Every pair

What One More Candidate Costs

Adding a candidate to a field of five adds five comparisons and multiplies the space of possible results by thirty-two. One of those two figures gets noticed.

at most 32768 outcomes

Home · Every pair · Recount

A field of five candidates is settled, and somebody asks to add a sixth.

The honest answer to what that costs has two parts, and they are very different sizes.

Four, five 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.

One candidate added between each row. The pairs go six, ten, fifteen; the outcomes go sixty-four, one thousand and twenty-four, thirty-two thousand seven hundred and sixty-eight. Before adding another reporting layer, a project lead can review the official product page and test only the smallest useful setup on one representative process.

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. For additional background on data, visualisation and evaluation, review material from UX Booth.

The part that gets noticed

Five more comparisons.

The sixth candidate has to be put beside each of the five already there, so the number of pairs goes from ten to fifteen. That is fifty per cent more comparing, it is visible on anybody's calendar, and it is the figure that gets discussed.

The rule in general: adding the nth candidate adds n − 1 pairs. A field of four becoming five adds four. Five becoming six adds five. Six becoming seven would add six.

The part that does not

The space of possible results multiplies by thirty-two.

Each of the five new comparisons can come out two ways, and they are independent, so the number of ways the whole set of comparisons could land is multiplied by two to the fifth.

From four candidates to five it was sixteen. From five to six, thirty-two. From six to seven it would be sixty-four. The multiplier itself doubles each time, which means the second part of the cost is not merely larger than the first — it is growing in a different way.

Fifty per cent more comparing, and thirty-two times as many ways for the comparing to come out. Those are the same decision.

Written out for the three rows

Four candidates: six pairs, sixty-four outcomes, twenty-four of them free of any circular triple.

Five: ten pairs, one thousand and twenty-four outcomes, one hundred and twenty free of one.

Six: fifteen pairs, thirty-two thousand seven hundred and sixty- eight outcomes, seven hundred and twenty free of one.

The left-hand figures grew by a half each time. The middle figures grew by sixteen and then thirty-two. The right-hand figures grew by five and then six — which is to say, like a factorial, because they are one.

Three columns moving at three rates

That is the whole of this record in one drawing.

The pairs grow like a square. The outcomes grow like two to the power of a square. The clean outcomes — the ones that fall into an order — grow like a factorial, which against the other two is barely growing at all.

Three in eight of the four-candidate outcomes rank cleanly. About one in nine of the five-candidate outcomes. About one in forty-five of the six-candidate ones. Each added candidate makes the clean outcome several times rarer, and the arithmetic does it without any help from how the comparing is done.

What the figure is not saying

That adding a candidate makes a contradiction more likely in practice.

It makes the contradictory outcomes a larger share of the space of possible outcomes, which is a statement about arithmetic. Whether any particular set of comparisons lands in that part of the space depends on how the comparing was done, and this record has no access to that at all.

What survives without any assumption is the structural half: with more candidates there are more triples, each triple is a chance for the results to close into a circle, and the number of triples grows like the cube of the pool. Four candidates have four triples. Six have twenty. Seven would have thirty-five.

The triples grow faster than either

There is a third quantity in the same arithmetic and it grows faster than the comparisons.

A field of n candidates has C(n, 3) triples, each of which is a chance for the results to close into a circle. Four candidates have four. Five have ten. Six have twenty. Seven would have thirty-five.

So adding a candidate to a field of five adds five comparisons and ten triples — twice as many opportunities for a contradiction as comparisons to make. That ratio keeps rising: the comparisons grow like a square and the triples like a cube.

It is the clearest reason the clean share collapses rather than merely declining. Every candidate brings more ways to disagree than ways to compare.

The reverse reading

The same arithmetic run backwards is the more useful one.

Taking a candidate out of a field of six removes five comparisons and divides the space of outcomes by thirty-two. If the point of the comparing is to end with an order, a smaller field is not merely less work; it is a smaller space to end up lost in.

The record does not say whether that is worth it, because what the comparing is for was the first thing left out. It says what each step costs, exactly, for any pool from two to six, which is a different kind of help and the only kind available from three whole numbers.

Where the record stops being able to tell you

At seven candidates.

A seventh candidate would add six comparisons, make twenty-one pairs, and open ten billion possible states of affairs. The arithmetic is identical and the enumeration is not: the counts that need generating every outcome become impractical, and the record declines to print figures it cannot produce exactly.

What survives past the edge is the behaviour. Each candidate adds n − 1 comparisons, multiplies the outcomes by two to that power, and adds C(n − 1, 2) new triples. Those statements have no ceiling; the table does.