[EM] Burial examples

[Kristofer Munsterhjelm]

On 04/01/2018 02:10 PM, Chris Benham wrote:
> On 1/04/2018 7:27 AM, Kristofer Munsterhjelm wrote:
> 
>> Passing dominant mutual quarter burial resistance is really hard, if 
>> even possible: 
> 
> I am sure that for a Condorcet method it isn't possible.
> 
> Meeting dominant mutual third burial resistance isn't so easy.  I don't 
> know of any Condorcet method
> that does that is better than the "Hare-Condorcet hybrid".
> 
> The version I like: voters strictly rank from the top as many or as few 
> candidates as they wish. If there is
> a CW (or single pairwise-undefeated candidate?) x then elect x.
> Otherwise eliminate the candidate that (among remaining candidates) is 
> voted top-most on the fewest
> ballots and if among remaining candidates there is pairwise-beats all 
> candidate (or single pairwise-undefeated
> candidate?) x then elect x.  If not, repeat and so on.
> 
> But it fails mono-raise and can fail to elect a positionally dominant 
> uncovered candidate, unlike my other
> favourite Condorcet methods.

My investigations seem to suggest that we can have mono-raise without 
affecting strategic susceptibility too much; or rather, without doing so 
as long as the Smith set is of three candidates or fewer.

Because of a combinatorial explosion, it's very hard to use the approach 
I am using to go beyond three-candidate methods, however, which limits 
what I can do.

I would ultimately want to find a method that is as strategically robust 
as a Hare-Condorcet hybrid while retaining clone independence and 
without having to give up mono-raise, and beyond that pass as many 
criteria as possible. (I'd also like to find out if Smith and 
mono-add-top are compatible.) Finding out if reversal symmetry is 
compatible with DMTBR and Condorcet would help narrow down the criteria 
such a "better method" could have.

My approach so far has been to look at what three-candidate methods my 
programs find, and then try to extend them. However, they are often hard 
to understand. Suppose the simulator finds out that a method where A's 
score in an ABCA cycle is (2 * fpA + fpB - 2 * fpC) is a good one (where 
fpX is the number of ballots ranking X first). What's going on here and 
how would it be generalized? There's no obvious answer. It gets even 
worse with nonlinear methods, as the methods may seem stranger still. 
E.g. the simulator might say that a method where A's score in an ABCA 
cycle is (max(C>A, B>C)/C>A) is a good one, but why?

Thus my attempt to find a criterion or criteria to distinguish good 
methods from bad ones. And thus my interest in DMTBR.

Would Uncovered,IRV be a better method? Or "eliminate and stop when one 
candidate covers everybody else"?

> It comfortably handles your four burial scenario examples. In the first 
> three the burying has no effect and in
> the fourth it backfires.
> 
> My other favourite Chicken Dilemma criterion complying Condorcet method, 
> MMLV(erw) Margins-Sort Elimination,
> also easily elects the sincere CW in the first 3 examples but the burial 
> succeeds in the fourt 
>
> https://wiki.electorama.com/wiki/Chicken_Dilemma_Criterion

Chicken Dilemma doesn't seem to be that criterion, at least not on its 
own, because as far as I understand it, Smith,Plurality passes it. And 
Smith,Plurality's performance is even worse than that of the methods 
that reduce to minmax.

Of course, it's possible to argue against my simulations by saying it 
counts what can be measured rather than measures what counts; more 
specifically, that CD violation is a far more serious problem as it 
induces a more destructive form of strategy. But that'd be more a reason 
to want both CD and DMTBR than to say CD is the criterion that 
distinguishes strategy-resistant methods from methods that aren't.

> Taking your first example:
> 
> 10: A>B
> 11: B>C (sincere is B>A)
> 03: C>A
> 
> C>A 14-10,   A>B 13-11,  B>C 21-3.
> 
> Scores:  B11   A10   C3
> 
> Both BA and AC are pairwise out of order.  The score margin between B 
> and A  is 1 (11-10)  and the score margin
> between A and C is 7 (10-3).   So the pair with the smallest margin swap 
> places and now no adjacent pair is pairwise
> out of order and only 3 candidates are on the list  (A>B>C) so A (the 
> sincere CW) now being on top of the list, wins.

That's interesting. Maybe I should try to make DMTBR failure examples 
for maximal sets of the voting methods implemented by LeGrand's voting 
calculator. In particular, Raynaud and Baldwin seem to have different 
enough logic that examples that work on the Plurality-like methods don't 
work on them.

Copeland and Small don't allow for any decisive (B wins with certainty) 
three-candidate examples, but arguably fail outright because B has a 
nonzero chance of winning as soon as the cycle is established.