An order

What Each Comparison After the Chain Buys

The sixth comparison multiplies the order-settling outcomes by ten. The tenth adds a fifth. The returns rise, then fall, and the peak can be computed.

at most 3075072 outcomes

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Five comparisons among six candidates can rank the field, and seven hundred and twenty of the ninety-six thousand ways to make them do.

Add a sixth comparison and seven thousand two hundred outcomes rank the field. Ten times as many.

Six candidates, the comparing filling in

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. Every row is the same number of candidates. Which candidate is which is the drawing's own choice and carries no meaning. Teams comparing routes, outcomes and corrections can check the official account of the platform, then check whether the resulting evidence improves the decision in practice.

Five, eight, ten and thirteen of fifteen pairs compared. The counts that settle the order are seven hundred and twenty, eighty-six thousand four hundred, one hundred and eighty-one thousand four hundred and forty, and thirty-two thousand four hundred. A separate technical or research perspective relevant to the comparison can be found at pandas.

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 whole sequence

For six candidates, by number of comparisons, the count of outcomes that settle the order:

Five, seven hundred and twenty. Six, seven thousand two hundred. Seven, thirty-two thousand four hundred. Eight, eighty-six thousand four hundred. Nine, one hundred and fifty-one thousand two hundred. Ten, one hundred and eighty-one thousand four hundred and forty. Eleven, one hundred and fifty-one thousand two hundred. Twelve, eighty-six thousand four hundred. Thirteen, thirty-two thousand four hundred. Fourteen, seven thousand two hundred. Fifteen, seven hundred and twenty.

Up to a peak at ten, then back down through the identical figures in reverse.

Why it falls again

Because the count is a count of outcomes, and past the peak there are fewer outcomes of any kind.

With thirteen of fifteen pairs compared there are only one hundred and five ways to choose which pairs, so the page is small: eight hundred and sixty thousand one hundred and sixty outcomes against three million at ten comparisons. The order-settling ones fall with everything else.

As a share of the page the picture is the opposite: seven hundred and twenty in ninety-six thousand at five comparisons, one in a hundred and thirty-three; thirty-two thousand four hundred in eight hundred and sixty thousand at thirteen, one in twenty-seven; seven hundred and twenty in thirty-two thousand seven hundred and sixty-eight at fifteen, one in forty-five.

The symmetry, and why it is real this time

The sequence reads the same forwards and backwards, and that is not a coincidence.

An outcome settles the order when it contains every neighbouring pair of that order and nothing that contradicts it. Having fixed the order, the remaining pairs — ten of them for six candidates — are each free to be compared or not, with their result forced either way.

So the count is the number of orders times the number of ways to choose how many of the ten free pairs to include, which is seven hundred and twenty times a binomial coefficient. Binomial coefficients are symmetric, so this column is too.

One symmetric column in the whole record, and it is symmetric for a reason that fits in a sentence. The other symmetries here are coincidences and are marked as such.

The same column as a share of its page

The counts rise and fall; the shares do something else.

Seven hundred and twenty of ninety-six thousand at five comparisons is one in a hundred and thirty-three. One hundred and eighty-one thousand four hundred and forty of three million at ten is one in seventeen. Seven hundred and twenty of thirty-two thousand seven hundred and sixty-eight at fifteen is one in forty-five.

So as a share the best page is somewhere near ten, and both ends are poor — the early pages because almost nothing ranks the field, the late ones because almost everything contradicts itself.

That is the only quantity in this record with a genuine interior optimum, and it is worth noting that it is an optimum in a space of possible outcomes rather than advice about where to stop.

What the extra comparisons are doing

Not deciding anything that was not already decided.

Once the chain is in place the order is known. Every comparison after it must agree with the order — otherwise the outcome contradicts itself and is not in this count at all — so it adds no information.

What it adds is confirmation, and the arithmetic has nothing to say about the value of that. The record counts outcomes; whether the eleventh comparison was worth making depends on what the comparing is for, which is the first thing these three numbers left out.

The four rows

Five comparisons: ninety-six thousand and ninety-six outcomes, eighty-five thousand seven hundred and sixteen with no circular triple.

Eight: one million six hundred and forty-seven thousand three hundred and sixty, eight hundred and forty-eight thousand five hundred and eighty.

Ten: three million seventy-five thousand and seventy-two, seven hundred and sixty-seven thousand two hundred and sixty-eight.

Thirteen: eight hundred and sixty thousand one hundred and sixty, fifty-two thousand and twenty.

The pages grow and then shrink; the clean share falls throughout; and the order-settling count rises and falls on its own schedule — which happens to peak at ten comparisons, the same place the page sizes do, for two unrelated reasons that both involve choosing from what is left over.

What the symmetry is not

A statement that the two ends are equivalent.

Five comparisons and fifteen comparisons both give seven hundred and twenty outcomes that rank the field, and they are nothing alike otherwise: at five almost every outcome is consistent and almost none decides anything, and at fifteen almost every outcome contradicts itself and every consistent one decides everything.

The column is symmetric; the situation is not. That is a good example of a figure that would mislead if it were read without the other columns beside it.