On 19.01.2022 08:10, Daniel Carrera wrote:

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.

Hi Daniel,

Le mercredi 19 janvier 2022, 01:10:59 UTC−6, Daniel Carrera dcarrera@gmail.com a écrit :

Yes, because if you want monotonicity you have to take a lot of care with the
effect of changes in a vote, so that it always works out OK. But runoff methods
are "messy." With IRV this is the price paid for not regarding voters' lower
preferences until the higher ones become unusable.

A finding of Woodall is that a method lacking both burial strategy and
truncation strategy (as IRV and FPP), which also elects from a "mutual majority"
or (similarly) satisfies "Clone-Winner" (as IRV does), can't be monotone.

I agree with you that failing participation ought to be considered bad. But it's
extremely limiting to insist on this. The only methods that satisfy
participation are simply adding up points from new ballots, without doing any
fancy math with the totals. The most exotic methods here are Woodall's DAC and
DSC, which sum points for each possible set of candidates in the top positions
of a ballot, and then run down the list doing set intersection.

I'm willing to look past monotonicity failures in order to explore other types
of methods. The failures are unsightly. But often they don't create strategies
that voters could actually exploit. And I think in some cases the unfairness of
monotonicity failures isn't obvious at first glance, which might (I'm not sure)
make it more acceptable.

Also, methods can vary in how egregiously or how often they can violate
monotonicity. One can design methods much worse than IRV here, although I guess
that may not be much consolation.

I think it depends what you mean. A lot of studies seem to assume all the
ballots must be complete orderings. In that setting, I think you must use a
method roughly like Condorcet//IRV because otherwise I (speaking personally as
a voter) will feel like I'm being explicitly asked to consider flipping some
orderings around in attempt to avoid helping the candidates that I believe I'm
trying to defeat.

At other times we have sometimes focused on resilience to burial strategy
specifically, not in the sense that the burial incentive isn't actually there,
but that due to certain expectations of truncation on the part of other voters,
voters may consider a burial strategy too risky, and truncate instead.

This is particularly expected when a faction supporting one of two frontrunners
would have to base a burial strategy on the assumption that their candidate
would receive some support from voters supporting the candidate's main rival.
If they don't get this support then they risk electing the candidate they used
for the burial.

To take this stance, you have to understand some truncation as benign or at
least inevitable. I also prefer this stance because the reduced burial incentive
of a method like Condorcet//IRV seems to translate to higher compromise
incentive, which I find unsightly, as it seems to undermine a major argument for
bothering with new methods in the first place.

Kevin

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@t-online.de wrote:

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.

Election-Methods mailing list - see https://electorama.com/em for list info

On 19.01.2022 19:48, Richard Lung wrote:

Unfortunately, without an implementation, I can't verify that claim; and
since no STV method I've seen so far have been proven to be monotone, I
would find it surprising if this were the case.

It would of course be good if it were true! But I have no way of
determining that.

-km

km,
You are repeating what the rest of my post already says -- the need for testing in realistic scenarios. -- Not so much to test monotonicity and strategy resistance. That is guaranteed by the removal of the ad hoc premature exclusion of candidates. And replacing it with an exclusion count, symmetrical to, or exactly the same as the election count,  That is a transferable vote, known to be monotonic -- and hence so, in an exclusion count as well as an election count.
I have actually given the odd example for illustrative purposes, which shows that when you swap preference transfers, it can never create a strategy incentive. That is because preference changes are always accompanied by a change in the respective candidates keep values. There is proper book-keeping of the electoral accounts!

You should be prepared to be surprised. Being surprised is no argument.

You raised a much more pertinent criticism of binomial stv, which needs to be tested by trial elections, to which I wrote an answer, not to hand. In any case trials would be good, instead of preconceptions, if it's not asking too much.

Richard Lung.

On 19 Jan 2022, at 6:59 pm, Kristofer Munsterhjelm km_elmet@t-online.de wrote:

Unfortunately, without an implementation, I can't verify that claim; and
since no STV method I've seen so far have been proven to be monotone, I
would find it surprising if this were the case.

It would of course be good if it were true! But I have no way of
determining that.

-km

On 19.01.2022 22:16, Richard Lung wrote:

You are saying that a particular procedural mechanism ensures
monotonicity and strategy resistance. I doubt that this can be inferred
simply from procedural design, because it's very easy to come up with
something that seems right but turns out to have a subtle weak spot.

I would like to have an implementation so that I could at least see
whether a computer search could find any monotonicity failures, and so
that the strategy resistance could be indicated in a similar way to what
Daniel is doing.

That's what I'm saying. Of course surprise is not an argument; I am
simply saying that, in the absence of anything more conclusive and
readily available, my experience is all I've got, and that experience
indicates that monotonicity in STV is very difficult, and that monotone
strategy resistance is even more difficult.

That is not a proof that your method isn't monotone, but I didn't claim
that it constituted such a proof. Just that when you keep saying that
your STV method (unlike other STV methods) is provably monotone, I would
like to see the goods for myself, as it were.

-km

Richard,

For the requested trial he needs pseudo code for Binomial STV!

El mié., 19 de ene. de 2022 1:16 p. m., Richard Lung <
voting@ukscientists.com> escribió:

km,

Well you've quoted from my explanation, some of it. My keep value
"book-keeping" is unique to binomial stv, so you can't use traditional
stv as a standard of comparison, which you seem to be doing.

Meek method employs keep values for elected candidates, but not for all
candidates, allowing a universal standard of comparison, and not for an
exclusion count, as well as an election count.

I fully agree with you that computational tests have their thoro part to
play. I think they would be useful for a previous consideration you
raised. Personally, I don't think they would unearth the present
problems, being considered. I regret if my three cited e-books are not
what you require.

Regards,

Richard Lung.

On 19/01/2022 21:38, Kristofer Munsterhjelm wrote:

Thank you, Forest,

Just to clarify, I'm not asking Kristofer personally, for a theoretical
test. I'm asking a group or groups tor actual binomial STV test
elections, with either of the hand count procedures, in my two booklets.
I admit these descriptions are sketchy. But at this stage, not a great
deal of detail is required, as it would be premature to set in concrete.
It's a dialog. You can always ask!

Regards,

Richard Lung.

On 19/01/2022 22:02, Forest Simmons wrote: