[EM] Quick and Clean Burial Resistant Smith, compromise

[Daniel Carrera]

On Sat, Jan 15, 2022 at 5:54 PM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> I've at times thought that a more realistic approach would be to use
> evolutionary stable strategies, e.g. suppose everybody is honest, then
> some small faction defecting can't grow in power to the point where
> everybody feels the need to vote tactically. But that's even harder to
> model, and some societies are used to tactical voting at the start (e.g.
> US Plurality elections), and might thus not fit that very well.
>

It would be a bit hard to model, but one could imagine allowing the
electorate to run the election many times and adjust their strategy, but
insert the assumption that if a voter will not get a better outcome by
being strategic, they'll choose to be honest. So you could model whether an
electorate that starts out with strategy evolves toward honesty.


Now I'm wondering if the strategically resistant methods mainly increase
> their resistance by making it impossible for trivial strategy to work in
> cases where it would otherwise work. From my experimentation, I've found
> out that reversal symmetry usually makes for a susceptible method, and
> that rev. sym. and Condorcet together are incompatible with dominant
> mutual third burial resistance.
>

This really got me curious. I've been reading up on these concepts and I
saw that you posted a proof of this in 2018. I still need to wrap my head
around these concepts. For example, it's not intuitively obvious to me why
reversal symmetry is important. If the voters are split into two equal
camps:

Camp 1: Loves A, tolerates B, hates C
Camp 2: Loves C, tolerates B, hates A

It's easy to see B as the right compromise winner, and if you reverse the
preferences that wouldn't change.




> Similar reasoning can be found in https://rangevoting.org/WVmore.html
> which seems to argue that DH3 is a problem even for Condorcet methods
> that pass DMTBR (like Smith//IRV, Benham, etc):
>

The wiki says that "a method that passes DMTBR is immune to electing the
dark horse", but I can easily devise a pathological example where the dark
horse is the Condorcet winner:

9: A > X > B > C
7: B > X > C > A
5: C > X > A > B

So I assume that the wiki most be defining Dark Horse in a narrower sense
than how I'm reading it.



In any case, as you say, strategy-resistant methods like the Smith-IRV
> hybrids will reduce the chance that strategy works, and so would give
> third parties some more room in which to grow.
>
> It would be interesting to do a test with Range; I imagine that it would
> be very susceptible to strategy, similar to Approval. And I would also
> imagine that STAR would do considerably better.
>

I just implemented Range and STAR and it seems that Range is indeed very
susceptible but STAR is not much better. I had to make some arbitrary
choices when implementing the scoring system. I made the score proportional
to the negative distance between a voter and each candidate, normalized to
span the 0-5 range. So every voter uses the entire 0-5 range, regardless of
whether they feel strongly about their preferences or not. I imagine that's
how most voters would behave. I chose a 0-5 range because that's what STAR
seems to use. I also had to make some decisions on how to implement each
strategy for a score ballot:

1) For the trivial strategy, I give c_k a score of 5 and w_A a score of 0,
and leave other scores the same.

2) For the reverse strategy, I compute the ballot that w_A would have cast,
I "reverse" the scores with "new_score = 5 - old_score". Then give a score
of 5 to c_k and a score of 0 to w_A.

3) For the moderate strategy, where I try every possible ranked ballot, I
was forced to find a way to convert a ranked ballot into a set of scores. I
decided to spread the scores uniformly from 0 to 5, so it's as if each
preference is equally strong.

N_elections = 20,000
N = 4, V = 99, C = 5
Method   , 95% c.i.     , trivial, reverse, moderate, majority
MiniMax  , 0.4019-0.4155, 0.895  , 0.0805 , 0.02447 , 0.219
Hare     , 0.0683-0.0752, 1.000  , 0.0000 , 0.00000 , 0.135
Benham   , 0.0470-0.0528, 0.979  , 0.0160 , 0.00502 , 0.132
Smith_IRV, 0.0474-0.0532, 0.944  , 0.0169 , 0.03888 , 0.137
Range    , 0.7828-0.7935, 0.832  , 0.1012 , 0.06667 , 0.601
STAR     , 0.7485-0.7621, 0.620  , 0.0775 , 0.30275 , 0.522

The 95% interval for Range is very close to the susceptibility that JGA
reports on Table 1 (not identical; he probably computed scores
differently). STAR adds very little protection from strategy. Both systems
are extremely susceptible to manipulation. STAR requires the most complex
strategies, but that should give us no comfort; the two score systems
succumb to the simplest strategy more often than other systems succumb to
any strategy at all.

STAR: 75% susceptible x (62% trivial + 8% reverse) = 52%

Benham: 5% susceptible x (98% trivial + 1.6% reverse) = 5%

Cheers,
-- 
Dr. Daniel Carrera
Postdoctoral Research Associate
Iowa State University
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