[EM] Re: Pairwise winner of the 2 least-beaten

[Mike Ossipoff]

Steve Eppley writes:
> 
> Mike O. wrote:
> >That wouldn't work out, picking the pairwise winner of the two
> >least-beaten candidates. Say Clinton is Condorcet winner in a
> >Buchanan, Clinton, Nader race. And say the Buchanan voters are so
> >devious that they rank Nader over Clinton, to create a circular tie,
> >hoping that Buchanan will win the circular tie.
> 
> To what did your message reply?  You changed the subject line and
> didn't quote anything from the original.


I'm going to start making more use of copies in my replies.

You suggested that circular ties be solved by determining which 2 
candidates are the 2 least-beaten ones, and choosing, as the winner,
the one of those two who beats the other in a pairwise comparison. It's
to that proposal that that letter is a reply.

> 
> >The Buchanan voters know that Buchanan beats Nader, and so they
> >make Nader beat Clinton real big.  So big that Buchanan & Nader are
> >the least beaten candidates.
> 
> How does Buchanan beat Nader if a majority rank Buchanan last?

No majority ranks Buchanan last. A majority ranks Clinton over Buchanan.
Buchanan beats Nader because either he has more 1st choice voters,
or he has help from the Clinton voters, or both.
If Buchanan has more 1st choice voters than Nader, then he of course
beats him unless the Clinton voters favor Nader sufficiently.

> 
> I tried playing with some numbers but didn't get them to come out
> meeting all your conditions, then got tired.  <grumble>  Could you
> post some vote numbers to make your examples (now and in the future)
> easier to follow?  They'll make good test cases that all the
> algorithms can be judged on.


Example:

40%: Buchanan, Nader, Clinton  (order-reversal cheating attempt)

25%: Clinton   (defensive truncation to thwart order-reversal)

35%: Nader, Clinton  (the side needing the compromise votes for him)

A 60% majority prefer Clinton to Buchanan. Buchanan beats Nader
40 to 35.

***

What happens in this scenario, using the 2 circular tie solutions,
Condorcet's method & the one that picks the pairwise winner of the
2 least beaten candidates?

Circular tie: Nader beats Clinton beats Buchanan beats Nader.

First, by Condorcet's method, the Buchanan cheaters can't do anything
about the fact that Buchanan has a majority against him--a 60%
majority (Clinton + Nader) have voted Clinton over him. Nor do they,
on their own, have the power to make anyone else be beaten with a majority
against him. By ranking Nader over Clinton they, with the Clinton voters'
help, cause Clinton to be beaten with 75% against him. That makes Clinton,
by far, the most beaten candidate. But, not having the help of the Clinton
voters, the Buchanan voters, who don't constitute a majority, have no
way of making Nader be beaten with a majority against him. So Nader only
has 40% against him. Therefore Nader, with only 40% against him, is the
the winner by Condorcet's method. 

The defensive truncation strategy by the Clinton voters has made it
impossible for the Buchanan voters to gain by order-reversal, and has,]
in fact soundly punished them for trying it. The mere knowledge that
the Clinton voters expect order-reversal, and (especially since they
have no real reason to vote for a 2nd choice) are going to truncate,
will deter order-reversal.

***

Now, what about the count rule that says to find the 2 least beaten
candidates, and pick as the winner the one who beats the other in a
pairwise comparison?

As I said, Clinton is, by far, the most beaten candidate, meaning that
Buchanan & Nader are the 2 least beaten ones. Of those 2, Buchanan
beats Nader, and therefore wins by that rule.

That means that the offensive strategy of the Buchanan voters, the
order-reversal cheating, has succeeded. The  mere fact that they
can beat Nader means that they can win in this way if they add their
votes to those of the Nader voters in order to make Clinton be
_badly_ beaten.

***

So my point was that the offensive strategy of order-reversal cheating
isn't rewarded in Condorcet's method, when a simple defensive strategy
is used. Since Clinton voters really have no reason to vote a 2nd choice,
no one would really order-reverse in this 3-way election, using Condorcet's
method, though they could get away with it much more easily in other methods.

***

Mike

> 
> --Steve
> .-
> 


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