Every pair

How Fast the Outcomes Grow

Two candidates allow three states of affairs. Six allow fourteen million. The growth is doubly exponential, and nothing about the pool looks like it.

at most 32768 outcomes

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Two candidates: one pair, and it can come out two ways.

Four candidates: six pairs, sixty-four ways. Six candidates: fifteen pairs, thirty-two thousand seven hundred and sixty-eight ways.

Two, four and six candidates, every pair compared

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 explore the platform before choosing before deciding which time and project records are genuinely necessary.

One comparison, then six, then fifteen. The figures on the right are how many of the outcomes contain no circular triple, which at full comparison is the number of orders there are. For an independent perspective on comparison, evidence or decision design, consult Nielsen Norman Group.

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 shape of that

Each added candidate brings n − 1 new pairs, and each new pair doubles the space.

Going from four candidates to five adds four pairs, so the space multiplies by sixteen. From five to six adds five pairs, so it multiplies by thirty-two. From six to seven it would multiply by sixty-four.

So the multiplier itself doubles with every candidate. That is a different animal from ordinary growth: the number of comparisons grows like a square, and the number of outcomes grows like two to the power of that square.

Written out

With every pair compared, a pool of two allows two outcomes, three allows eight, four sixty-four, five one thousand and twenty-four, six thirty-two thousand seven hundred and sixty-eight.

Allowing pairs to go uncompared, the same pools allow three, twenty-seven, seven hundred and twenty-nine, fifty-nine thousand and forty-nine, and fourteen million three hundred and forty-eight thousand nine hundred and seven.

Neither list has any steadiness in it. The second is the first with three in place of two, and the gap between them widens at every step.

The two ways of counting, side by side

Allowing uncompared pairs changes the character of the growth as well as its size.

With every pair compared, the count is two to the number of pairs, and the number of pairs is a square. With pairs allowed to go uncompared, it is three to the same power. Both are doubly exponential; the second simply has a larger base.

What that means in practice: the ratio between them is three halves to the power of the number of pairs. For four candidates that is about eleven. For five, about fifty-eight. For six, about four hundred and thirty-eight. The gap between "what could have happened" and "what could have happened if everybody had been compared" widens at the same furious rate as everything else here.

What stays small

The number of orders.

Four candidates can be put in twenty-four orders. Five in one hundred and twenty. Six in seven hundred and twenty. That is a factorial, and factorials are famous for growing fast — but against two to the power of a square they are slow.

Six candidates: seven hundred and twenty orders against thirty-two thousand seven hundred and sixty-eight outcomes. The orders are one outcome in forty-five.

The outcomes that rank cleanly are exactly the orders, and the orders are a vanishing fraction of what the comparisons can do.

That is the central result of this record and it gets a part to itself. What matters here is the rate: the two quantities are not in a race, because one of them is growing in an entirely different way.

The record's ceiling, and why it is where it is

Six candidates, and no further.

Seven candidates would mean twenty-one pairs and ten billion possible states of affairs, which is past the point where the figures can be produced by enumeration and far past the point where a reader could check one.

The arithmetic has no ceiling. The table does, it is at six, and every summary figure on this site is a fact about the table rather than about comparing people.

The three rows

Two candidates with their one pair compared: two outcomes, and both of them rank cleanly, because one comparison cannot contradict anything.

Four with all six pairs compared: twenty-four of the sixty-four rank cleanly.

Six with all fifteen: seven hundred and twenty of thirty-two thousand seven hundred and sixty-eight.

All of them, then three in eight, then about one in forty-five. The first row is not a special case being generous; it is what the collapse looks like before it has started.

Why this is worth knowing before comparing anything

Because the decision that sets the size of the space is how many candidates go into it, and that decision is usually taken for reasons that have nothing to do with the arithmetic.

Adding a seventh candidate to a field of six does not add a seventh of anything. It adds six comparisons and multiplies the space of possible results by sixty-four. Whether that matters depends on what the comparing is for, which this record does not know — but the figure is available before anybody starts, and it takes one multiplication to get.

What does not grow

The number of results a single comparison can have: two, always.

That is the one fixed quantity in this part, and everything else here is built on it. The pairs grow like a square because of how many ways two candidates can be picked. The outcomes grow like two to that square because each pair contributes a factor of two. The third state, a pair nobody compared, turns the two into a three.

So three constants — two results, three states, and one pair per pair of candidates — produce every figure in this record, and not one of them is a measurement.