[EM] Hybrid/generalized ranked/approval ballots

[Kristofer Munsterhjelm]

Peter Zbornik wrote:
> Dear all,
>  
> Please let me return to an older discussion (see emails below).
> The issue of the hybrid ballot A>B=C>D.
> Just an idea on this topic, which might be worth mentioning.
> It could be a way to handle the problem of bullet voting.
> Ant it could be a way to disband the dichotomy between different 
> criterias of winning in condorcet elections (margins, winning votes, 
> quotas losing votes).
>  
> 1] IRV-based elections:
> Basically in IRV-based STV, when arriving at an equal sign in the 
> ballot, the ballot could simply be split into the number of 
> candidates with equal preferences and re-weighted accordingly (i.e. for 
> instance A=B=C would give three ballots, A>B=C, B>A=C, C>A=B, each with 
> weight 1/3 of the original weight).

This sounds a lot like Woodall's concept of "symmetric completion". A 
method passes symmetric completion if truncated ballots are split into 
ballots with the latter (truncated) preferences filled out, for all 
possible ways those can be filled out, and with the same cumulative 
power. E.g. with candidates A,B,C,D and a method satisfying symmetric 
completion,

1: A>B

is the same as

0.5: A>B>C>D
0.5: A>B>D>C.

Unless I'm mistaken, you're generalizing symmetric completion to equal-rank.

Woodall writes about symmetric completion here: 
http://www.votingmatters.org.uk/ISSUE3/P5.HTM , where he also shows that 
STV does not obey that criterion, but that IRV does. In another Voting 
Matters article (http://www.votingmatters.org.uk/ISSUE14/P1.HTM ), he 
shows how STV can be made to obey symmetric completion, but says that 
doing so isn't a good idea.

In a more general sense, there are two possible ways to handle equal 
rank in a weighted positional system. I think the first has been called 
"whole" and the second "fractional" on the list - that is at least the 
names I use in Quadelect.
If the method is "whole" (or ER-, e.g. ER-Plurality), equal ranks give 
the same point value to every candidate that is equal ranked. With 
ER-Plurality you can simulate approval, for instance, by simply voting 
all approved candidates equal first, ahead of all not-approved candidates.
If the method is "fractional", equal ranks distribute the point score 
over all the candidates equally ranked. Equally ranking k candidates 
first in Plurality would give each 1/k of the ballot's weight, and if 
I'm not mistaken, this is equivalent to generalized symmetric 
completion. You can simulate cumulative voting with fractional Plurality.

> Condorcet-based elections:
> In Condorcet elections (including STV) then A=B would simply mean 0.5 
> wins for A>B and 0.5 wins for B>A.

That's what Margins does. As a consequence, methods based on Margins can 
meet symmetric completion, but methods based on WV can't. However, 
Margins methods can't meet the Plurality criterion whereas WV can.

> Kevin Venzke wrote in his mail below (May 9th 2010):
>> 35 A>B
>> 25 B
>> 40 C
>> A will win. This is only acceptable when you assume that the B and C
>> voters meant to say that A is just as good as the other candidate that
>> they didn't rank. I don't think this is likely to be what voters expect.
>> It seems misleading to even allow truncation as an option if it's treated
>> like this.
> End of quote
>  
> Well I think think that as a voter I would indeed be pleased if A would 
> win and not C.

The example above shows how Margins can fail to meet Plurality. The 
Plurality criterion says that if some voter X has more first place votes 
than Y has *any* place votes, then Y shouldn't win. Yet that's what 
happens above:
C has 40 first place votes. A has 35 any place votes, yet A wins. 
Margins elects A. Any other method that does, also fails Plurality.