Comparing less

Which Pairs Get Compared Beats How Many

Five comparisons among six candidates rank the field in seven hundred and twenty of ninety-six thousand ways. The number is not the problem; the arrangement is.

96096 outcomes

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Five comparisons can rank six candidates. Ninety-five thousand three hundred and seventy-six arrangements of five comparisons cannot.

The number was right. The arrangement was not.

Six candidates, five pairs compared

solid wedge: the row came out ahead of the column
blank cell: the column came out ahead of the row
small ring in both cells: the pair was never compared
dash: the diagonal: nothing is compared with itself

Six candidates and 15 possible pairs. Five of them compared. No circular triples. One of the 96096 outcomes this page allows. When a team turns an abstract comparison into an operating review, employee workload visibility can be evaluated beside outcomes, delays and direct staff feedback.

Row beats column. Five comparisons is exactly the fewest that could rank six candidates, and this outcome does not rank them: the comparisons do not form a single chain, so two candidates have no path between them. For an independent perspective on comparison, evidence or decision design, consult UC Berkeley.

One of the outcomes this page allows, drawn in full. Which candidate sits in which row is the drawing's own choice; the counts are of outcomes, not of anybody.

What the right arrangement looks like

A single chain through the whole field.

Each comparison has to connect somebody new to what is already connected, and no comparison may be spent on a pair that is already linked by a path. Five comparisons, six candidates, nothing wasted.

Spend two of the five on the same two candidates and the field cannot be covered. Spend them in two separate clusters and the clusters float free. Spend them all on one candidate against the others and five candidates are left unplaced relative to each other.

How few of the arrangements work

Of the ninety-six thousand and ninety-six outcomes with five comparisons among six candidates, seven hundred and twenty rank the field.

That is about one in a hundred and thirty-three. The other ninety-five thousand three hundred and seventy-six either contain a ring — twelve thousand one hundred and forty-four of them do — or are consistent and leave the field open.

The drawing

The matrix shows one such outcome: five comparisons made, no circular triple, and no ranking.

The filled marks say which way each comparison went; the small rings mark the ten pairs nobody looked at. Following the marks shows the problem at once: what was compared falls into separate pieces with nothing joining them.

A reader can check in a few seconds what no figure on the page reports: this outcome settles part of the field and says nothing about the rest.

Why this matters more than the count

Because the number of comparisons is the thing people plan and the arrangement is the thing that decides.

Two people each making five comparisons on the same field of six can end with completely different amounts of information: one with a full ranking, one with two pairs placed and four candidates floating. The record counts both as five comparisons, which is exactly the limitation of counting by number.

Spending the right number of comparisons is necessary and almost worthless on its own. What they were spent on decides everything, and the record cannot see it.

The one thing the arithmetic does say

Which arrangements can work, in general terms.

A set of comparisons ranks the field only if every candidate is joined to every other by some path, and only if no path doubles back. The first takes n − 1 comparisons at the least. The second is what the walk tests.

The first of those two is a property of the plan alone, so anybody who has decided which pairs to look at can tell beforehand whether the plan could finish. That is the one piece of planning this record supports.

Two plans, same count, different fields of view

One comparer spends five comparisons on a chain through all six candidates and ends with a ranking.

Another spends five on the same candidate against each of the others and ends knowing exactly where one candidate stands and nothing about the order of the remaining five, which could be in any of one hundred and twenty orders.

A third spends five on two separate groups and ends with two fragments that cannot be joined.

All three made five comparisons. The record counts all three as the same page, reports ninety-six thousand and ninety-six outcomes, and cannot tell them apart — because telling them apart requires knowing which pairs, which is the fourth number this record does not have.

What it will not do

Tell anybody which chain to use.

The chain that ranks the field is the order, and the order is the thing being looked for. Any method that picks the comparisons in advance is picking blind; any method that picks them as it goes is using the results, which the record does not model at all.

So the honest statement is narrow: five comparisons might rank six candidates, four certainly cannot, and whether any particular five do depends on something outside these three numbers.

The fourth number the record does not have

Which pairs.

Three integers make a shape here, and a fourth would be the set of compared pairs itself — not a number but a choice among thousands. A record indexed by that would have thirty-two thousand seven hundred and sixty-eight rows for a field of six alone, and nothing in it could be checked by hand.

So the record counts over all choices instead, treating them as equally reachable, and says plainly that the figures therefore describe the space rather than any plan inside it.

What this costs the record

Its usefulness as a plan.

Everything here prices a quantity, and the skill in comparing anything lies in the choice of what to compare next. The record is silent on that by construction, and a reader looking for a method should take the floor, the walk and the table of sizes, and look elsewhere for the rest.