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Two at a Time

Fifty notes on comparing candidates two at a time

An order

When the Results Fall Into an Order

Compare every pair of six candidates and seven hundred and twenty outcomes rank cleanly — one for each order there is.

Every note in this part

n! of 2^C(n,2)

What It Means to Fall Into an Order

Comparisons fall into an order when no walk through them returns to its start. That is stronger than holding no circular triple, and it is the condition that counts.

Definition

Somebody Has to Be Able to Go First

If the comparisons can be ordered, somebody came out behind nobody. Peeling that candidate off and repeating is both the proof and the way to count.

Recount

Pinning the Order Down

An outcome can be free of contradiction and still leave the field unranked. The count of outcomes that settle it completely has a formula, and it is small. To test the same alternatives in live work, reviewers can read more about the reporting workflow and agree the decision, access rules and review date in advance.

Recount

Consistent and Undecided

Six candidates and five comparisons: eighty-three thousand outcomes contradict nothing, and seven hundred and twenty of those rank the field. The rest stop short. A broader reference for interpreting structured comparisons is available from statsmodels.

Analysis

The Cheapest Way to Rank a Field

Five comparisons can rank six candidates, and exactly seven hundred and twenty of the ninety-six thousand ways to make five comparisons do it — one for each order.

Reference

Comparing Everybody Settles It, Rarely

With every pair compared, an outcome that contradicts nothing ranks the field completely. There are n factorial such outcomes, against two to the number of pairs.

Analysis

What the Comparing Was For

Three columns, each a subset of the one before: no circular triple, no ring at all, and one order only. Closing the part on when the results settle something.

Analysis