[EM] Re: optimal Condorcet truncation

[Chris Benham]

Russ,
In my last post in this thread, I wrote:

> "A more useful criterion is the normal (as opposed to Mike-style)
> criterion taken from Blake Cretney's website:
>
> Name: Secret Preferences Criterion: SPC
> Application: Ranked ballots
> Definition:
> If alternative X wins, and some of the ballots are modified in their
> rankings below X, X must still win.
>
> Condorcet does not pass this criterion, which tells us that voters have
> incentive to truncate in some cases if not routinely."
>
> Woodall splits this somewhat oddly-named criterion into two fairly 
> self-explanatory others:
>
> "Later-no-Harm: adding a later preference to a ballot should not harm 
> any candidate already listed", and
> "Later-no-Help: adding a later preference to a ballot should not help 
> any candidate already listed".
>
> Condorcet passes neither of these, but your conclusion  only applies 
> to Later-no-Harm.
> In WV Condorcet (BP/RP/MM/River), the two LNHs are not in balance 
> (adding a later preference is more likely
> to help than harm an already listed candidate)  so that in the 
> zero-information case there is a random-fill incentive.
>
> As Kevin Venzke just more-or-less pointed out, the right 
> zero-information strategy in WV is to equal-rank the candidates
> above some ("the") approval cutoff point and to strictly rank 
> (random-filling if necessary) all the candidates below it. 

You responded:

> Interesting. Your telling me that adding a preference is more likely 
> to help than harm a higher-ranked candidate?

Yes.

> Can you prove that or point me to a proof?
>
>
In the great EM  Margins versus Winning Votes (formerly called 
"Votes-Against") debate (mainly between Blake Cretney
and Mike O.)  it was an undisputed point on the Margins side  that in WV 
in general  adding more votes to the winning side
of a pairwise comparison harms the loser more than it helps the winner.
Suppose there are 3 candidates and the voting method is MinMax (WV)  or 
one of the equivalent methods. If the voter's
favourite has a pairwise loss, then that candidate can only win if each 
of the other candidates also have a pairwise loss and
if  the pairwise comparison lost by the voter's favourite is the one 
with the fewest votes on the winning side.  So in that case
a faction of  voters that is indifferent between their two 
non-favourites does better to not truncate because by increasing
the "strengths" of  the non-favourites' defeats they might cause their 
favourite's defeat to be the weakest.

This scenario is more likely than the one in which with truncation there 
is a cycle that is won by the faction's favourite, but
random filling causes the candidate that pairwise beats the faction's 
favourite to also beat the other candidate and so become
the voted CW.

Methods that have the two LNHs probabilistically out of balance will 
either have a 0-info. random-fill incentive or else
(say for voters with a big enough gap in their sincere ratings of the 
candidates) a 0-info. truncation incentive.
One of  Woodall's criteria/"properties" is  "Symmetric-Completion".

"Symmetric-Completion:  a truncated ballot should be treated in the same 
way as its symmetric completion.
(The symmetric completion of a ballot is obtained by replacing it by all 
possible completions of it with equal weight, chosen so
that the total weight is 1. For example, if there are five candidates 
a,b,c,d,e, then the symmetric completion of a ballot marked
ab consists of six ballots, each with a weight of 1/6, marked abcde, 
abced, abdce, abdec, abecd, abedc.)"

IMO this isn't really a big deal in itself, but it seems easy to check 
and it implies No Zero-Information Strategy (NZIS), without
being a prerequisite for it.  It is met by Margins and IRV, as well as 
my two current favourite Condorcet plain ranked-ballot methods:
SCRIRVE and  Woodall's  CNTT,AV.

> And what if equal rankings are not allowed? 

Then WV Condorcet couldn't meet  Steve Eppley's "Non-Drastic Defense" 
criterion (and probably a similar one of Mike O.'s).

*/Non-Drastic Defense/:*  Each voter must be allowed to vote as many 
alternatives as s/he wishes tied for top, and if more than half of the 
voters 
vote some alternative /y/ (tied for) top, then no alternative voted 
below /y/  
by more than half of the voters may be chosen.

http://alumnus.caltech.edu/~seppley/Strategic%20Indifference.htm

I'm not sure that would be a huge loss.


Chris Benham
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