2015 22:17:11 +0200

stevebosworth@hotmail.com; election-methods@lists.electorama.com;
jgilmour@globalnet.co.uk; fredgohlke@verizon.net

[EM] The 'post mortem' discussions on UK radio (from Steve)

Hi
Kristofer,
Perhaps to continue our
dialogue most efficiently, please tell me if I have properly
understood
your use of
'asymptotically' below.

S: Because you want
both a 'simple' and 'fair' electoral system, perhaps

would like to consider APR. It is referred to in my next comment to

would be

constituencies that remained

election. Consequently, rather than having to rank

than one candidate, each citizen would still have the option of

At the

guaranteed to continue

legislative assembly. APR seems to offer

throughout'. APR is almost as simple as party-list

systems but puts each citizen in control of to which
representative's

added.

K:
Unfortunately, because APR reduces to IRV, I can only consider it

asymptotically fair (that is, when you have enough seats compared to
the

S: Please correct me if
I have misunderstood you to mean the following:  The more reps to be
elected from a multi-winner district (from many more candidates) by
an electorate of many millions of citizens, the probability of
monotonicity failures (and thus 'unfairness') would becomes almost
fanishingly small.

If so, this would be
the case with APR because its general election, in effect, elects all
reps from one district (i.e. the whole country, even though it is
initially administer through all the single-member district in the
country).  Moreover, the probability of monotonicty failures
occurring with APR is made even smaller by its giving 'weighted
votes' to each rep in the assembly, i.e. the transferring of
so-called 'surplus votes' is not a part of APR's modified use of STV.

What do you think?

Steve

Could you please use the standard mail quoting format used on this list,
as given by bottom/interleaved style on
https://en.wikipedia.org/wiki/Posting_style ? That would render your
replies easier to read, be consistent with how I (and most of the other
participants of this list) quote, and in my case, would also make my
mailing program do the right thing when viewing or replying to your mail
in turn.

That is, instead of using S, K, and >, use > for the mail you're
replying to, > > (or >>) for what it replied to and so on. There is no
need to preface your new text with anything; just write it directly as
in the Wikipedia examples.

If you need to separate chunks of text, just add newlines as I have done
below.

Thank you :-)

Now onto the posting!

On 06/22/2015 03:03 AM, steve bosworth wrote:

I don't know how APR aggregates single-member districts into a national
election, so I can't comment on that. I know that the way that
single-member districts are aggregated can be very important; for
instance, regular old single-member plurality is not proportional while
biproportional representation (and Balinski's FMV) would be proportional.

I don't think it's the surplus transfers in STV that causes
nonmonotonicity. The general pattern for an IRV monotonicity failure is
(in the three-candidate case) that in the original election, X gets
eliminated and then Y wins against Z, but if you raise Y on some Z>Y>X
ballots, then Z gets eliminated instead and Y loses to X.

So it's the elimination that causes nonmonotonicity, and the
nonmonotonicity is evidence of a greater problem, which is a kind of
butterfly effect. I'll call this "sensitivity to initial conditions"
even though it's not precisely the same thing as this refers to in
dynamic systems.

In an election where B is in last place in the first round with 100
first preference votes and C is next to last by 100 + 10^-10, B would
get eliminated, and this could lead to a completely different chain of
eliminations than if B started with 100 votes and C with 100 - 10^-10.
See e.g. http://tinyurl.com/p3793pm which, though it focuses on full
STV, also emphasizes the effect of elimination to this sensitivity or
quasi-chaos[1].

Now, STV's surplus transfers will in effect transform a single STV
election into multiple related IRV elections. STV "sweeps across" the
IRV landscape with different ballot sets related to each other by the
transfers. It might be the case that the sweep makes STV hit multiple
IRV monotonicity failures which would make STV more prone to
monotonicity failures than IRV is. But it could just as easily be the
other way around (that the sweep makes STV dodge multiple monotonicity
failures), so I can't speculate on how often monotonicity failures occur
in STV versus IRV.

As for asymptotic fairness, I was thinking of weighted voting as being
similar to party list PR methods (without explicit thresholds). In a
party list method, if there are a fixed number of parties, then even
with an unbounded number of voters, as the number of seats approaches
infinity, you can get as close to complete fairness as you'd like[2].

More properly speaking, with a fixed number of voters and candidates,
the more seats you allow, the closer to complete fairness do you get
until you get complete fairness at number of seats = number of voters.
So perhaps "asymptotic" was the wrong word when we're dealing with a
finite number of voters.

For party list PR, there are two effects that contribute to this. First,
with more seats, you lessen quantization effects: the fraction of the
seats belonging to party A gets closer and closer to the fraction of the
voters who voted for A. Second, more and more parties get included since
there are room for more of the minor parties when you have lots of seats
and no threshold.

For weighted voting methods, the first factor is a non-issue because the
weights are exact. The second factor still holds. The more positions and
hence winners you have, the more candidates you can include. Unless
you're using an unusual way of assigning weights to candidates, it
should be clear that if the number of winners is equal to the number of
candidates, then you get fairness (by weight, at least) because every
candidate is represented, so it doesn't matter what sort of distortions
the voting method that picks the winners might have.

To be more concrete in the IRV setting: if you have 10 candidates and 10
winners, it doesn't matter if a distortion in IRV makes what should have
been the winner drop to fourth place. He's going to be included anyway.
If you have 10 candidates and 9 winners, only a distortion that pushes
someone who shouldn't have been last into last place matters, but if you
have 10 candidates and 3 winners, any candidate who should have been in
the top three but are pushed below the third candidate will suffer[3].

Whether this will make IRV-APR less prone to errors that matter would
depend on which force pulls strongest. As you make a district larger,
you get more seats (more winners), which would tend to point towards
distortions mattering less. On the other hand, you also add more people,
which means that there may be more candidates to begin with. If the
increase in the number of candidates grows faster than the increase in
the number of seats, then going to larger districts (or a nationwide
district) is not going to help. But if you get more winners relative to
candidates, it could.

9 winners out of 10 candidates behaves better than 3 winners out of 10,
but 9 out of 30 might not. So I prefer using a method where there are
fewer distortions to think about to begin with. If the method is
monotone and doesn't exhibit sensitivity to initial conditions, such
errors simply don't happen.

Also note that what I.D. Hill call "premature exclusion"[4] and others
call center squeeze may push candidates with broad support almost to
last place in IRV. So if you wish to preserve those, you might need a
lot of seats, whereas better methods do, well, better at that. That, in
my view, is another point against IRV, although I won't get into it in
detail here as it's kind of a tangent to my answer about asymptotic
fairness.

--

[1] On a side note, there's also something a little weird with IRV's
logic in that it implies first preference count is not a good way to
determine winners (or it'd just be Plurality) yet it is a good way to
determine losers.

[2] assuming you define complete fairness in the way that every voter
gets a representative from the party he voted first. If you want
proportional representation of coalition power instead (e.g. by Banzhaf
index), the picture's different, but I won't go into that here.

[3] I am not sure if "one winner and ten candidates" will have more
potential for error than "three winners and ten candidates". On the one
hand, any error might propagate up to the winner in the former case
whereas errors that shuffle the order of the three winners are
inconsequential in the latter case. On the other hand, there's much more
freedom in the latter: only situations involving raising the winner
counts as a monotonicity violation in the former case, but in the
latter, one may raise any of the three winners. So I don't know if the
convergence to fairness is straightforward.

[4] Quoting Hill: "Premature exclusion of a candidate occurs when
someone is the lowest because hidden behind another who, in the end, is
also not going to succeed. If A, who would otherwise have been elected,
fails because B stood and was elected instead, it is bad luck for A but
there is nothing disturbing about it in principle. If, however, A fails
because B stood, but then B does not get in either, that is disturbing."
[ Voting matters, Issue 2, September 1994:
http://www.mcdougall.org.uk/VM/ISSUE2/P2.HTM ]. Hill also makes the
point I made in [1].

On 06/24/2015 01:40 PM, Kristofer Munsterhjelm wrote:

Replying to myself here because I see that my email client made a
mistake of mixing blockquote and >. I'm going to set it to only reply as
plain text to fix that problem in the future. If you (Steve) need a
properly rendered version of the mail for quoting purposes, let me know
and I'll send you one.