Nondictatorial & Nonmanipulable axioms (was Re: York's

[Bruce Anderson]

On May 10,  5:37am, Steve Eppley wrote:
> Subject: Re: Nondictatorial & Nonmanipulable axioms (was Re: York's
> Bruce Anderson wrote:
> [snip]
> >In terms of Ordeshook's structure (which you wisely provide below),
> >Condorcet's method violates A4 (Independence).  This axiom is
> >frequently called the Independence of Irrelevant Alternatives axiom
> >(or criterion).
> [snip]
> 
> If an alternative is part of a circular tie, can it be an irrelevant
> alternative?  (I.e., what about Smith//Condorcet?)
> 
>-- End of excerpt from Steve Eppley

In a nutshell, that's the whole point of my research on Arrow's impossibility 
theorem.  In fact, Arrow's theorem essentially requires that Smith winners be 
considered "irrelevant."  I have proven, roughly speaking, that if Smith winners 
are not allowed to be irrelevant, but that any other candidates can be called 
irrelevant, then all of Arrow's conditions (suitably modified) can be satisfied. 
 This is a major reason why I strongly favor single-winner voting methods that 
satisfy Smith's generalized Condorcet criterion (like Smith//Condorcet, Kemeny, 
Copeland, and Regular-Champion) over those that don't, (like "plain" Condorcet 
and Black).

As an aside, I recommend against using the term "circular tie" without 
definition.  To see why, consider the following example that Mike and I worked 
out.  Suppose that there are 10 candidates, A through J.  Suppose that A beats B 
and C, and ties everyone else.  Suppose that B beats C who beats D who beats B, 
and they tie everyone else (except for B and C losing to A).  Suppose that E 
beats F who beats G who beats E and that they tie everyone else, except that G 
beats H, and that H ties everyone else.  Finally suppose that I beats J and they 
tie everyone else.  Is there a circular tie here and, if so, who is in it?

(All 10 candidates are Smith winners here.)

Bruce