[EM] (3) Vote-wasting questions: Steve 3rd dialogue with Kristofe

[Kristofer Munsterhjelm]

On 01/22/2016 11:28 PM, steve bosworth wrote:
> 
>  Re: [EM] (3) Vote-wasting questions: Steve 3rd dialogue with Kristofer
> 
>  
> 
>  
> 
>> From: km_elmet at t-online.de
>> Subject: Re: [EM] (2) Vote-wasting questions: Steve 2nd dialogue with Kristofer
>> To: stevebosworth at hotmail.com; election-methods at lists.electorama.com
>> Date: Thu, 7 Jan 2016 01:16:23 +010

>> 
>> For something like
>> 
>> 10: A>B
>> 11: C>B
>> 12: D
>> 
>> and two seats, electing A and C wastes votes (12 of them to be exact),
>> but electing B and D doesn't.
> 
>  
> 
> S: No.  In this case, APR would elect C with a “weighted vote” of 11 and
> D with a weighted vote of 12.  The 10 votes given to A would be wasted
> only by ordinary IRV using “weighted votes”.  APR would not waste these
> 10 because it gives each citizen who fails to rank any candidate that is
> elected the option of requiring her 1^st choice but eliminated candidate
> to transfer her one vote to the elected candidate who that eliminated
> candidate trusts most (e.g. see the Sample Secret Ballot at the end of
> the article).  However, perhaps I have never sent you a copy of my
> article that systematically explains APR:  “Equal Voting Sustained”. 
> Separately I will send it to you now and to anyone else who might
> request it.  Perhaps together with the above additional explanation, my
> APR proposal will be clearer?

Thanks for doing so. So your definition is, if I've understood it right:

A voter's vote is not wasted if either
1. one of the candidates he ranked above last is a winner, or
2. none are, and the voter lets his first preference choice give his
vote to someone who is a winner.

(end definition)

Suppose for simplicity that every voter lets his first preference choice
delegate his vote in that manner (otherwise, it's impossible to not
waste votes), and that every voter's first preference has a winner that
he can tolerate (again, otherwise it's impossible).

In the past posts, I first said asked about examples since it seemed to
me that the condition of not wasting votes was weak enough to let many
methods could pass it. Then you showed what you meant in a more detailed
way, to which I replied that the resulting property was both impossible
to satisfy completely, and in some cases undesirable even if relaxed.

Now you've further specified the criterion, and I hope the definition
above is accurate. That further specification does solve the issue of
impossibility and undesirability, but it simply moves the problem back
into the old situation where "not wasting votes" is underspecified (i.e.
the property is weak).

Consider the Plurality alternative where you mass-eliminate the n-k
candidates with fewest Plurality votes. That is, if there are 20
candidates and 10 seats: the method eliminates the 10 candidates with
the fewest first preference votes in one blow at the first round, then
elects the 10 remaining as winners.

You correctly said that such a method can waste votes (as I understood
it in the original definition) because there may be voters who voted
only for the 10 that got eliminated. However, given the fix above, that
"each citizen who fails to rank any candidate that is elected [is given]
the option of requiring her 1^st choice but eliminated candidate to
transfer her one vote to the elected candidate who that eliminated
candidate trusts most", it's pretty easy to patch the Plurality method
too. You end up with something like:

1. Eliminate the 10 candidates with fewest first preferences.
2. Assign each voter who ranked any of the 10 remaining winners to the
winner he ranked first.
3. Ask each voter who failed to rank any of the 10 remaining winners the
option of requiring the first ranked on his ballot to transfer his votes
to one of the 10 remaining candidates.
4. Once done, no votes are wasted.

Even a most unsatisfying "upside-down" method could be made to waste
no votes:

1. Eliminate the 10 candidates with the *most* first preferences.
2. Assign each voter who ranked any of the 10 remaining winners to the
winner he ranked first.
3. Ask each voter who failed to rank any of the 10 remaining winners the
option of requiring the first ranked on his ballot to transfer his votes
to one of the 10 remaining candidates.
4. Once done, no votes are wasted.

Thus, clearly this fixed criterion can not distinguish good methods from
bad ones by itself. It is weak (or ambiguous) in the sense that it
admits both good and bad methods. It's not a bad thing for a property to
be weak in itself - for instance, the monotonicity criterion is quite
weak - but that does mean that it cannot alone be used to determine
whether a method is good or not.

You could try to firm up the property by counting direct non-wasted
votes more strongly than indirect ones, and then say that directly
non-wasted votes are better. That would automatically render upside-down
methods like the on above inferior, because the good methods would have
much fewer voters ask their favorites whom to delegate to. However, then
we're back to the previous case again: that APR is not optimal and that
being optimal leads to very strange destinations.

In short, not wasting votes becomes either impossible or not always
desirable (if using the previous definition); or very weak (if using the
current definition). To show just how general it is, consider that it
technically lets single-winner elections be free of vote-wasting:

1. Elect a single winner by your method of choice.
2. Ask the voters who failed to rank this winner to let their first
preference choice say "I choose the winner" and thus transfer the
voter's vote to the winner.
3. No votes are wasted.