Condorcet(x( ))

[Steve Eppley]

Mike O wrote:
>Steve E wrote:
>> Here's an illustrative example, a 3-way race between Good, Evil1, and 
>> Evil2.  One voter's ballot: {Good > Evil1 = Evil2}
>> I chose the candidate's names from this one voter's perspective.  
>> Other voters might see things differently, producing a close election 
>> and maybe a circular tie.  Not counting ballots like this against both 
>> Evil1 and Evil2 in their pairing might throw the election from Good 
>> to one of the evils.  
>
>Good. I fully admit that different results are gotten when we
>count un-expressed preferences. If you voted only for Good, and
>no one else, your truncated ballot could create a circular tie,
>even though Evil1 was Condorcet winner, and would have won
>had you voted it 2nd.

By Condorcet winner here, I assume you mean the beats-all-pairwise
candidate.  I'm not sure why this is relevant, since whether or not
Evil1 beats-all is independent of x.  I think what's relevant is
whether the effect of x on who wins circular ties is enough to cause
voters to change their votes (vote defensively).  So let's go on to
your next paragraph.

>If we count you as not voting any preference that you didn't
>_willingly_ vote, then Evil1 can't have a majority against it. If
>we invent a preference vote by you for Evil2 against Evil1, then
>you (unwillingly) can be making Evil1 be beaten with a majority
>ranking Evil2 over it. 

There's nothing in the definition of Condorcet that explicitly cares
about a majorities.  Is this relevant?  (Is it related to an emergent
property of Condorcet's method?)

Counting the Evil1=Evil2 vote against both Evil1 and Evil2 doesn't
change the winner of the Evil1 vs Evil2 pairing, but if Evil2 beat
Evil1 pairwise it would increase the size of this Evil1 defeat.  If
this defeat is Evil1's worst defeat then counting the vote this way
could affect the result of the circular tie breaker.

>Result: As you said, Good might win. That's bad. It means that your
>truncation has stolen the election from Condorcet winner Evil1. It
>means that I would have to quit saying that no defensive strategy is
>needed unless order-reversal is attempted.

To help me think about this more clearly, I'm going to substitute
Dole=Good, Clinton=Evil1, and Nader=Evil2.  Let's try the old example:
  46 Dole
  20 Clinton
  34 Nader>Clinton>Dole
                             x=0           x=.5            x=1
  Dole loses to Clinton    46 to 54      46 to 54        46 to 54
  Clinton loses to Nader   20 to 34      43 to 57        66 to 80
  Nader loses to Dole      34 to 46      44 to 56        54 to 66

In this example Clinton wins with x=0.  Dole wins with x=.5 or x=1.

>The possibility of mere truncation would be enough to make it
>necessary for Evil2 voters to rank Evil 1 equal to their favorite,
>insincerely. Drastic defensive strategy needed even without anyone
>using order-reversal.

This decision by Evil2 voters wouldn't hinge on having implausibly
precise knowledge of the expected votes, would it?  In this case, an
expectation of massive truncation by the Dole voters?  

In this example, the tally using x=.5 is very close.  The worst
defeats for each candidate are within a few votes (or percentage
points) of each other.  In the past, you've said it's okay to ignore
scenarios where implausibly precise knowledge is needed to affect
the voting.  This isn't one of those scenarios?  

If you were a Nader voter here, you'd vote Nader=Clinton>Dole just
because the Dole voters *might* truncate en masse?  You'd protect
Clinton even though Nader has a good shot at winning?  What if a few
Dole voters vote Nader>Clinton or a few Clinton voters vote
Nader>Dole, which is all it would take for Nader to win?  Are 
you saying these possibilities are less likely than the massive
truncation?

Perhaps we need a slightly different example, where the Nader support 
is much less, to illustrate why the Nader voters would be strongly 
tempted to ranked Clinton=Nader if x>0:

  48 Dole
  24 Clinton
  28 Nader>Clinton>Dole
                             x=0           x=.5            x=1
  Dole loses to Clinton    48 to 52      48 to 52        48 to 52
  Clinton loses to Nader   24 to 28      48 to 52        72 to 76
  Nader loses to Dole      28 to 48      40 to 60        52 to 72

If x=.5, here the Nader voters have a stronger incentive to abandon
their hopes for Nader and insincerely rank Clinton=Nader, if they
have enough predictive knowledge.  They may foresee a close race
between Dole and Clinton even though Clinton loses to Nader pairwise.  
So I guess I'm down to one question: does is take an implausible
level of information for the Nader voters to resort to drastic
defensive strategy, if x=.5 for the equally last-ranked pair? 

--Steve