[EM] APR: Vote-wasting questions from Kristofer

[Kristofer Munsterhjelm]

On 11/19/2015 07:30 PM, steve bosworth wrote:
>  
>> To: stevebosworth at hotmail.com
>> From: km_elmet at t-online.de
>> Subject: Vote-wasting questions
>> Date: Mon, 16 Nov 2015 22:36:08 +0100
> 
> Kristofer asked Steve:
>  
>> - What does it mean for a method to not waste any votes, as you
>> define it?
>>
> I see a citizen's vote as wasted to some degree to the extent that
> it does not do what APR does.  APR enables each voting citizen to
> guarantee that her one vote will continue to count in the legislative
> assembly both quantitatively and qualitatively. Firstly, it will count
> mathematically through the weighted vote earned by the one rep who that
> citizen has helped to elect.  Consequently, this rep is also most likely
> to be seen by this citizen also as most accurately (qualitatively)
> representing her own scale of values.

If not wasting votes is a good thing that APR has and others do not,
then it's hard to know what makes APR good in this sense if you say
"votes are wasted to the extent that the method does not do what APR
does". But you do say that it matters that a voting citizen's vote
"continues to count both quantitatively and qualitatively", and that
gives me something to go on.

It does, however, seem to be underspecified in a sense that many methods
could claim to do so. Consider the following five alternate methods for
electing 10 winners from 100 candidates:

1. Count every candidate's first preference. Eliminate all but the 10
with the highest first preferences. Each of the 10 remaining gets weight
according to how many first preferences he has on the ballots where the
90 who didn't make it have been eliminated.

2. Count the Borda scores for each candidate. As long as there are more
than 10 candidates in the running, eliminate the candidate with worst
Borda count. Once only 10 are left, they get get weights according to
how many voters ranked them first among the 10.

3. As APR, but repeatedly eliminate the MAM loser rather than
the Plurality loser until only 10 candidates remain[1].

4. Each voter ranks the candidates and gives each candidate either a
thumbs up or down. Eliminate candidates with fewest thumbs up until only
10 remain. The remaining candidates get weight according to how many
voters ranked them first among the 10.

5. Each voter rates the candidates. Start the first round with no
winners. In each round, each voter gives his vote to the winner he rates
highest, and a winner's weight so far are the number of people who give
a vote to him.
Repeatedly find the candidate that would get the greatest weight
("steal" the most votes) if he were to be added to the group of winners.
Add him to the group of winners and go to the next round. Break ties by
total rating. After the tenth round, you have 10 winners  and their weights.

If "a citizen's vote [is] wasted to some degree to the extent that [the
method] does not do what APR does", then these should waste votes where
APR does not. But how do they waste votes? How are they less
proportional than the method you're using for APR?

I'm not saying these methods are good or bad - those are just ones that
came to mind - but in every method there, each voter contributes to the
weight of the candidate he prefers. The objection to Plurality and
unweighted vote methods thus doesn't apply.

If you have more candidates than winners, you have to narrow down the
field somehow. No matter how you do so, you can later weight those who
remain according to how many voters ranked or rated them first among the
10. So what makes one method of narrowing down the field better than
another in a vote-wasting sense? It would seem that the idea of
"guarantee[ing] that her one vote will continue to count in the
legislative assembly both quantitatively and qualitatively" is
independent of the exact way you do the mathematical count so long as
the method is at least somewhat reasonable.

> In this context, each citizen's votes in plurality systems like those in
> the USA and UK can be guaranteed only of her one vote being recorded as
> cast for one of the candidates running in her electoral district.   Her
> vote may have been given to a losing candidate, in which case she will
> not be represented in the assembly.  Even if her vote has helped the
> winning candidate to win, she may not see that candidate as the one in
> the country who would represent her most accurately.  Even if she does
> like the elected candidate, her vote would not give this representative
> any more power in the assembly if this winning candidate received more
> than one vote more than he needed to win the plurality election. 

True, Plurality is infamous in that regard, and APR is better than
single-member Plurality. But so are a lot of methods.

There would also be situations where no method, not even APR, can ensure
full representation. E.g. two winners:

10: A
9: B
8: C > D

However you do it, at least 8 voters will end up not being represented.
Doesn't that also represent an (albeit unvoidable) waste of votes? Yet I
think it would be wrong to insist upon full ballots (as Australia does)
even if it would waste fewer votes.

-

[1] Such a method might use a logic that if MAM is better at finding a
deserving single winner than Plurality is, then MAM run in reverse
should also be a better method at finding a justified loser than
Plurality is. In addition, unlike Plurality, if you ask the voters to
rank from best to worst and then run the ballots through MAM, the
candidate who comes last according to MAM will always be the candidate
who comes first if you ask the voters to rank from worst to best and run
*those* ballots through MAM (and the voters have the same opinion in
both cases). So in one sense, the MAM loser is more consistent than the
Plurality loser. Furthermore, if you run method 3 until there is only
one candidate left, that candidate is always the MAM winner assuming no
ties.