[ER] IRO fails plurality CWs too.

Mike Ositoff ntk at netcom.com
Wed Oct 14 19:11:46 PDT 1998


I'm replying farther down in this message:

> I haven't had a chance to respond to anything in a few days, but let's
> see what we have:
> Mike Ositoff wrote:
> > 
> > Hi--
> > 
> > Before I get to the example, let me mention that IRO's proponents
> > often say that it gets rid of the lesser-of-2-evils problem. They're
> > talking about a particular special situation, like:
> > 
> >  47  45  8
> >   A   B  C
> >   B      B
> > 
> > The more extreme candidates are smaller--candidate size tapers
> > toward the extremes. But the extreme candidate is still big enough
> Questionable statement.  The middle may have "larger" candidates, or it
> may have a lot of smaller candidates.  I'll go along with the former for
> now, just to see where it goes.
> > to tip the scales between the more middle candidates when its votes
> > transfer inward.
> > 
> > Ok, then let's keep those assumptions, but say that there are
> > more candidates. For example:
> > 
> >  60 70 100 83 75
> >   A  B   C  D  E
> >   B            D
> > 
> > (I've only listed the 2nd choices that affect the result. Only
> > the extremes need get eliminated to establish C as a loser.
> > 
> > Note that in this example not only does IRO dump a CW, but
> > it dumps a  CW who has a plurality. So much for IRO advocates'
> > favoriteness standard.
> So since when do we care about the plurality winner?
> Actually, what is clear from the example is that there are three main
> factions; namely B, C, and D.  A and E are fringe groups that have no
> other choice but to join with B and D.  So in effect you have:
>   130  100  158
>    B    C    D
> This is a fairly typical IRO example.  C loses, but it has more than
> enough votes to tilt the election from B to D, depending on what C's
> voters prefer.
> Would Approval or pairwise have different results?  Under those methods
> the default strategy for B and D would be to truncate, assuming both
> believe that they have a good choice of winning.  A and E would still be
> eliminated in favor of B and D, and again C would potentially determine

Approval & Condorcet don't have eliminations, but it's true that
A & E would be unlikely to win.

> the outcome between B and D.
> If B believes a loss to D is likely, and thinks that C is a reasonable
> second choice, then B is capable of throwing enough support to C to
> ensure C's win.  Under IRO, 30 of B's voters could switch and rank C

The problem is when B voters mistakenly believe that they're
in the position shown in your example. Then, in Approval they
have reason to give a vote to C. In IRO they need to vote C 1st,
above B. In Approval it takes twice as many B voters making that
mistake to give the election away. In IRO, when they do, it's
an equilibrium--B voters will never know how numerous they are,
since they're voting like C voters. 

> over B.  This should be enough to make sure C is a finalist even if D

Yes, and it should be enough to give the election away twice as
easily as in Approval, when the B voters mistakenly believe they
need to protect C.

> tries to use the pushover strategy.  Assuming another 29 or so of the
> remaining B voters rank C as second choice, C's win is guaranteed.  So
> the worst-case final result would be:
>   100   30  100  130   28
>    B    C    C    D    B
>    C    B              D
> Under Approval, a similar strategy is possible.  If B believes a loss to
> D is likely, then 59 or more of B's voters can also give approval to C. 
> Actually more than 59 would probably be needed, since we don't know how
> many of C's backers also gave approval to D; because of this it may
> actually be _more_difficult_to guarantee a Condorcet winner under
> Approval than it was under IRO.  Of course, D's backers could have given

In Approval the supporters of the middle candidate of 3 have
no reason to vote for anyone else. Either one of the extremes has
a majority, in which case it makes no difference how C voters
vote, or else C is Condorcet winner, in which case one of the
extremes needs C, and should be the one to share votes. Ands
that extreme would suffer more from the election of the opposite
extreme than the C voters would.

> some votes to C, but I am assuming D's voters know that it is in their
> interest to truncate.
>   130   29   71  158
>    B    C    C    D
>    C         D
> In this example, all of B's supporters (including those originally
> voting for A) are required to ensure a Condorcet winner if 71 of C's
> supporters give approval to D.  Any more than 71 and the CW loses.

Again, there's no reason to send Approval votes outward to the most
extreme candidate, for the reasons stated above.

> I would give a pairwise example, but tactical voting would probably
> result in a circular tie, and I'm not sure I even understand what the
> results mean at that point -- I suppose you can devise a tiebreaker to
> do whatever you want.  I assume that's where Votes Against comes in.

> Bart
> > 
> > And that isn't a rare special example. Though the middle candidate
> > of 3 will sometimes be the smallest, a voter distribution that
> > gives tapering support toward the extremes is especially plausible
> > & likely and in keeping with a normal distribution of voters
> > on the political spectrum.
> > 
> > So IRO not only fails the Condorcet Criterion & Monotonicity,
> > but it also fails the main standard that its proponents defend
> > it with: favoriteness.
> > 
> > Mike

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