[EM] Tactical voting under a jury model

[Kristofer Munsterhjelm]

On 06.03.2022 14:41, Colin Champion wrote:
> I wrote in another thread that "under a jury model with tactical voting,
> I have some evidence that the best methods are the ones with the worst
> reputations: FPTP and IRV and their extensions". Perhaps some people
> would like to see the evidence. The following table gives mean valence
> losses (so smaller is better) under a jury model with voters attempting
> a burial strategy. It has to be viewed in a fixed-width font.

That makes sense because FPTP and IRV have in common that burial doesn't
really work: either it does nothing or A>W voters burying W makes
someone else than A win. And it also seems reasonable that the minmax
methods do worse than Condorcet-IRV hybrids since the latter pass DMTBR
and the former don't.

For compromising strategy, I imagine that FPTP and IRV would be
considerably worse than the Condorcet methods.

I suppose you could try Daniel Carrera's trivial strategy, but I'm not
sure how you would translate it into a score. A method is susceptible to
trivial strategy in an election if there exists a candidate X different
from the winner W so that when all the voters who prefer X to W move X
first on their ballots and W last, then X wins.

But if the strategy can be pulled off for more than one X, then what
election counts for calculating the score of the method? I'm not sure.

I'd guess that the trivial strategy would also put the IRV-likes ahead
of (better than) the rest.

What's most surprising, IMHO, is that Condorcet-IRV hybrids do worse
than just plain IRV. James Green-Armytage gives a result where
Condorcetifying a method (electing the CW if there is one) can't
increase the proportion of elections where strategy works, if a majority
can force an outcome for the method in question. But apparently this
does not extend to gracefully degrading: your results seem to show that
even if there are fewer elections where strategy is possible, then under
this model, the strategy does more harm on average where it is possible
to execute.

> 
>            random   fptp    sptp     av   sinkhorn borda     mj coombs
>           115.6767  4.2340 15.4462  3.4934 22.3533 21.3425    - 36.3451
>          condorcet benham   btr    nanson minimax minisum    rp river 
> schulze   asm
>              -     3.5469  4.6945  9.5228  7.7089  8.0696  8.4014
> 8.3864  9.6397  9.8513
> condorcet+ random   fptp    sptp     av    borda
>            9.9406  4.9425 11.5035  3.5441 14.2516
>  copeland+ random  fptpf   fptpr    sptp    avf     avr    bordaf bordar
> minimaxfminimaxr
>           11.4423  9.6349 10.7388 12.4150  9.5974 10.5371 14.4200
> 12.7967  8.7613 12.0312
>     smith+ random  fptpf   fptpr    sptp    avf     avr    bordaf bordar
> minimaxfminimaxrtideman   q&dc
>            9.6397  4.9559  4.9856 11.5026  3.5442  3.5532 14.2516
> 10.0643  7.7106  7.7106  4.0854  7.3363

Some of the columns seem to almost run into each other; perhaps it would
be better to split them over two rows.

That Copeland does so badly also seems to tally well with my own
results. If I'm correct, then Landau should be slightly worse than
Smith, but not anywhere near Copeland.

Are the Smith+ methods Smith// or Smith,?

> AV (=IRV) seems to be best, while most Condorcet methods do badly and
> the Borda count does appallingly.
> 
> These results are from my own evaluation software; full details are at
> https://www.masterlyinactivity.com/condorcet/condorcet.html  The call is
> "condorcet 5 101 100000 jury:2+bur". Obviously the correctness of my
> code is not guaranteed.
> 
> I've never seen an evaluation under a jury model, even assuming sincere
> voting. I certainly don't rate such models highly, but I think
> discussions of voting need a model and spatial models aren't the whole
> truth, so it helps to keep alternatives in mind.

Another non-spatial/jury-like possibility is the Kemeny model: there's a
distinguished "ground truth" order of the candidates. Each voter casts a
vote according to this ground truth, but for every pairwise preference,
flips them around with some probability p.

So the procedure would be to take the ground truth, and rejection sample
applying the independent noise to the pairwise components, until the
result is transitive (i.e. no voter can cast a cyclical ballot, so just
keep trying until it isn't).

E.g. if the G.T. is A>B>C>D, then a ballot may become
	   A>B A>C A>D B>C B>D C>D
truth        T   T   T   T   T   T
corrupted    T   F   T   F   T   T

C>A>B>D, instead.

Then the voting method's performance is given by the Kendall tau
distance between the ground truth and the social order produced by the
method when given these ballots.

And if you want to assign scores to impartial culture, then Durand's
hypersphere model might work. I'm not entirely sure how that works (or
how you would sample from it), but Forest showed that it's a natural
extension of impartial culture.

-km