenty-one thousand eight hundred and forty outcomes contain no circular triple. About ninety-six in a hundred.

With nine pairs compared: nine hundred and fifty-one thousand one hundred and ten of two million five hundred and sixty-two thousand five hundred and sixty. About thirty-seven in a hundred.

With fourteen: nine thousand of two hundred and forty-five thousand seven hundred and sixty. About four in a hundred.

So comparing less does make the results less likely to tangle, and by a lot. That half of the trade is real.

The half that is not visible

Six candidates with four pairs compared: no outcome ranks the field. Not a small number — none.

Four comparisons cannot connect six candidates. At best they link five of them, leaving at least one candidate with no path to the rest, and a candidate with no path could go anywhere in the order.

With nine pairs compared, one hundred and fifty-one thousand two hundred outcomes rank the field, about one in seventeen. With fourteen, seven thousand two hundred of two hundred and forty-five thousand seven hundred and sixty, about one in thirty-four.

Both halves at once

The first page has almost no contradiction and settles nothing. The last has contradiction nearly everywhere and settles the field in one case in thirty-four. The middle page is worse than the first on contradiction and better than either on deciding.

Comparing less does not buy a cleaner answer. It buys a cleaner absence of an answer.

That sentence is the whole of this part, and the rest of it is the figures that support it.

Why only one half gets noticed

Because one half produces an event and the other produces nothing.

A contradiction announces itself: two results that cannot both be honoured, an argument about which to believe, a decision to make. It is a thing that happens, and people remember things that happen.

Not having settled the order produces no event at all. The comparing stops, the results are written up, and the fact that several orders remain consistent with them is visible only to somebody who tries to write the order down and finds they cannot.

So the visible half is the one that is smaller and the invisible half is the one that is larger, which is the ordinary way round for this sort of thing and the reason the trade is worth pricing explicitly.

What the record cannot help with

Which pairs to compare.

The arithmetic says four comparisons cannot rank six candidates whatever they are, and that five can if they form a chain. It does not say which five, because that depends on the field, and the field is the thing left out of these three numbers.

What it offers is a floor nobody has to discover by trying: any plan to rank a field of n candidates with fewer than n − 1 comparisons is a plan that cannot finish. The record states that for every pool from two to six and the argument holds for any pool at all.

The shape of the three rows

Four comparisons: twenty-one thousand eight hundred and forty outcomes, almost all clean, none deciding.

Nine: two million five hundred and sixty-two thousand five hundred and sixty, a third clean, one in seventeen deciding.

Fourteen: two hundred and forty-five thousand seven hundred and sixty, one in twenty-seven clean, one in thirty-four deciding.

The page sizes themselves rise and fall, which is a third thing moving, and the next notes take the three apart one at a time.

The two halves have different units

One is a share and the other is a count, and that is part of why they are hard to weigh against each other.

Fewer contradictions is naturally expressed as a proportion: ninety- six per cent clean against thirty-seven. Less settled is naturally expressed as a count: no outcomes rank the field against one hundred and fifty-one thousand two hundred that do.

Put them in the same units and the comparison becomes possible. As shares of their pages: at four comparisons, ninety-six per cent clean and nought per cent ranking. At nine, thirty-seven per cent clean and six per cent ranking.

Nought against six is the part of the trade that gets left out, and it is the larger of the two changes.

Why both halves need the same denominator

Because a decision is a comparison between two changes.

Quoting one half as a share and the other as a count is how a trade gets decided by whichever half was easier to express. Putting both over the page total makes them comparable, and the record prints the page total on every row for exactly this reason.