[EM] (2) UK 'post mortem': Steve's 2nd dialogue with Kristofer

[Kristofer Munsterhjelm]

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:
>  > Date: Thu, 18 Jun 2015 22:17:11 +0200
>  > From: km_elmet at t-online.de
>  > To: stevebosworth at hotmail.com; election-methods at lists.electorama.com;
> jgilmour at globalnet.co.uk; fredgohlke at verizon.net
>  > Subject: Re: [EM] The 'post mortem' discussions on UK radio (from Steve)
>  > On 06/18/2015 05:49 PM, steve bosworth wrote:
>
> Hi Kristofer,
>
> Perhaps to continue our dialogue most efficiently, please tell me if I
> have properly understood your use of 'asymptotically' below.

> > Unfortunately, because APR reduces to IRV, I can only consider it
> > asymptotically fair (that is, when you have enough seats compared to the
> > number of candidates).
>
>
> 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.

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].