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

[Kristofer Munsterhjelm]

On 24.05.2022 21:05, Richard Lung wrote:
> 
> The snag is that these and other criteria were invented for what
> amounts to uninomial elections, that is elections that don't have both,
> or either, a rational election count and a rational exclusion count.
> Together they make possible the application of the binomial theorem, to
> higher order counts. My binomial STV hand count is just a first order
> binomial count of one election count and one exclusion count. 

The criteria are method-agnostic: for any ranked voting method (in this
case, that supports truncation), if someone gives you a failure example,
you can verify if the method passes or fails the criterion without
knowing anything about the internals of the method.

Put differently, suppose that in a scenario perhaps reminiscent of
Roadside Picnic, a mysterious device falls out of the sky. And it turns
out that this mysterious device calls elections: you can input ranked
ballots with a set of buttons and get the results shown as a series of
lights on the other end.

Then as long as it allows for ballots with truncation, it's possible to
check if a particular ballot where A-first voters truncate can be used
to induce a later-no-harm failure.

Whether the strange technology that makes up the device implements
rational election and exclusion counts doesn't matter. As long as it's a
ranked voting method outputting winners and supporting truncation, the
question "does this pass later-no-harm?" makes sense.

The same goes for things like monotonicity, participation, consistency,
Smith, Condorcet, etc. The criteria say something about the desired
behavior of a method. Nothing about the inner workings makes the
criteria inapplicable (apart from some exceptions like the polynomial
runtime criterion).

Without a mathematical description of the method, you couldn't be sure
it actually passes later-no-harm or later-no-help, but as soon as you
found a counterexample, that would settle the question in the negative.

> I am not aware of any untoward effects of tactical voting on the bstv
> system. I am aware of it doing away with residual irrationalities to
> traditional stv, including Meek method. Tho I accept that traditional
> stv (zero-order stv in relation to binomial stv) is a robust system,
> in practise, as the Hare system of at-large stv/pr.

As a ranked method, it must fail IIA, which means that strategy must
sometimes be possible. And as it fails Condorcet, the obvious starting
place to look is for an election where it doesn't pass Condorcet. For
instance, this:

549: A>B>C
366: B>A>C
366: B>C>A
366: C>A>B

A is the Condorcet winner. The first preferences are:
	A: 549, B: 732, C: 366
and last preferences:
	A: 366, B: 366, C: 915

so the ratios are:
	A: 366/549 = 0.67
	B: 366/732 = 0.5
	C: 915/366 = 2.5

so B wins. Then the C>A>B voters have an incentive to vote A>C>B instead
(compromising), after which the counts are:

	A: 366/915 = 0.29
	B: 366/732 = 0.5
	C: 915/0   = infinity

and A wins. The C>A>B voters prefer A to B, so the strategy is to their
benefit.

It can be shown that, for methods where a majority can always force an
outcome by coordinating how they vote, then modifying the method so that
it elects the Condorcet winner if there is one never increases the
proportion of elections where strategy is useful, and may indeed reduce it.

> BSTV counts require values for all preference positions, which are
> equal to the number of candidates. Any preference position may be an
> abstention. A citizen who never voted but made an exception of their
> dislike for Donald or Hilary could abstain on their first preference
> but vote for either on their second preference, effecting an
> exclusion, because there is only one vacancy.

> That is the theory of it. I don't know how well it would work in
> practise, because there never has been a practise. But I do know that
> democracy is minimised, and evidently works badly, based on single
> vacancies, in the Anglo-American systems.

> Fully fledged binomial stv, FAB STV, does not work on less than 4 or 
> 5 member constituencies, the minimum requirement for a democracy of
> all the people being represented by their choices.

> Thank you for your examples. They have helped clarify my thinking -- somewhat!
> According to my (accident-prone) working, A wins on a keep value of  38957/58966.
> B also has a less than unity keep value of 38957/39366. The 
> difference is that one can say A has been elected on a quota of
> 48961.5, with 58966 first preferences.
> But B has not reached the elective quota. Tho B has not reached the
> exclusion quota, that only says B has not been excluded.

So by the keep values: A's first preference count is 58966 and last
preference count is 38957, since the keep value is 38957/58966.

You say that B's keep value is 38957/39366, i.e. first preference count
of 39366 and last preference count of 38957. But that seems to be in
reverse order. Indeed, your HTML page shows that it is 39366/38957.

>From the keep values, it seems that truncations are not included when
counting last preferences. I was pretty sure that BSTV would fail
later-no-harm because the standard way of counting truncations, as STV
does, is to consider everybody not ranked to be equal-ranked for last;
and if you had done that, then it would be possible to induce later-no-harm.

The good news is that you avoid this particular problem if you count
anything past truncation simply as abstentions. So I guessed wrong,
which was then cleared up by the example, which shows how useful they
are :-)

However, instead it seems that you get later-no-*help* failure. Consider
this modified election:

18125: A
20035: A>B>C
18722: A>C>B
34488: B>A>C
38634: C>B>A

By my count, the first preferences are: A: 56882, B: 34488, C: 38634
and the last preferences are:           A: 38634, B: 18722, C: 54523
and the last to first ratios are:       A: 0.68,  B: 0.54,  C: 1.41

so B wins. But if now the A voters fill out their ballot by voting
A>C>B, then B's last preference count changes to 36847 and A wins
instead. This is a violation of later-no-help.

Ordinary STV passes both.

I should note that Condorcet methods, that I prefer, fail both. My point
isn't as much that later-no-harm and later-no-help are intrinsically
good, as that it's much easier to check a claim by concrete evidence
than by references to personal terminology (which may be hard to
understand for others or take a lot of time to get acquainted with).


On a final note, I would say that always counting truncation as
abstention could lead to an unknown candidate problem: suppose there's a
candidate who nobody has heard of and thus nobody bothers to rank. But
he has a dedicated following all of whom rank him first. If nobody
obtains a majority, then this candidate could win, e.g. something like:

3300: A>B>C
3300: B>C>A
3200: C>A>B
   2: D

I'm also not entirely sure what's going on with the quota transfers. If,
in the single-winner case, someone who exceeds the quota is
automatically elected, then there's no need for any transfers. However,
if passing the quota doesn't guarantee victory, then later-no-harm
failure might actually be possible. Suppose A is just above the quota
and B is just below it (with B closer to the majority line), then if the
A voters only vote for A, A might win; but if they vote A>B, then the
surplus might be transferred to B and make B win. Perhaps. As I said,
I'm not sure how the logic works in that case.

-km