[EM] Suppose, for a moment, there were never any cycles...

Kristofer Munsterhjelm km_elmet at t-online.de
Sat Jan 21 14:18:41 PST 2023

On 21.01.2023 20:49, robert bristow-johnson wrote:
> So I am trying to grok this Benoit paper
> https://people.cs.pitt.edu/~kirk/CS1699Fall2014/gibbard-sat.pdf and I
> might be able to do it.  But I would like to ask the authorities here a
> simple hypothetical question.
> We know that Condorcet cycles can occur and in the U.S. there is one
> known RCV election that demonstrated a cycle, the 2021 Minneapolis Ward
> 2 City Council election.  It was a simple Rock-Paper-Scissors cycle;
> Smith set of 3.
> I think it's the case that we know that the necessary ingredients
> for  a cycle is a very close 3-way race *and* that there is at least a
> 2-dimensional political spectrum (Nolan chart) with candidates and
> voters spread out all over the map in 2 dimensions, *not* in mostly a
> linear spatial distribution.  There has to be a sorta political
> schizophrenia where a lotta voters are saying something like "If I can't
> have my favorite, Bernie, then I'm voting for T****."

There could also be simple noise, i.e. some candidates are extremely 
close, the voters randomly fill in what's really equal-rank, and the 
random fill happens to create a Condorcet cycle.

IIRC, Worlobah and Arab were very close in the Minneapolis election, so 
the cycle could be a result of noise, but I don't know the circumstances 
well enough to say for sure.

> Here is my question: Suppose that cycles *never* happened.  Suppose
> we were somehow guaranteed that there is *always* a Condorcet winner.
> And also suppose we're far enough from a cycle that no collective
> strategy would succeed at pushing the election into a cycle.  Then,
> *if* there is always a CW, is there any strategy that will serve a
> voter's political interest  than better than ranking the candidates
> sincerely?
If there is a CW, then IIA always holds, yes. Suppose X is the CW and Y 
is someone else. Then running the election without Y doesn't change 
anything: X still beats everybody else pairwise and wins.

Thus if somehow you could prevent both sincere and tactical Condorcet 
cycles, the method would be strategy-proof. (Strictly speaking, this 
doesn't invalidate Arrow's theorem because a method that magically 
forbids Condorcet cycles would fail universal domain.)


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