User:Andrewa/Condorcet and New York

See also User:Andrewa/Condorcet and New York simplified.

It has been suggested that the question of whether New York State or New York City is the primary topic of New York, or whether there is no primary topic, may be an example of a Condorcet paradox.

The evidence is that it is not, and it is easily tested if there is doubt. But like all mathematical paradoxes, Condorcet paradoxes have some tricky underlying logic. So hold on to your hatnotes...

The decision we need to make is between three conflicting views: and it appears from past discussions that no one of these three has a clear majority over the other two. This is indeed one of the necessary conditions for a Condorcet paradox (from now on abbreviated Cp) to exist. But it is not the only one.
 * It is best for New York to point to New York State (from now on abbreviated NYS)
 * It is best for New York to point to New York City (abbreviated NYC)
 * It is best for New York (abbreviated NY) to point to a DAB (or BCA)

To decide whether this is a Cp situation, we need to also consider the three possible two-way outcomes (well, six really):
 * A: It is better for NY to point to a DAB rather than NYS (or not, i.e. it is better for NY to point to NYS rather than a DAB, abbreviated /A)
 * B: It is better for NY to point to NYC rather than a DAB (or not, /B)
 * C: It is better for NY to point to NYS rather than NYC (or not, /C)

Anyone with a view on which is best believes at least two of these six: and may (or may not) have a view on the third two-way decision but it does not affect their voting in a three-way poll. And that is the critical point that makes a Cp possible.
 * It is best for NY to point to NYS: C and /A
 * It is best for NY to point to NYC: B and /C
 * It is best for NY to point to a DAB: A and /B

Now, we have tested A and C in an RfC. It is not possible to have NYS at the base name, as it is not the primary topic. This affirms A and denies C as possible outcomes if we were to hold three two-way polls (evaluating them according to the closing instructions of course, as was done with the RfC).

And we need go no further. This is not a Cp. There are only two possible Cp scenarios:


 * A, B and C
 * /A, /B and /C

and we can have neither of these. QED

It is remarkable that none of those who have suggested that a Cp may exist have said which of these two very different Cp scenarios they think may be the case here. Perhaps they should be asked?

If a specific Cp is still claimed, it can be further tested by a two-way poll on any one of the three two-way outcomes that are necessary parts of the alleged paradox. It should not be hard to choose the weakest link.

Failing that, the proposed move if successful confirms A. This seems a good way forward.

It does not answer B or try to. A simple RM cannot do both at once.