[EM] Poking about Heaviside methods, again

[Kristofer Munsterhjelm]

On 1/14/23 01:18, Kevin Venzke wrote:
> Hi Kristofer,
> 
>>> If we compare to methods that people actually propose, it seems true that non-Condorcet
>>> methods don't tend to have huge burial issues. Main exceptions would be Borda, MMPO, and
>>> iterated DSV methods that are nearly Condorcet-efficient. Methods like DSC and Chain Runoff
>>> do have burial incentive, but their compromise incentive is probably a much bigger concern.
>>   
>> Some LNHelp methods have burial problems, though it's usually easier to
>> just truncate instead of bury.
> 
> Do you say this because you conceive of burial as simultaneously adding one candidate while
> lowering another, or because you include altering relative order within the ranking and not
> just at the bottom?

Perhaps an example is better, for Bucklin:

Suppose that every ballot is complete (i.e. no truncation) and A wins 
after first and second preferences are counted, being ahead of C by just 
one point. And suppose that if it went to the third round (counting all 
three ranks), then C would have won.

Then some C>A>... voters could change their vote to C>...>A and C would 
win. That's just plain burial of A.

For Bucklin it's possible to do the same thing by just bullet-voting for 
C. From the perspective of my simulator, that doesn't mean that this 
election isn't susceptible to burial, but rather that it's also 
susceptible to truncation. When it's checking for burial in isolation, 
it doesn't know if there's also a possibility to truncate.

>> My general direction of thought was: it would be useful to identify just
>> what contributes to low burial incentive because all the strategy
>> resistant methods I know of have mainly low burial incentive.
> 
> I guess from my perspective I don't have a good way to merge different forms of strategic
> incentive into a single resistance metric. Even if I could do exhaustive searches it seems
> like different strategies are qualitatively different, for example in how intuitive they
> are for a voter to apply.

Yes, I'm just considering all strategies to count equally much at the 
moment. This is probably overly generous and makes methods look worse 
than they are, but I don't know how to add different weighting in a way 
that won't introduce too much subjectivity. E.g. whether a party can 
coordinate all of its voters to rank its candidate top and the honest 
winner bottom, whether all the voters would go along with this, and 
whether the party gets the honest winner right, and so on.

>> As I mentioned in a private mail, it should retain monotonicity because
>> A and B share the pool that's compared against the quota. A will get A>B
>> from any B where A and B's first preferences combined exceed the quota,
>> so raising A to top on a B ballot can't harm A.
> 
> After playing with the H(fpA + fpB - fpC - fpD) method (max or sum, no Condorcet check) I
> find a problem:
> 
> Suppose that the first preference count order goes ABCD, and the proportions are such that
> the majorities are AB, AC, and AD. Say that B wins. If B obtains new first preferences at
> the expense of C, the proportions could change so that fpB + fpD is now a majority. This
> could allow D to be elected on the basis of a strong D>B gross score which wasn't
> admissible before.

That's a good point. As a shortcut, I was looking at how f(A) grows when 
A is raised. What I really should be looking at is f(A) - f(B) for any 
other B. Since I didn't, I didn't see that monotonicity violation. (It 
also suggests that there are monotone methods that I don't detect, too, 
because while dA/d(raise) < 0, it might be that dA/d(raise) - 
dB/d(raise) >= 0 for all other B -- to misuse differential notation a bit.)

But with the way things are set up, evaluating the growth of f(A) - f(B) 
would be very hard. I probably have to implement a monotonicity checker 
and either prescreen methods by checking them against it or combine the 
checker with the bandit system so that the moment a monotonicity 
violation is detected, it disqualifies the method in question.

>> But if it's necessarily true that you can only have burial resistance
>> with elimination, then a direct consequence is that we can't have burial
>> resistance and summability.
> 
> I have recently found a couple of exceptions to the first part although not the second.
> That's to say LNHarm+LNHelp methods that can't be explained in terms of eliminations. I'm
> not sure it will be any use to you, but it's still interesting. I'm sort of checking my
> work and looking for any "nearby" findings before posting.

Now that I think about it... of course. I must've been not thinking far 
enough ahead. Plain old Plurality qualifies: it has zero burial 
resistance as it passes both LNHarm and LNHelp, and it's summable. For 
that matter, the f(A) = max X: A>X method has low burial resistance and 
is also summable.

The difficult part is that neither has low strategy resistance in 
general. I guess there has to be some kind of "compromise resistance" 
criterion that's weaker than the strong FBC, and that IRV passes but the 
other methods don't.

IRV passes mutual majority, which would at first seem like a good 
candidate. But Woodall showed that we can't have all of mutual majority, 
LNHelp, LNHarm, and monotonicity.

-km