Allow indifference? (was Re: [EM] Condorcet for public proposals - Tournament)

[Steve Eppley]

Kevin Venzke wrote:

> > > BTW:  Debatable whether voters should be permitted to 
> > > rank candidates as equal.
-snip-
> > That doesn't make any sense to me.   If two candidates are 
> > ranked, I think that neither should get the win -- 
> > at least if we're doing winning votes (wv)
> 
> Giving half-votes is the Margins (and Symmetric-Completion)
> interpretation. I suggest the same method be used for both
> expressed and truncated equal rankings.  I can't think why 
> there should be a difference in treatment. 

My Feasibility criterion requires allowing each voter to 
leave as many candidates unranked as she wants, since there 
might be a huge number of candidates. (Feasibility also 
requires the time to tally the votes must be a small 
polynomial function of the number of voters and the number 
of candidates.)

To satisfy certain criteria (e.g., my Minimal Defense 
criterion which is based on Mike Ossipoff's Strong 
Defensive Strategy criterion) and assuming voters may only 
express orders of preference, it's necessary to (at least) 
allow each voter to rank as many candidates as she wants at 
the bottom (or leave them unranked, which is to be treated 
as the bottom) and to count 0 votes, not 1/2 vote, for and 
against each candidate in the pairings of bottommost 
candidates.

It's possible to satisfy an even stronger criterion, and 
without having to worry about the "wv" vs "margins" issue,
if voters are allowed to insert a special dividing line in 
their orders of preference:

   Sincere Defense criterion
   -------------------------
   For all subsets of alternatives X and Y, if more than 
   half of the voters prefer every alternative in X
   over every alternative in Y, then there must exist 
   a set of admissible voting strategies for that majority 
   that does not require any misrepresention of preferences
   yet ensures all of Y will be defeated (and finish below
   all of X if the method produces a social ordering).

Both margins and "wv" methods are capable of satisfying 
Sincere Defense, if they're designed right.  For example, 
MAM can be tweaked as follows: 

   (See www.alumni.caltech.edu/~seppley for the definition 
   of MAM, a list of criteria it satisfies, and proofs.)

   Let SpecialMajorities denote the subset {(x,y) in
   Majorities such that the number of votes that rank 
   x over the dividing line over y exceeds the number 
   of votes that rank y over the dividing line over x}.

   When sorting the majorities (primarily from largest to 
   smallest in untweaked MAM) adjust the sort order so that
   the special majorities precede all other majorities
   (which means the majorities are now sorted secondarily
   from largest to smallest).

It is easy to show that the set of special majorities is 
acyclic, which means none of them needs to be discarded. 
(They can all be "affirmed", in MAM-speak.) 

I'm reluctant to propose the dividing line "enhancement" 
for use in public elections anytime soon, because I'm 
concerned many voters would try to use it as some sort of 
"sincere approval" threshold, rather than as the strategic 
device it's intended to be. (Note that the concept of 
"sincere approval" cannot be derived from conventional 
"rational choice" models of individual preferences.)  I 
would wait to propose it until years after society adopts 
good voting methods, to allow time for misleading 
absolutist concepts like "approval" to fade from the 
culture.

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