[EM] Strategy-proof vs Monotone

[Richard Lung]

Treating an election as a statistic, binomial stv is monotonic and strategy-resistant. I would guess that all run-off methods, which actually includes traditional stv, are non-monotonic, in principle. But the more proportional the count in greater multi-member constituencies, the test, of times tried, shows it doesn't matter in practise. It's routine for traditional stv to elect high proportions of first preferences, with most of the rest being high preference elections. Remember, this is from a much better range of choice, offered by large, compared to small, constituencies.

It is essential to actually test elections, in realistic scenarios, not merely to analyse them. I would certainly say that is the case with my own invention of binomial stv.  

Regards,
Richard Lung.  



On 19 Jan 2022, at 2:57 pm, Kristofer Munsterhjelm <km_elmet at t-online.de> wrote:

> On 19.01.2022 08:10, Daniel Carrera wrote:
> So recently I've been posting a lot about using simulations to estimate
> which voting systems are most vulnerable or resistant to strategy. That
> was certainly interesting. But as Colin pointed out, strategy resistance
> is not the only goal. One issue that keeps coming to mind is that, I
> think..., all the Condorcet-IRV systems are non-motone. Am I right about
> that? I think all (most?) runoff-based methods non-monotone.
> 
> So I guess I have two questions:
> 
> 1) How important do you think monotonicity is? I'm not comfortable with
> the idea that you can harm a candidate by ranking him higher, but I
> would say the same thing about failing the participation criterion yet
> all Condorcet methods fail (for reasons I still don't fully understand).
> 
> 2) Does anyone know a different class of Condorcet systems that are also
> resilient to strategy?

My optimal strategy simulations seem to indicate that (in the model
we're using) monotonicity is essentially free: you don't lose any
strategic susceptibility by insisting that the method should be
monotone. At least not for c=3 or v<5, c=4. (See below for my results.
If you have Gurobi, perhaps I could send you some of my MIPs to see if
it can solve them quicker; I only have access to Cbc and CPLEX myself.)

However, monotone strategy resistant Condorcet methods are very hard to
understand. I'm still trying to devise a method that's Condorcet, DMTBR,
and monotone, but I haven't had much luck yet. I know through (more or
less) exhaustive search that for three candidates, fpA-fpC and
Smith,Carey are close to optimal. But as Craig himself pointed out,
generalizing to more than three candidates is very tough.

I'd share your position -- I probably would put it like this: common
sense criteria shouldn't be broken unless there's a very good reason for
them to be. In the case of participation, there's a good reason in that
Condorcet is more important; but for mono-raise, it doesn't seem to be
giving up too much.

Of course, the same can be said about Smith and mono-add-top. I suspect
that in the case of complete ballots, Smith is not incompatible with
mono-add-top, although the question remains open (and is seemingly very
hard to prove or disprove). However, Smith and the plurality criterion
combined are incompatible with mono-add-top. If asked, I would say that
the former is better than the latter, but I couldn't justify this
particular decision. There's ultimately some aspect of value judgement
to all of this.

So I guess I would say Mazur's quote about number theory applies to
voting methods too :-)

-km



Results:

If "method" is a set (e.g. "anything goes" or "Smith-efficient") then
the result is for the optimal method that's constrained to electing from
that set. Otherwise, it's a particular voting method. (MC) means
Monte-Carlo, i.e. not exact, and the significant figures encompass the
95% c.i. The model is impartial culture.

For the exact results, I have the susceptibility values in fraction
form, but I've written them in decimal here. There's a tradeoff between
decisiveness and strategic resistance: trivially, a method that returns
every candidate tying for first can never be manipulated. So I've marked
(NT) where I've constrained the method to a minimal number of ties
beforehand: this is necessary to stop "Anything" from having
manipulability zero (due to the aforementioned pathological solution).

When I've included both monotone and nonmonotone methods, the
nonmonotone ones are marked *.

Potentially nonmonotone, 3 candidates:

Method          5 voters   7 voters   9 voters   11 voters   13 voters
--------------  ---------  ---------  ---------  ---------   ---------
Anything(NT)    0.0925926  0.1140261             intractable intractable
Condorcet       0.0925926  0.1140261  0.1993789  0.1208137   0.1356658
fpA-fpC         0.0925926  0.1455333  0.2226342  0.1211194   0.1620844
*Benham         0.0925926  0.1455333  0.1993789  0.1211194   0.1606773
*IRV            0.0925926  0.1455333  0.1993789  0.1211935   0.1626362
IFPP            0.0925926  0.1500343  0.2203871  0.1212722   0.1626776
Ext-Minmax      0.3472222  0.4365998  0.5005727  0.5394900   0.5716314

The optimal monotone Smith-efficient method is below IRV in every case,
and fpA-fpC does worse than IRV only with 9 voters. I seem to have lost
the data for the nonmonotone optimal methods, but I also seem to recall
that they have the same manipulability as monotone ones.

Four candidates:

Method          3 voters   4 voters   5 voters
--------------  ---------  ---------  ---------
Anything(NT)               intract.   intract.
Smith-eff.      0.2673611
Schwartz        0.2673611  <=0.4635
Landau          0.2673611             <=0.229
Copeland                   0.6004743  <=0.380
*Smith//IRV     0.375      0.4900174  0.1866319
*Landau//IRV    0.375      0.4900174  0.2622252
*Copeland//IRV  0.3506944  0.6684028  0.3174450
Smith//E-Minm.  0.3506944  0.6579861  0.5287905

Again the optimal monotone methods do better than IRV (where known).

"intractable" means I couldn't even load the MIP model without going out
of memory. (This was before my recent upgrade, so perhaps I should try
it again...) For the blank fields, I couldn't find my data for these or
there is none because it's intractable.
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