Going round

What Comparing Everybody With Everybody Produces

Four candidates, all six pairs, sixty-four outcomes. Five candidates, all ten pairs, one thousand and twenty-four. The clean share falls as the field grows.

at most 32768 outcomes

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Comparing every pair is the thorough thing to do, and it is where the arithmetic is least forgiving.

Four candidates: six comparisons, sixty-four outcomes, twenty-four of them free of any circle. Five: ten comparisons, one thousand and twenty-four outcomes, one hundred and twenty free. Six: fifteen comparisons, thirty-two thousand seven hundred and sixty-eight outcomes, seven hundred and twenty free. The comparison described here can be extended with work pattern analysis across teams, provided the team states what the resulting evidence can and cannot prove.

Every pair compared, three sizes of field

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. A separate technical or research perspective relevant to the comparison can be found at The R Project.

The exhaustive case. In each row the figure on the right is how many outcomes contain no circular triple at all, which here is also the number of orders the field can be put in.

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 clean count is the number of orders

Twenty-four, one hundred and twenty, seven hundred and twenty. Those are four factorial, five factorial and six factorial.

That is not a coincidence and the argument is short. With every pair compared, an outcome with no circle can be arranged in an order: put the candidate who came out ahead of everybody first, then the one ahead of everybody remaining, and so on. The arrangement is forced at every step, so each clean outcome gives exactly one order.

Going the other way, each order gives exactly one clean outcome: compare every pair and let the earlier one win. So the clean outcomes and the orders are the same things counted twice, and there are n factorial of them.

When every pair has been compared, an outcome that contradicts nothing is an order, and an order is an outcome that contradicts nothing. They are one thing under two names.

The share, and how it falls

Twenty-four of sixty-four is three in eight. One hundred and twenty of one thousand and twenty-four is fifteen in one hundred and twenty-eight. Seven hundred and twenty of thirty-two thousand seven hundred and sixty-eight is forty-five in two thousand and forty-eight.

As decimals: about thirty-eight per cent, about twelve, about two. Each added candidate divides the clean share by something between three and six, and the divisor grows.

Three candidates, for completeness, give six of eight — three quarters. That is the last field where the clean outcome is the likely one.

Why it collapses

Because the two quantities are growing in different ways.

The orders grow like a factorial: times four, times five, times six. The outcomes grow like two raised to a quantity that is itself growing like a square, so they go times sixteen, times thirty-two.

A factorial against a doubly exponential is not a contest. By six candidates the clean outcomes are one in forty-five; by seven they would be one in four hundred and sixteen; the record stops before that but the arithmetic does not.

What this does not establish

That exhaustive comparing is a mistake.

It establishes that exhaustive comparing of a large field, if every outcome were equally reachable, would almost never produce a clean order. Real comparing is not like that: a comparer with a view of the field produces results that agree far more often than chance would give.

What the figure does is set a floor under how much agreement is being claimed. A clean order over six candidates is one outcome in forty-five by the arithmetic alone, and whatever produced it was doing something other than tossing coins.

What happens to the other outcomes

They contain at least one circle, and most of them contain several.

Six candidates fully compared: seven hundred and twenty outcomes have no circle, and the other thirty-two thousand and forty-eight have between one and eight. The commonest number is six.

So on a fully compared field the question is not whether the results will tangle but how much. That is a statement about the space of possible outcomes rather than about any comparer, and the convention that makes it a statement about likelihood is the one the record declares and does not defend.

The smaller reading, which is the useful one

For four candidates the clean share is three in eight, and four candidates is a field anybody can picture.

Six comparisons, sixty-four ways for them to land, twenty-four of which rank cleanly and forty of which do not. That is not a remote or asymptotic statement; it is a complete description of a small situation, and a reader can check every part of it with a pencil in a few minutes.

The figure behind all three rows

Two to the number of pairs, which is the number of ways a set of comparisons can come out when every pair has been looked at.

Six for three candidates, sixty-four for four, one thousand and twenty-four for five, thirty-two thousand seven hundred and sixty-eight for six. Each is the previous one multiplied by two to the power of the comparisons added.

Against that, n factorial. The whole of this note is those two sequences read side by side, and the reason the record returns to them is that nothing else in it is quite so easy to get wrong by intuition alone.

What a reader should check first on this page

The right-hand figure against a factorial.

Twenty-four for four candidates, one hundred and twenty for five, seven hundred and twenty for six. If the clean count on a fully compared page is not a factorial, something is wrong with it — and the check needs no table, no enumeration and about four seconds.