Condorcet Truncation Example

[Hugh Tobin]

Markus Schulze wrote:
> 
> Here is an example to show why voters would truncate, if
> a Condorcet Criterion Method is used:
> 
Thank you for posting the useful example, but I think it fails to show
as much as you claim. See below.

> Case 1:
> 
> 47 voters vote ABC.
> 10 voters vote BAC.
>  8 voters vote BCA.
> 35 voters vote CBA.
> 
> A:B=47:53.
> A:C=57:43.
> B:C=65:35.
> 
> B wins against A and against C in the pairwise comparison. Thus B
> is the Condorcet winner.
> 
> Case 2:
> 
> Now the 47 voters, who prefer A most, do truncate.
> 
> 47 voters vote A.
> 10 voters vote BAC.
>  8 voters vote BCA.
> 35 voters vote CBA.
> 
> A:B=47:53.
> A:C=57:43.
> B:C=18:35.
> 
> Now there is a tie between A, B, and C. Whether A is elected,
> depends on the used tie breaker. But if Condorcet/Smith is
> used, then A is elected.
> 

A is elected if the tiebreaker is based on margin of defeat, or if it
counts the truncated ballots as half-votes each way between B and C.
(Using the tiebreak most popular on this list, B still would win in Case
2.) The Smith criterion, as I understand it, does not break ties; it
defines who is tied, and in your case all candidates are in the Smith
set.  If there were others (D,E and F) each of whom lost to each of A,B
and C in pairwise races, then Smith would say that whatever tiebreaker
is used must decide only among A, B and C.

But the main question is, does this example show that truncation is a
rational strategy, assuming a tiebreak system such that A wins in your
example?  I submit that is does not.  If I am an A voter, and I am
confident enough of A's plurality and of the support for A over C among
B voters, so that I want C to beat B for tactical reasons even though C
is my last choice, then surely I will vote ACB.  Truncating is a halfway
measure, which requires much greater participation among the A voters in
order to be successful.  To put it in concrete terms: suppose my wife
and I deviously truncate (vote A only) in hopes of winning a circular
tie, and one occurs, but C's margin over B turns out to be one vote less
than B's over A.  Wouldn't we kick ourselves for not using our voting
power for second place to give maximum help to A, i.e., for not voting
ACB?  And if we (or A's campaign staff) are smart enough to think of
voting tactically, wouldn't we think of that possibility in advance?

Of course, if I have certain knowledge of all votes by B and C
supporters, and if I control absolutely the 47 A votes, then I can
truncate those votes, or reverse order (ACB) on all 47, or reverse order
on half of them, all with the same result.  But as a voter in the real
world I face uncertainty -- a probability distribution of expected
totals in the pairwise races.  Given that uncertainty, I want to vote so
that my candidate will have the maximum probability of winning, adjusted
for the risk of electing my least favorite.  If the expected benefit of
changing from ABC to A is worth the marginal increased risk of electing
C, then how is it that the expected benefit of changing from A to ACB is
not worth it?  
This is another way of putting the question that I posed to Steve
Eppley: what plausible probability distribution of expected votes by
others could I have that would make it rational for me (in Condorcet
with a margins-of-defeat tiebreaker) to use truncation as a strategy
instead of reversing order?  This question still awaits an answer.  

I believe the question with regard to tactical voting in Condorcet is
whether opportunities for order-reversal, given the great risk involved
in that strategy, are serious enough so that some feature should be
added to the system to reduce them (the possibility of tactical voting
cannot be excluded in any system, and with the possible exception of
Smith//Random I believe the other systems discussed on this list create
more incentives to insincere voting).  Even with a tiebreaker that is
not, as Steve Eppley would say, "truncation resistant," truncation as a
tactic has no significance independent of order-reversal, simply because
it is equivalent to order-reversal by only half the voters who
truncate.  Therefore short ballots are much more likely to reflect true
indifference than tactical voting, and what is called "truncation
resistance" in a tiebreaker really should be thought of as a penalty
against the voter for not making up his or her mind about a second or
lower choice.

Nonetheless, if you consider deterrence against truncation as a tactic
important, then I believe you should consider endorsing what has been
called "Smith//Condorcet [EM]" on this list, in which the tiebreak is
based on the total number of votes against (without half-votes for
truncated ballots), and which would elect B, not A, in your truncation
example (Case 2) above -- thereby penalizing the sincerely indifferent A
voters (as I believe them to be) for not making tactical second-place
choices.

I have previously explained why I think Condorcet would want A to win in
your Case 2, and the reasons why I agree.  

Regards,
-- Hugh Tobin

> I believe, that to every tie breaker method it is possible to
> create an example, where truncation makes sense.
> 
> Markus Schulze (schulze at speedy.physik.tu-berlin.de)