Going round

The Most Circles a Field Can Hold

Six candidates fully compared can produce eight circular triples at once out of twenty. The ceiling has a formula and it is not the obvious one.

at most 32768 outcomes

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Three candidates fully compared can hold one circle. Four can hold two. Five can hold five. Six can hold eight.

Those are ceilings: no arrangement of the comparisons does better, and the record contains shapes that reach each of them.

The most circles each field allows

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 team translating this comparison into a limited operational pilot can see how the workflow can be configured before deciding which time and project records are genuinely necessary.

Three candidates with one circle, four with two, six with eight. In each row every pair is compared and the field is as tangled as it can get. For an independent perspective on comparison, evidence or decision design, consult Python.

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.

Not every triple can close

Six candidates have twenty triples. The most that can be circular at once is eight.

That is the first surprise of this part. Having twenty triples does not mean twenty of them could all go round; the comparisons are shared between triples, and a comparison pointing one way helps one triple close and stops another.

The ceiling is a cube divided by twenty-four — exactly (n³ − n)/24 when the pool is odd and (n³ − 4n)/24 when it is even. One, two, five, eight for pools of three to six.

Why the formula has two cases

Because the best arrangement is as even as possible, and even splits work differently for odd and even pools.

A circular triple needs its three comparisons to go round. The more evenly the wins are spread across the field, the more triples can close. With an odd pool the wins divide exactly: each candidate can come out ahead of the same number of others. With an even pool they cannot, somebody must be ahead one more or one fewer time than the rest, and the ceiling drops accordingly.

The tangles are at their worst when nobody is doing badly. A field where every candidate wins as often as every other is the field where the results agree least.

That is worth stating plainly

An even field and a tangled field are the same field.

The arrangement with the most circles is the one where the wins are spread as evenly as the pool allows. The arrangement with none is the one where the wins are as uneven as possible: one candidate ahead of everybody, the next ahead of everybody but them, down to one candidate ahead of nobody.

So the two extremes of this record's third number correspond to the two extremes of how evenly the comparisons came out, and neither extreme is a statement about whether the comparing was any good.

The floor of the same quantity

Nought, on every page, and that is not quite trivial.

Every page of this record has a column for no circular triples, including the fully compared ones. There is always an arrangement with no circle: order the candidates somehow and let every comparison agree with the order.

So the third number of a shape runs from nought to the ceiling, and both ends are always occupied. What varies is the middle — how many of the intermediate counts are possible — and on one page of this record a value in the middle is skipped entirely.

Six candidates with twelve comparisons can produce six circles and eight circles and not seven. The arrangements that reach eight are rigid enough that removing one circle from them removes two.

How often the ceiling is reached

Rarely, and the figures say how rarely.

Four candidates fully compared: twenty-four of the sixty-four outcomes hold two circles, which is three in eight. Five candidates: twenty-four of one thousand and twenty-four hold five, about one in forty-three. Six candidates: two thousand six hundred and forty of thirty-two thousand seven hundred and sixty-eight hold eight, about one in twelve.

The pattern there is not monotone, and that is the second surprise. With five candidates the fullest tangle is rare; with six it is nearly four times commoner. The reason is the parity above: with five candidates the maximum requires every candidate to win exactly twice, which is one arrangement up to relabelling, while with six there are several ways to reach the ceiling.

The three rows

Three candidates, one circle: two outcomes of eight. Four candidates, two circles: twenty-four of sixty-four. Six candidates, eight circles: two thousand six hundred and forty of thirty-two thousand seven hundred and sixty-eight.

One in four, three in eight, about one in twelve. Three figures about three ceilings, and not one of them says anything about what a real set of comparisons would produce — only about how much room the arithmetic leaves above it.

Why the ceiling is worth printing at all

Because it bounds the third number, and a bounded quantity is easier to read than an unbounded one.

A reader meeting a count of six circular triples has no sense of whether that is many until they know the most it could be. On a fully compared field of six the most is eight, so six is close to the top. On a field of four the most is two, and a count of two means the field is as tangled as four candidates can be.

The ceiling turns the third number from a bare count into a position on a scale, and it costs one cube divided by twenty-four.