Method Y
For now this is is a manual Plurality runoff method:
At each runoff stage let Y be the candidate with the fewest votes.
Then by majority decision either eliminate Y or else elect Y by eliminating
all other candidates.
That's it.
Note that majority decision by rational informed voters will never
eliminate Y when it is the sincere CW of the remaining candidates.
Very simple, but monotonic, and burial resistant ... not to mention
strongly Condorcet efficient.
In fact, it could accurately be called Sincere Benham (pending permission
from Chris) because if there is a sincere CW at any stage, optimal strategy
requires informed rational voters to elect it ... while if only Y is
eliminated at each stage, the ordinary Plurality runoff candidate wins.
Note that since the sincere CW wins whenever there is one, there can be no
burial of a sincere CW ... which is the only kind of burial that concerns
us.
The main drawback is the potentially large number of manual votes required.
Is there a DSV version that gets around this problem?
Quickly finding the Smith set by some elegant manual method would largely
solve the problem for many deliberative assemblies including parliaments,
senates, summits, etc.
A Coombs version of this method is equally burial resistant, monotone,
Condorcet efficient, etc, while more decisive... less likely to tie for Y.
[Bottom two sequential pairwise elimination with takedown and majority
pivot-save should also work except perhaps for the monotonicity.]
fws
On 8/4/23 14:36, Forest Simmons wrote:
For now this is is a manual Plurality runoff method:
At each runoff stage let Y be the candidate with the fewest votes.
Then by majority decision either eliminate Y or else elect Y by
eliminating all other candidates.
That's it.
Note that majority decision by rational informed voters will never
eliminate Y when it is the sincere CW of the remaining candidates.
Very simple, but monotonic, and burial resistant ... not to mention
strongly Condorcet efficient.
I think this would only be monotone in a "manual DSV" sense where the
honest ballots are common knowledge.
The simplest IRV monotonicity case is where we have an ABCA cycle and
some BAC ballots become ABC ballots, then C survives to the final round
and beats A pairwise because B lost enough votes to be pushed down to
third place.
If we have manual DSV, then in the first round, A's supporters would
coordinate and decide that they're going to "donate" some first
preferences to B so as to push C down to third place, then in the final,
a majority (sincerely) indicates it prefers A to B, hence A wins.
But to me, that's kind of like the proof that a Condorcet winner is a
Nash equilibrium under Approval: it requires too much coordination.
Perhaps in a legislature, but in a public election, I wouldn't think
this would be monotone.
It might be possible to make it more strongly monotone by doing all the
manual DSV automatically, like a refinement of my "Contingent vote with
donation" method. The calculations could become rather hairy, though,
for more than three candidates.
But the idea that a majority will sincerely say "yes" when the CW
appears is good! Perhaps something revelation principle based could be
used to lessen the demands on the voters' calculation ability and the
degree to which the honest ballot set needs to be common knowledge.
In fact, it could accurately be called Sincere Benham (pending
permission from Chris) because if there is a sincere CW at any stage,
optimal strategy requires informed rational voters to elect it ... while
if only Y is eliminated at each stage, the ordinary Plurality runoff
candidate wins.
Note that since the sincere CW wins whenever there is one, there can be
no burial of a sincere CW ... which is the only kind of burial that
concerns us.
The main drawback is the potentially large number of manual votes required.
Is there a DSV version that gets around this problem?
Quickly finding the Smith set by some elegant manual method would
largely solve the problem for many deliberative assemblies including
parliaments, senates, summits, etc.
A Coombs version of this method is equally burial resistant, monotone,
Condorcet efficient, etc, while more decisive... less likely to tie for Y.
Antiplurality isn't burial resistant, so I wouldn't imagine this to be
either - in a Condorcet cycle, at least.
Consider a "fixed Benham" version of the above, with some election
method being used to set the elimination order, then we go from lowest
ranked to highest, asking "elect or eliminate".
If there is an honest CW then any ordering will work: for every other
candidate A that's checked before the honest CW, a majority will know
that if they hold out for longer, they can say "yes" for the honest CW
and then he wins.
If there's an honest cycle, then all bets are off. The benefit of using
interleaved Plurality over something like (fixed or interleaved) Ranked
Pairs or Antiplurality is that if there is an honest cycle, then you
have some measure of burial resistance. It won't be perfect, as you
can't have that and retain Condorcet, but you'll have some protection.
Just what amount of resistance you'll have depends on the original
method, I think. Your initial suggestion is Benham; so there should be
some kind of DMTBR analog - something like, if there's an honest cycle,
then nobody who prefers a candidate outside the DMT set can benefit by
behaving like they prefer some third candidate to the candidate who
would otherwise win, when they do not actually prefer this third candidate.
I think. I'm not entirely sure.
-km
If there is an honest cycle, then there can be no burial of the (non
existant sincere CW) the only kind of burial we should worry about until
we have that completely under control, which (almost) no other proposed
method claims to do.
On Fri, Aug 4, 2023, 11:58 AM Kristofer Munsterhjelm km_elmet@t-online.de
wrote:
On 8/4/23 14:36, Forest Simmons wrote:
For now this is is a manual Plurality runoff method:
At each runoff stage let Y be the candidate with the fewest votes.
Then by majority decision either eliminate Y or else elect Y by
eliminating all other candidates.
That's it.
Note that majority decision by rational informed voters will never
eliminate Y when it is the sincere CW of the remaining candidates.
Very simple, but monotonic, and burial resistant ... not to mention
strongly Condorcet efficient.
I think this would only be monotone in a "manual DSV" sense where the
honest ballots are common knowledge.
The simplest IRV monotonicity case is where we have an ABCA cycle and
some BAC ballots become ABC ballots, then C survives to the final round
and beats A pairwise because B lost enough votes to be pushed down to
third place.
If we have manual DSV, then in the first round, A's supporters would
coordinate and decide that they're going to "donate" some first
preferences to B so as to push C down to third place, then in the final,
a majority (sincerely) indicates it prefers A to B, hence A wins.
But to me, that's kind of like the proof that a Condorcet winner is a
Nash equilibrium under Approval: it requires too much coordination.
Perhaps in a legislature, but in a public election, I wouldn't think
this would be monotone.
It might be possible to make it more strongly monotone by doing all the
manual DSV automatically, like a refinement of my "Contingent vote with
donation" method. The calculations could become rather hairy, though,
for more than three candidates.
But the idea that a majority will sincerely say "yes" when the CW
appears is good! Perhaps something revelation principle based could be
used to lessen the demands on the voters' calculation ability and the
degree to which the honest ballot set needs to be common knowledge.
In fact, it could accurately be called Sincere Benham (pending
permission from Chris) because if there is a sincere CW at any stage,
optimal strategy requires informed rational voters to elect it ... while
if only Y is eliminated at each stage, the ordinary Plurality runoff
candidate wins.
Note that since the sincere CW wins whenever there is one, there can be
no burial of a sincere CW ... which is the only kind of burial that
concerns us.
The main drawback is the potentially large number of manual votes
required.
Is there a DSV version that gets around this problem?
Quickly finding the Smith set by some elegant manual method would
largely solve the problem for many deliberative assemblies including
parliaments, senates, summits, etc.
A Coombs version of this method is equally burial resistant, monotone,
Condorcet efficient, etc, while more decisive... less likely to tie for
Y.
Antiplurality isn't burial resistant, so I wouldn't imagine this to be
either - in a Condorcet cycle, at least.
Consider a "fixed Benham" version of the above, with some election
method being used to set the elimination order, then we go from lowest
ranked to highest, asking "elect or eliminate".
If there is an honest CW then any ordering will work: for every other
candidate A that's checked before the honest CW, a majority will know
that if they hold out for longer, they can say "yes" for the honest CW
and then he wins.
If there's an honest cycle, then all bets are off. The benefit of using
interleaved Plurality over something like (fixed or interleaved) Ranked
Pairs or Antiplurality is that if there is an honest cycle, then you
have some measure of burial resistance. It won't be perfect, as you
can't have that and retain Condorcet, but you'll have some protection.
Just what amount of resistance you'll have depends on the original
method, I think. Your initial suggestion is Benham; so there should be
some kind of DMTBR analog - something like, if there's an honest cycle,
then nobody who prefers a candidate outside the DMT set can benefit by
behaving like they prefer some third candidate to the candidate who
would otherwise win, when they do not actually prefer this third candidate.
I think. I'm not entirely sure.
-km
On 8/4/23 22:54, Forest Simmons wrote:
If there is an honest cycle, then there can be no burial of the (non
existant sincere CW) the only kind of burial we should worry about
until we have that completely under control, which (almost) no other
proposed method claims to do.
In that case, you should be able to use any ordering you'd like for the
elimination process. Even a random one disclosed ahead of time should
work! (And with random, all you'd have to ask the voters is one binary
question per round.)
I'm curious if there's a backwards induction argument that traditional
exhaustive runoff should also elect the honest CW. I'm thinking
something like: every voter who prefers the outcome of continuing the
process to the current candidate will vote against the current
candidate, but when the current candidate is the CW, that's not
feasible. All that the supporters of whatever the result is when
continuing have to do, is to vote for someone else than the current
honest Plurality loser.
-km
On Fri, Aug 4, 2023, 2:41 PM Kristofer Munsterhjelm km_elmet@t-online.de
wrote:
On 8/4/23 22:54, Forest Simmons wrote:
If there is an honest cycle, then there can be no burial of the (non
existant sincere CW) the only kind of burial we should worry about
until we have that completely under control, which (almost) no other
proposed method claims to do.
In that case, you should be able to use any ordering you'd like for the
elimination process.
True, except we still have to deal with sincere cycles no matter how rare
they may be. My point is that insincere cycles are much more likely to be
brought about by burial of a CW than some other kind of burial.
The simplest policy (imho) is to treat all cycles as though they were
either subverted CW's or sincere.
Sincere cycles should be resolved by Classical Condorcet. Insincere cycles
should be resolved by some method akin to Method Y.
Here's how to do it when the ballot Smith set has exactly three candidates
X, Y, and Z:
Suppose without loss that X is the candidate whose max pairwise defeat
(among the three Smith members) is minimal.
Then the manual runoff should be between Y and Z unless a majority of
voters prefer X to the Y vs Z runoff.
This ceremony will elect the Sincere CW if there is one ... otherwise it
will elect X.
To see the truth ofthis fact suppose that there is no sincerely unbeaten
candidate among the three. Then there is a sincere cycle XYZX or its
reverse ZYXZ.
In the first case, if X is passed up, then the winner is Y, the runoff
winner between Y and Z. But X is preferred pairwise over Y, so a well
informed electorate would probably not pass up the opportunity to elect X.
The other case is similar ... but this time if X is passed up, Z wins ...
but X (in the sincere ZYXZ cycle) is preferred over Z ... so it would be a
mistake to pass up the chance to elect X (mod self hating voters).
In sum, when the Smith set is a triple, and X is the MinMax PairwiseDefeat
candidate (according to the actual preference ballot set) of the three,
then a manual runoff between Y and Z if a majority decides to not elect X
... will elect the sincere CW if there is one ... else it will break the
sincere top cycle at the weakest link.
Pretty nifty?
fws
Even a random one disclosed ahead of time should
work! (And with random, all you'd have to ask the voters is one binary
question per round.)
I'm curious if there's a backwards induction argument that traditional
exhaustive runoff should also elect the honest CW. I'm thinking
something like: every voter who prefers the outcome of continuing the
process to the current candidate will vote against the current
candidate, but when the current candidate is the CW, that's not
feasible. All that the supporters of whatever the result is when
continuing have to do, is to vote for someone else than the current
honest Plurality loser.
-km