[EM] Generalized symmetric ballot completion

[Kristofer Munsterhjelm]

Peter Zbornik wrote:
> Juho,
>  
> summarize my argument concerning generalized ballot and generalized 
> ballot completion and in the end of this email I suggest a new 
> single-member Condorcet election system.
>  
> Nomenclature: I think that "null-candidate" (marked "X") is a fitting 
> name for voting for not filling a seat. The other names given do not 
> have that chique mathematical sound: "White", "None of the Above", 
> "Re-open nominations", "Ficus (the plant)", etc.
>  
> In the discussion, I think I showed the following
> If blank voting ("null candidates") is not allowed, then 
> truncated/incomplete ballots give different election results for winning 
> votes and for margins.
> Compare Kevin Venzke's example:
> 35:A>B
> 25:B
> 40:C
> If we complete this election (Woodall's original proposal) to
> 35:A>B>C
> 25:B>A=C
> 40:C>A=B,
> then the election gives different results whether the candidates in the 
> ties are resolved as 0.5 vs 0.5 (margins - A winner) or 0 vs 0 (winning 
> votes - B winner)
> (compare the results of the election at 
>  http://condorcet.ericgorr.net/ and 
> http://www1.cse.wustl.edu/~legrand/rbvote/calc.html)
>  
> For margins, Woodall's plurality criterion is violated.
>  
> If the same election is completed to allow for blank voting:
> 35:A>B>X>C
> 25:B>X>A=C
> 40:C>X>A=B,
> then the election gives same result (B - winner) both for margins and 
> for winning votes and the parwise comparison matrix will be identical 
> for both methods if a an equality  awarded 0.5 votes for both candidates.
>  
> Thus, truncated/incomplete ballots can be completed using the 
> following generalized symmetric ballot completion algorithm, in order to 
> give same election results for margins and winning votes and to not 
> violate Woodall's plurality criterion for margins:
> 1.  add s "null candidates" under the ranked candidates, where s is the 
> number of seats
> 2.  rank the unranked candidates equally and under the "null candidate".
> 3.  equalities are resolved by giving each candidate 0.5 votes in the 
> pairwise comparison.
>  
> If margins are used in Condorcet elections with generalized symmetric 
> ballot completion, then Woodall's plurality criterion is not violated, 
> since the "blank votes" are actually represented and the ballot is complete.
>  
> Maybe the entry in Wikipedia could be updated, where we read "Only 
> methods employing winning votes satisfy Woodall's plurality criterion 
> <https://mail.google.com/wiki/Plurality_criterion>."
> http://en.wikipedia.org/wiki/Condorcet_method#Defeat_strength
>  
> I think an equality on the ballot between two candidates A=B should 
> intuitively mean nothing else than giving half a vote to A>B and B>A, 
> i.e. the pairwise comparison matrix should not change and Woodall's 
> plurality criterion should be kept at the same time. This is only 
> possible if the generalized symmetric ballot completion algorithm is used.
>  
> The rule of requiring the candidate to score more than 50% in a pairwise 
> comparison which I proposed in a previous email is enforced if 
> generalized symmetric completion is used.

How would you deal with Condorcet cycles? Is the rule "must score more 
than 50% in at least one comparison"? If not, there could be problems 
when no beats-all winner exists.

> If we use generalized ballot completion, then the null-candidate wins in 
> a Condorcet election (but not in an IRV election):
> 45:A>X>B
> 40:B>X>A
> 15:X>A=B
> Woodall's plurality criterion is not violated because X is not a 
> candidate to win a seat.
>  
> Introducing a cutoff, like saying that "a winning candidate needs to be 
> explicitly ranked on 50% of the ballots" maybe is equivalent to the 
> generalized ballot completion algorithm (I don't know). However such a 
> cutoff doesn't allow for ranking between disfavoured alternatives, which 
> the generalized ballot does.

It seems you've rediscovered the concept of an approval cutoff, but in 
another context. There are two sorts of approval cutoff: implied and 
explicit. In the former case, truncation means that you don't want those 
you didn't rank to win. In the latter, there's an explicit mark - your X 
- after which the same holds true.

Implied cutoff methods are less expressive but they can guard against 
burial. Kevin Venzke shows how this works in Condorcet//Approval: you 
can't effectively bury a candidate because ranking him last means you 
give him approval, which strengthens his position. I imagine that like 
with explicit cutoff, you could bury more powerfully by saying something 
like A>B>D>X>C instead of a honest A>B>C>D (or strategic A>B>D>C).

As for whether X wins whenever the winner is not ranked above him on a 
majority of the ballots, that would be a majority version of the Pareto 
criterion. "If a majority ranks A above B, then B must not win". But we 
know that it's possible for a majority to rank A above B, another 
majority rank B above C, and a third majority rank C above A. Therefore 
I don't think it's equivalent, although I can't find an example.
(In other words, a winning candidate could be not-X even when X is 
ranked above him by a majority, but then this candidate must be in a 
cycle with X.)

> I aggree that it is better to require the voter to rank all candidates, 
> as an incomplete ballot is completed in any case and the voter might not 
> know the ballot completion algorithm.

On the other hand, forcing voters to rank all candidates can lead them 
to taking the easy way out - in the short term by donkey-voting and in 
the long term by encouraging (or at least not resisting) a change of the 
ballots to include above-the-line voting. Both of these things have 
happened in Australia, where the type of IRV they use means that voters 
have to rank every candidate, and the consequence is that the parties 
wield greater power.

> I don't think that introducing a null candidate in a Condorcet election 
> has any impact on its violation of Later-no-harm, i.e..the incentive of 
> the voter to bullet-vote to maximize the success of "His" 
> candidate. Even if the equalities and null candidates would be 
> disallowed on the ballot, later-no-harm would still not hold for 
> Condorcet elections and burying would still be an efficient strategy 
> (slightly OT: the claim that Condorcet methods elect centrist canidates 
> is questionable, since the centrist candidate will be the prime target 
> for burying attempts, since he/she has the highest chance of winning, 
> thus losing his "centricity" even before it is measurable in a election).

I think I found out earlier that to immunize completely against burial, 
you'd need both LNHarm and LNHelp, because there are two ways burial 
could work, and one is blocked by LNHarm whereas the other is blocked by 
LNHelp. The Condorcet criterion is incompatible with both, in any event.

Your OT would probably hold no matter what tendency Condorcet has. Say 
that Condorcet tends to elect winners of type K. Then type-K candidates 
are the most visible targets, and thus could be buried if the electorate 
is strategic enough.

> Some ideas:
> An other interesting issue, is if election systems with several election 
> election rounds can improve results in Condorcet elections, for 
> instance, an STV Condorcet election could be held with three seats.
>  
> Those who get one of the seat go through to the second round (which 
> maybe can be automatical), where one of the candidates is elected in a 
> Condorcet election, where a Condorcet winner is guaranteed.
>  
> Maybe an election type could be devised which makes a bottom-up 
> proportional ranking. At the start of the election, as many seats as 
> there are candidates are elected, then in each subsequent round one 
> candidate is dropped util we have a Condorcet winner. 

I have a hunch (as you put it :-) that you can't both have Droop 
proportionality and monotonicity. My conjecture is based on that the 
apportionment/party list version of these methods (largest remainder 
Droop) fails population-pair monotonicity, i.e. transferring votes from 
one party to another can in some cases make the former party gain seats. 
I have not been able to prove this, but if I'm right, that means that 
runoff methods based on STV (even Condorcet STV, if it passes the Droop 
proportionality criterion) would be nonmonotone.

Further, such methods wouldn't be cloneproof. To slightly alter an 
example of Woodall's, consider:

700: x
300: y
   1: z

and two winners should elect x and y. But now clone x:

350: x>x'
350: x'>x
300: y
   1: z

Then the DPC forces the two winners to be x and x', so cloning can push 
other candidates out of the runoff. Perhaps a better runoff algorithm 
would be something akin to LeGrand's minmax approval. Minmax approval 
picks the set of candidates for which the candidate who disagrees the 
most about the composition of the council, as measured by Hamming 
distance, disagrees the least. I'm unsure how to generalize it, however.

> Do you or anyone else around on this list have a reference to where the 
> debate between IRV and Condorcet stands today (pros and cons of the 
> methods respectively)?
>  
> Personally I am not yet convinced that Condorcet is a "better method" 
> than IRV when it comes to resisting tactical voting.

I don't have a reference. You're probably right that IRV in general 
resists strategy better than does the ordinary Condorcet methods 
discussed on this list (like Schulze, Minmax, or Tideman); but note that 
James Green-Armytage's paper shows Smith-limited IRV to be as good as, 
if not better than, ordinary IRV with respect to strategy resistance, 
while always choosing winners from the Smith set (and thus always 
electing the CW where it exists).

His working paper is here: http://www.econ.ucsb.edu/~armytage/hybrids.pdf

(One does pay for this resistance by having a method that fails 
monotonicity, though.)