universal domain

In social choice, the requirement that a procedure should be able to
produce a definite outcome for every logically possible input of individual
preference orderings.

So, all ranked equal is a "logically possible preference ordering."

The main thing I'm wondering is how to modify ASM (Approval Sorted Margins)
to make it more broadly acceptable ... and to perhaps comply with Universal
Domain as a bonus.

Here's my best attempt so far:

FIASM Fractional Implicit Approval Sorted Margins: Ballots are ranked
preference style with equal rankings and truncations allowed. Each
candidate's fractional implicit approval score is the number of ballots on
which it is ranked equal top plus half the number of ballots on which it is
ranked above at least one candidate, but not ranked top.

The candidates are listed in fractional implicit approval order. While
there is any adjacent pair where the fractional implicit approval order
contradicts the pairwise (head-to-head) win order, transpose the members of
the out-of-order pair with the smallest absolute discrepancy in fractional
implicit approval.

The resulting list is a social order that satisfies a reverse symmetry
property ... reversing all of the ballot ranking inputs (so that equal top
becomes equal bottom [or truncated] and vice versa) reverses the social
order output.

Does this method satisfy Universal Domain?

Now, what if optional explicit cutoff marks were allowed to demarcate the
three levels (0, 1/2, or 1) of fractional approval. Would that violate
Universal Domain?

If so, then the Oxford definition quoted above is inadequate, since it does
not logically rule out optional marks when the lack of any optional mark
defaults to a standard ranking, and the only stated requirement is that no
standard ranking be unusable.

Thoughts?

El vie., 10 de sep. de 2021 10:56 p. m., Kevin Venzke stepjak@yahoo.fr
escribió:

El 10 sep. 2021 10:56 p. m., "Kevin Venzke" stepjak@yahoo.fr escribió:

To my mind Implicit Approval (as a method in itself) only satisfies it if
you can define the ballot format while discussing only relative rankings.
So, for example, if the voter ranks all candidates totally equal to each
other (no matter whether they are explicitly so ranked, or the ballot is
submitted with all preferences truncated), this can only be allowed to mean
that all are approved or that none are approved, since there is no way to
differentiate these two stances using relative rankings only.

Kevin

Le vendredi 10 septembre 2021, 21:49:14 UTC−5, Forest Simmons <
forest.simmons21@gmail.com> a écrit :

Does Implicit Approval satisfy Universal Domain?

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

Evidently, at some point it became apparent that complete rankings (or
symmetrically completed partial rankings) were inadequate for defense
against certain kinds of offensive maneuvers ... we needed to have partial
rankings to distinguish between winning votes and margins to measure defeat
strength, etc.

But, it seems to me that we have to go further if we want a method that can
reasonably defend against both Burial and Chicken attacks.

The classic ballot profile

49 C
26 A>B
25 B

is well known as a Chicken attack against the sincere Condorcet candidate A
by the B faction when it's honest preferences were 25 B>A. So this ballot
set must elect C or reward the delinquent faction B, or fail Plurality by
electing A.

But the same profile could arise just as well from a truncation attack
against the sincere Condorcet candidate B by the C faction when its honest
preferences were 49 C>B. So this ballot set must not elect C if it is to
disappoint its naughty gambit.

In summary, to satisfy Plurality, we cannot elect A. To deter Chicken
attacks on a possible sincere CW we cannot elect B, and to deter
truncation/burial attacks on a possible sincere CW we cannot elect C.

But Universal Domain says our method has to pick one of the three
candidates.

What would IRV do?

It would happily eliminate B and then elect C, the same as it would do if
the true preferences were 49 C>B: defeat of the CW by truncation or burial
is not a problem for IRV: it never made any promises about Condorcet.

However the 51 IRVvoters that preferred B over C would be highly
disappointed by this outcome ... in fact, it is quite likely that some of
the A>B faction would forestall it by insincerely reversing their
preference to B>A.

How could relaxing Universal Domain slightly get us out of this dilemma?

I think the answer is to use the traditional Sequential Pairwise
Elimination factorization of a Condorcet method into two parts ... one part
for setting an  agenda ... and the other part for sorting the agenda
pairwise (always giving priority to rectifying the order of the
out-of-order pair nearest the least promising end of the agenda).

It is only the agenda setting part that requires going slightly beyond
"Universal" in Universal Domain. For example, setting the agenda by some
kind of approval, implicit or otherwise.

The other factor, the Pairwise win/loss/tie matrix is completely determined
by the ordinal information in the ballots.

In our example, what if the A faction could distinguish the Chicken attack
from the other scenario by use of an explicit approval cutoff: 26 A>>B ?

This is enough to change the SPE (Sequential Pairwise Elimination) agenda
so that B with the least approval is pitted against A and so is eliminated
first and does not get rewarded for the attack.

In the second scenario the default/implicit approval cutoff (truncation) is
assumed which gives B the greatest approval ... pitting A against C, and
then C against B, making the sincere Condorcet candidate B the winner.

It seems to me that this factorization idea is the safest and most
transparent way of resolving this dilemma. Since it (SPE) is an ancient
method with lots of pragmatic use in all sorts or traditional "deliberative
assemblies" (Robert's Rules terminology) we should not feel too timid in
proposing it for public elections.

What say ye?

El sáb., 11 de sep. de 2021 9:01 p. m., Kevin Venzke stepjak@yahoo.fr
escribió:

Dear All,

Dare I say it, it's true anyway, FAB STV can and does differentiate the "two stances" approved and disapproved, "using relative rankings only."
FAB STV ballot looks like any other preference vote. But a last preference counts just as much against a candidate, as a first preference counts for a candidate -- and so on, relatively speaking.
FAB STV has the same or symmetric counts for election and exclusion. They are combined to give an overall result, in terms of keep values -- an extension of the Meek method practise.

FAB STV doesnt require all preferences to be given. Blank preferences count towards a  NOTA quota, an unfilled seat. In the unlikely event of your just wanting to exclude a candidate, a last preference, leaving the rest blank, would count as much as a first preference, the other way.

Regards,
Richard Lung.

On 12 Sep 2021, at 3:13 am, Forest Simmons forest.simmons21@gmail.com wrote:

universal domain
In social choice, the requirement that a procedure should be able to produce a definite outcome for every logically possible input of individual preference orderings.

So, all ranked equal is a "logically possible preference ordering."

The main thing I'm wondering is how to modify ASM (Approval Sorted Margins) to make it more broadly acceptable ... and to perhaps comply with Universal Domain as a bonus.

Here's my best attempt so far:

FIASM Fractional Implicit Approval Sorted Margins: Ballots are ranked preference style with equal rankings and truncations allowed. Each candidate's fractional implicit approval score is the number of ballots on which it is ranked equal top plus half the number of ballots on which it is ranked above at least one candidate, but not ranked top.

The candidates are listed in fractional implicit approval order. While there is any adjacent pair where the fractional implicit approval order contradicts the pairwise (head-to-head) win order, transpose the members of the out-of-order pair with the smallest absolute discrepancy in fractional implicit approval.

The resulting list is a social order that satisfies a reverse symmetry property ... reversing all of the ballot ranking inputs (so that equal top becomes equal bottom [or truncated] and vice versa) reverses the social order output.

Does this method satisfy Universal Domain?

Now, what if optional explicit cutoff marks were allowed to demarcate the three levels (0, 1/2, or 1) of fractional approval. Would that violate Universal Domain?

If so, then the Oxford definition quoted above is inadequate, since it does not logically rule out optional marks when the lack of any optional mark defaults to a standard ranking, and the only stated requirement is that no standard ranking be unusable.

Thoughts?

El vie., 10 de sep. de 2021 10:56 p. m., Kevin Venzke stepjak@yahoo.fr escribió:

El 10 sep. 2021 10:56 p. m., "Kevin Venzke" stepjak@yahoo.fr escribió:
To my mind Implicit Approval (as a method in itself) only satisfies it if you can define the ballot format while discussing only relative rankings. So, for example, if the voter ranks all candidates totally equal to each other (no matter whether they are explicitly so ranked, or the ballot is submitted with all preferences truncated), this can only be allowed to mean that all are approved or that none are approved, since there is no way to differentiate these two stances using relative rankings only.

Kevin

Le vendredi 10 septembre 2021, 21:49:14 UTC−5, Forest Simmons forest.simmons21@gmail.com a écrit :

Does Implicit Approval satisfy Universal Domain?

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

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

On 9/11/21 4:48 AM, Forest Simmons wrote:

As I understand it, universal domain means that the method should be
able to provide an outcome for a set of untruncated ranked ballots, and
do so for every possible set.

Like Kevin says, implicit Approval should pass UD if the implied
approval is defined for such untruncated elections. I think that most
methods would consider every candidate to be approved in that case.

There's a stronger sense where the only input allowed is untruncated
ranked elections, but every method that allows for equal rank or
truncation will fail that stronger version. This stronger sense is how
Wikipedia defines it, here:
https://en.wikipedia.org/wiki/Unrestricted_domain

In addition, while IA methods may formally pass the (weaker) universal
domain, they could very well still be awful methods under the restriction.

Consider e.g. the following method: "Let the preliminary winner set be
the set of candidates whose implicit approval score is maximum.
Eliminate every candidate not in the set, then elect the Borda winner
from the reduced election". If the election features no truncations at
all, the method is just Borda, and Borda is not particularly good.

And then there's the muddling of honesty and strategy that comes from
rejecting universal domain, which gets worse the stronger the violation
is. Maybe I should write a separate post on that :-)

-km

Kevin, thanks for your clarifications about Universal Domain; they
confirmed my worst fears ... that all of my favorite methods violate it,
and that it is likely impossible (as Kristofer has helped me understand) to
make a clone proof, monotone agenda based Banks or Landau method without
violating it.

In particular, all of the good approval variants ... chiastic approval,
midrange approval, etc violate Universal Domain or, like fractional
implicit approval (FIA), are not truly compatible with both monotonicity
and clone independence. Agendas based on the pairwise matrix, seem to
always fail mono raise because raising x from below to above y always
entails lowering y from above to below x ... increasing an entry in x's row
decreases an entry in y's row.

For ordinary round robin team tournaments, this no problem ... team x can
get another point against team y without y decreasing its entry in column
x, and the clone dependence of Borda doesn't seem to matter ... mostly
because there are no ballots to worry about in this sports tournament
context. So the agenda is set by the total number of points scored by a
team in the entire tournament minus the total points scored against it. The
logical way to bring this agenda into pairwise conformity is by Borda
Sorted Margins ... try to beat that with Kemeny-Young! You get Borda from
the pairwise matrix (row sum minus column sum), but how do you get K-Y?

Anyway if in general we are to separate the strategic burden into the
formation of the agenda, and have nearly complete rankings for the
formation of a truly faithful pairwise matrix, we either need two sets of
rankings, or one set  of rankings annotated by approval cutoffs, ratings,
or other markings not permitted by Universal Domain.

Until academicians get a better appreciation of the importance of defensive
levers against offensive attacks like Chicken and Burial, I'm afraid we can
expect our non-Universal Domain innovations to be marginalized by academia.
It may be a long row to hoe!

Other such innovations are proxy methods, including Asset Voting, lottery
methods, oracles, voting for a public ranking, use of ballot space and
candidate space metrics, binary decision tree ballots, etc. And of course
multi-winner PR methods ... all are wide open!

El sáb., 11 de sep. de 2021 11:51 p. m., Forest Simmons <
forest.simmons21@gmail.com> escribió:

Excellent points and clarifications, Kristofer!

It is instructive to see how close the UD compliant FIASM comes to
resolving the dilemma ballot set we were talking about.

49 C
26 A>B
25 B

FIA's for A, B, and C, respectively are 26, 38, and 49. Candidate B's score
is closer to C's than to A's, so the list becomes
B>C>A with B's chicken gambit rewarded.  However, one A>B voter truncating
B, moves B's score closer to A's resulting in the final order C>A>B.

If the faction sizes were 48 C, 28 A>B, and 24 B, the respective FIA scores
would be 28, 38, and 48, with B exactly halfway between A and C.

One A>B voter could tip the scale in favor of B or C by changing to A=B or
by truncating B, respectively. This shows that FIASM is as close as
possible to resolving the dilemma without violating Universal Domain.  Pity
it cannot effectively satisfy clone dependence while clones straddle the
boundaries demarcating the three fractional approval zones ... which is one
important way to discern ranks within individual clone sets. Explicit
approvals would resolve that annoyance, but only by sacrificing Universal
Domain.

El dom., 12 de sep. de 2021 6:06 a. m., Kristofer Munsterhjelm <
km_elmet@t-online.de> escribió:

On 9/12/21 3:43 PM, Forest Simmons wrote:

I'm not entirely sure I see the implication. But assuming the worst,
that there are no such methods, that still leaves the possibility for
non-agenda methods.

(I do agree, though, that using implicit approval for the agenda would
lead to a bad method; with untruncated elections, every candidate would
be tied and the agenda would have no power at all.)

And even if you can't have all of Landau, monotonicity, and clone
independence, River's "strongly dominated alternatives" comes quite
close, if I recall correctly.

Is Random Ballot agenda any good, if we allow for some chance?

-km