[EM] Re: Election-methods Digest, Vol 2, Issue 44

[Alex Small]

>Message: 1
>Date: Sun, 29 Aug 2004 19:03:58 +0100
>From: "James Gilmour" 
>Subject: RE: [EM] recommendations

>The facility for party voting in the Australian Federal Senate STV-PR elections is a 
>gross perversion of STV. It has reduced STV to just another party list PR system.
 
Correct me if I'm wrong, but my understanding is that people have the option of voting either their own preference order, or else a preference order that a party decided upon in advance.  I was under the impression that it was a response to the complexity of the system.
 
I understand all of the principled objections to party list systems, but in practice I think that the open list methods (especially the Swiss version) are good enough.  Or, put another way, I'd regard an open list system as a huge improvement over what we currently have in the US.  Progress is about improvement, not perfection.
 
 
 
 
As to Steve Eppley's comments on resolvability:
 
I like the definition from the social choice literature, but sometimes I prefer to work with continuous models rather than discrete models.  The definition that you posted really only works for discrete models.  Here's what I mean by a continuous model:
 
For N candidates I will consider an N!-dimensional space where each coordinate corresponds to the fraction of the electorate voting with a given preference order, and I'll restrict my analysis to an affine N!-1 dimensional manifold in that space where all of the coordinates are non-negative and sum up to 1.  I find that it makes a geometrical approach easier.
 
To ensure resolvability, I stipulate that the method must yield a definite result for every point on that manifold except a sub-manifold of N!-2 dimensions or lower.  Or, I'll use a short-hand that the method must give a definite result except "on a set of measure zero", with all necessary caveats and definitions implicit in that short-hand.


 
Date: Sun, 29 Aug 2004 11:46:25 -0700
From: "Steve Eppley" 
Subject: Resolvability (was Re: [EM] Re: Election-methods Digest, Vol
2, Issue 42)
To: election-methods at electorama.com
Message-ID: <4131C211.25986.5D9890 at localhost>
Content-Type: text/plain; charset=US-ASCII

Alex S wrote:
>> Steve Eppley wrote:
>>> Aren't all the voting methods we've been promoting 
>>> both anonymous and neutral? Doesn't that mean
>>> none of them are entirely non-random?
>
> My understanding is that anonymous and neutral methods only 
> need a non-deterministic component to break ties. When I 
> analyze methods mathematically (e.g. my never-ending quest 
> to prove that Strong FBC is impossible) I simply leave aside 
> the issue of ties by specifying that the method yields a 
> definite winner determined uniquely and deterministically 
> from the ballots submitted except "in rare cases", 

It can be misleading to leave aside the issue of ties
and tiebreaking. There may also be negative consequences
if, as a result, a poor tiebreaker is chosen. In 
particular, suppose the tiebreaker is not independent
of clones. Even though ties will be rare, it won't be 
known at nomination time whether the vote will be a tie, 
so it would be rational for every faction to nominate 
a slew of clones just in case there's a tie.

> where "rare cases" is made more precise depending
> upon the mathematical framework that I'm using. 
-snip-

There's a criterion in the social choice literature, 
resolvability, that's defined precisely and is useful for
distinguishing which methods rarely involve randomness. 
(For instance, Tideman provides a definition in his 
1987 paper on clone independence.)

Resolvability
-------------
A voting method is resolvable if and only if, 
for each possible collection of votes, 
one of the following two cases holds:

1. There exists a candidate that would be 
elected with certainty given those votes.

2. There exists a candidate x and an expression v
such that v would be an admissible vote if 
some voter voted v, and x would be elected
with certainty given the votes with v added.

In other words, if the voting method is resolvable, 
then every election is at most one vote away from an
election that would be tallied entirely non-randomly. 
It follows that if the number of voters is large,
then the fraction of possible elections that would
require some randomness to elect a single winner 
is very small.

I slightly prefer a slightly different definition 
of resolvability, changing the wording of case 2:

2. For each candidate x that has a non-zero chance
of being elected given those votes, there exists 
an expression v that would be an admissible vote 
if some voter voted v, such that x would be elected
with certainty given the votes with v added.

>>> Man, this stuff has got me really worried. It's just 
>>> scary how unconcerned the Republicans seem about 
>>> having a verifiable vote-counting process. 
-snip-

I didn't write that. Alex must have quoted someone else.

--Steve



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