2nd example of manipulation in Condorcet's method

[Steve Eppley]

Bruce A wrote:
--snip--
> 26,000:  A,(B&C)
>  2,000:  (A&B),C
> 20,000:  B,A,C
> 14,000:  (B&C),A
> 31,000:  C,(A&B)
>
>  6,000:  B,C,A
>If those 6,000 voters vote honestly, then their second choice, C,
>will win.  If those 6,000 voters do not vote at all in that election
>(e.g., they "stay home"), then their first choice, B, will win.

I've worked through the example's arithmetic and my results agree 
with Bruce's:

Sincere:          Stay home:
A>B  26           A>B  26
B>A  40           B>A  34*

A>C  48           A>C  48*
C>A  51*          C>A  45

B>C  28           B>C  22
C>B  31*          C>B  31*
------------      ------------
C beats all.      B breaks tie.
(Numbers in thousands.  The * marks a pair-loser's worst votes-against.)


Whether they stay home or vote {B>C>A}, B beats A and C beats B.  
But the stay-or-vote decision affects the A vs C pairing: voting
makes C beat A, which makes C the beats-all winner.  Staying home
lets A beat C, a circular tie where B wins because there are only
31000 votes for C>B (B's worst defeat).

Note that 40000 voters rank B=C, a large number which is more than
rank B>C or C>B.  It's this large indifference to B vs C which keeps
the size of C>B (B's worst defeat) small.  But if a significant
fraction of these B=C voters had slightly changed their votes so 
the 40000 instead breaks down as, say:
  32,000: B=C
   4,000: B>C       (Note: I'm not showing their complete ballots.)
   4,000: C>B
then by staying home the 6000 would throw the election to A, their 
*worst* result.  

I don't know how disturbing I should find this scenario.  I wasn't
bothered by Bruce's previous example, in which two voters had
unusual access to last minute accurate polling data, because it 
was sufficiently implausible imho.  

This new example also raises questions of plausibility: could the
6000 keep their poll results secret, and keep their strategic voting
plot secret, from the other 93000?  Can the 6000 really trust the
"40000 B=C" poll data to hold up unchanged in the actual voting (a
high risk proposition)?  

In the following quotes, Bruce wrote 2000 but I'll assume he meant 
6000.

>If they decide to vote, but to truncate instead of voting honestly
>(i.e., to vote:
>  2,000:  B,(A&C)
>instead of:
>  2,000:  B,C,A,
>even though they strongly prefer C over A), then B wins again.

This is a more important scenario than the stay-home scenario, imho.
But like the above, it's risky to rely on the "40000 B=C" poll data.

Sincere:          Truncate:
A>B  26           A>B  26
B>A  40           B>A  40*

A>C  48           A>C  48*
C>A  51*          C>A  45

B>C  28           B>C  28
C>B  31*          C>B  31*
------------      ------------
C beats all.      B breaks tie.

If 9001 of the 40000 B=C voters instead vote C>B, then A wins if the
6000 truncate.  How much reliance can be placed on '=' rankings
shown by polls?  Less than '>' and '<' rankings, I'd assume.

Nevertheless, I'm bothered by this scenario.  It looks like it
doesn't fit Mike O's description of what it takes (massive,
implausibly secret order-reversal) to beat a sincere beats-all
winner.  I'd like to hear Mike's comments on this; is it an 
implausible scenario?

>Of course, if they truncate, they would be the only voters who
>don't express a preference between A and C, and so they might decide
>to reverse C and A on their ballots instead of voting honestly
>(i.e., to vote:
>  2,000:  B,A,C,
>instead of:
>  2,000:  B,C,A,
>even though they strongly prefer C over A).  B wins in this case
>also.

Sincere:          Reverse:
A>B  26           A>B  26
B>A  40           B>A  40*

A>C  48           A>C  54*
C>A  51*          C>A  45

B>C  28           B>C  28
C>B  31*          C>B  31*
------------      ------------
C beats all.      B breaks tie.

Looks to me like Bruce included this reversal scenario more for
completeness than to make a point, since reversal may be even more
suspicious than truncation.  Although, there are also 20000 sincere
{B>A>C} voters so these reversers may blend in well.

My comments and questions regarding the reversal scenario are
basically the same as in the truncation scenario above. 

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)