[EM] Notes on a few Later-no-harm methods

[Kristofer Munsterhjelm]

On 14.05.2022 10:33, Richard Lung wrote:
> 
> Just a quick reply. I'm not much familiar with notation.
> 
> Binomial stv is a statistical count that doesn't apply for very small
> numbers. For that, there is non-parametric statistics. There is no hard
> and fast rule. I'd say about 32 votes minimum. but that's just a
> minimum. There is a law of large numbers for better approximations.

Alright, that's not important to my examples either, so I can easily
just multiply the numbers.

> I forget the meaning of truncated, kindly explained to me. If you mean
> what happens with abstentions, they are counted towards the quota for a
> vacancy.

Later-no-harm is a criterion that restricts what happens when people who
leave some candidates unranked, later go on to rank them. For a method
to pass later-no-harm, it needs to support ballots where not all
candidates are ranked. Such ballots are usually called "truncated".

A truncated ballot is the kind of ballot that's allowed in optional STV
such as in New Zealand but disallowed in non-optional STV and AV such as
the ones used in Australia.

So suppose that we have three candidates running for election: A, B, and
C. A truncated ballot is one where, for instance, the voter ranks A and
B and leaves C off the ballot. This is represented in EM notation by

1: A>B

for a single voter, or

1000: A>B

if a thousand voters voted that way. Similarly, a voter who votes:

1: A

is simply expressing a preference for A, being indifferent between B and
C but considering both to be lower ranked than A. And a voter who votes:

1: A=B>C

is expressing indifference between A and B, but consider both to be
better than C.

> With regard to equal preferences, my position is one of both principle
> and expediency. They are counted by the multinomial theorem. But to me
> that is an "illection" count, a choosing-in of candidates, not a
> choosing-out or election.

Could you give an example? I find concrete examples much easier to work
with.

In the election:

3000: A=B>C
5000: B=C>A
13000: C=A>B

who wins single-winner Binomial STV, and what are the keep and exclude
values?

Note that the previous election isn't really relevant to my question, so
if you'd prefer not to answer that one, that's okay. But I would very
much like to know the keep and exclude values, and winner, for this
election:

3000: A>B>C
5000: A>C
13000: C

That is: 3000 voters rank A first, B below A, and C below B;
5000 voters rank A first, C below A, and have not expressed any opinion
about B except that he's (implicitly) ranked below C;
and 13000 voters rank C first and have expressed no opinion about
whether A is better or worse than B, only that both are worse than C.

Now, perhaps your practical implementation of Binomial STV hasn't
defined what happens in the case of truncation. But then its
later-no-harm compliance is undefined, not applicable, because
everything that either fails or passes later-no-harm has to produce
outcomes when given truncated ballots.

As a final note, if determining the keep and exclude values requires
some calculation that's very complex in the number of voters, then feel
free to replace 3000 with 3, 5000 with 5, and 13000 with 13. I'm just
interested in the outcome of some concrete elections with truncation in
them, so I can get a handle on Binomial STV's behavior. It doesn't
*have* to be the ones I listed.

-km