Method X
Kevin is absolutely right ... all the more reason to heed Toby Pereira's
advice of using high definition ratings or else rankings with approval
cutoffs for Condorcet methods.
If you have a monotone one at a time elimination method, you can make it
Landau efficient by "covering takedown:"
Every time you eliminate a candidate take down with it every candidate that
is covers.
Although this disturbs the one-by-one property, the transitivity of
covering ensures that the order of take down of those candidates covered by
X does not matter: if X covers Y so Y is eliminated with X, and Y covers Z,
so Z is eliminated with Y, then Z is also covered by X, so Z is already
taken down by X without waiting for Y to be eliminated.
On Sun, Aug 6, 2023, 2:38 PM Kevin Venzke stepjak@yahoo.fr wrote:
Hi Forest,
Le samedi 5 août 2023 à 20:32:28 UTC−5, Forest Simmons <
forest.simmons21@gmail.com> a écrit :
Kevin,
Copeland is the simplest monotone Landau method.
Do we have a clone free version of Copeland that is for sure monotone?
This I don't know, but such a simple method as Copeland satisfying Landau
is quite
interesting, and I suppose we have to somehow proceed from Copeland in
searching for
additional methods.
Here's another one that (unlike Copeland) is definitely clone free as
well as monotonic
(if my proof holds warer):
Initialize a candidate variable X as the highest approval candidate.
Then while X is
covered, update X to be the most approved candidate that covers the
recent value of X that
we are updating.
In a UD context relying on implicit approval, this doesn't work wrt
monotonicity, because
a raised winner can obtain approval at the expense of another candidate.
For example say A is the approval winner and B covers A and wins. Then
some ballots are
changed from ...A>B to ...B>A (these are the bottom of the ranking) so
that A is losing
approval to B, and now some C is the approval winner, and B does not cover
C.
If I am not mistaken Agenda Based Chain Climbing is monotonic in the
sense that if the winner
moves "up" the agenda without disturbing the relative agenda order of
the other candidates ...
then the winner will still win.
If the agenda is based on approval scores, it seems to me that this
requirement should be met.
Am I wrong?
With implicit approval I guess the issue appears.
Kevin
votingmethods.net
On 8/8/23 01:17, Forest Simmons wrote:
I think it depends on the person. Myself, I find ranking easier
than rating, because I'm always trying (and failing) to find some
natural calibrated scale when rating, but ranking is easy: just "do
I prefer a world with X to one with Y?". And then if it's below my
JND, equal-rank.
To me "Below my JND" is the same as "infinitely close." And "I strongly
approve X" means I consider X to be infinitely close to my favorite. "I
strongly disapprove Z" means I consider Z to be infinitely close to my
anti-favorite.
Yes, that's an intuitive idea. But after further consideration, I think
it also depends on the effort, which brings a sort of "paradox of
voting" logic to it.
Suppose that I'm an agrarian leftist voter. Suppose that candidate X has
a strong agrarian position while candidate Y has a strong leftist
position. On intial examination, I find that they're about equally good,
so I would rank X and Y equally.
So far so good, "very close".
But let's say I were instead part of a deliberative body (something like
a citizens' assembly) writing a reference on the different candidates'
policies for voters to consult. Then I might investigate the candidates'
past records, the results of policies they supported, and so on, because
the assembly is smaller and the effect of getting it wrong is more
serious (assuming the guide would be used by the voters). And after
careful investigation, I might find out that, in my opinion, X is better
than Y.
Because the stakes are higher, I would make an additional effort to
distinguish X from Y. While complete instrumental rationality is
completely unrealistic for elections (or nobody would vote), there's
some part of it to ranking otherwise very close, or very hard to tell
apart, candidates.
These candidates may not even be an epsilon apart in the limit of time
spent scrutinizing them going to infinity. I just can't determine what
the actual distance is at a glance. So my equal-rank is an expression
that I trust the rest of the electorate enough, and that it would not be
worth it to spend excessive effort trying to determine if X is really
better than Y.
I'm kind of mixing "personal preference" (i.e. what I like the most) and
"best for society" (what candidate would be best for society), but it's
the best I can do at getting at what my intuition says.
Perhaps I would disapprove of the other end of the scale from where
my preferences lie, but if you were to add a (hypothetical)
Stalinist party and a Norwegian NSDAP (to mirror the Stalin and
Hitler example above), then my disapproval thresholds would
probably change so that I would disapprove of those two and approve
of all the democratic parties. >
And what that suggests to me is that when multiparty rule happens
and there's more of a gradual scale, then it gets harder to place
dividing lines [...]
You seem to be forgetting that strong approval and strong disapproval
are optional designations. If you do not feel strongly about approving
or disapproving a candidate, then you cannot honestly use those
designations.
In infinitesimal calculus, you are not required to classify every number
you use as infinitely large, infinitesimal, or neither ... but it is
nice to have those options.
I think what I was trying to say is that it seems on principle very hard
for a method to infer anything consistent from the approval cutoffs, due
to the voters' differences in idea about where they should lie. Like I
said in my quick and dirty STAR post, even if we assume consistent
utilities in a vNM sense, the voter-dependent affine scaling values
makes it very difficult to compare my expression of a cutoff to someone
else's.
With ranking, there's no problem, because the affine transformations are
all monotone.
The relative difficulty in a gradual setting (like multiparty democracy)
also makes sense in that context. Suppose U(v, x) is voter v's utility
if x is elected, and suppose that we have two voters with rating functions:
R(v1, x) = a_1 * U(v1, x) + b_1
R(v2, x) = a_2 * U(v2, x) + b_2
and some very large threshold values A >> B so that a voter v strongly
approves of every candidate for which if R(v, x) >= A, and strongly
disapproves of every candidate for which R(v, x) <= B.
Then if we suppose that the values of a and b are bounded in magnitude,
something kinda like Balinski and Laraki's "common language" idea, then
as long as candidates are easy to tell apart, then you can compare
different voters' below-B/above-A statements. In addition, the voters
can more easily classify them, particularly if the difference between
the sides are clear; if U(v, "my side") >> U(v, "their side"), then the
distinction is natural.
On the other hand, if U is a sliding scale, then either it's very
difficult to say just where the cutoffs should be, or the meaning won't
be preserved.
I have a kind of vague feeling that to the degree the meaning isn't
clear, honest voters will be incentivized to strategize because there
are multiple honest ballots. So the harder it is to understand, the
harder it is to "just stay honest".
Or in the terms of the above: if I'm only willing to disapprove of
totalitarian dictators, and I see that nobody on the ballot is a
totalitarian dictator, then I may start thinking "what other use could I
put this cutoff to?". Which seems to go against the purpose of elections
as information gathering.
I'm repeating myself, but maybe it'll give a better idea of the hard to
express intuitive feeling I have that approval cutoffs are hard and have
very unclear interpretations. Maybe it'll give both of us a better idea,
even!
-km
See if any of your worries apply to the main application I am proposing for
use of Strong Approval and Strong Disapproval:
Ballots are ranked preference style with two virtual candidates ... the
Strong Approval Cutoff and the Strong Disapproval Cutoff.
The default positions of these respective cutoffs are immediately below the
top (or equal top) position, and immediately above the bottom position that
harbors any and all candidates that do not outrank any candidate(s).
For each candidate X, let SA(X) and SD(X) be the respective number of
ballots on which X is above (respectively below) the Strong (Approval
respectively Disapproval) cutoff.
For each ballot B, let f(B) (respectively f'(B)) be the candidate strongly
(approved. resp disapproved) on ballot B that is strongly (approved resp
disapproved) on the greatest number of ballots.
We refer to f(B) and f'(B) respectively as the favorite and anti-favorite
candidates, respectively of ballot B.
The swap cost of converting one permutation P of the candidates into
another such permutation Q is the sum of the costs of the elementary swaps
needed to effect the conversion.
The swap cost of a single swap
AB --> BA is b'a, where b'=f'(B) and a=f(A).
Example.
48 C
28 A>B
24 B
What is the swap cost of converting C>A>B to its reverse order B>A>C?
The set of elementary swaps required are CA to AC, CB to BC, and AB to BA.
The total swap cost is given by
ca'+cb'+ab' where c=48, a=28, b'=0, and
a'=24+48=72
So the swap cost is=3456.
48×72+48×0+28×0=48×72
How about converting BAC to its reverse?
ba'+bc'+ac' = 24(72+28)+28×28=3184
We see that it is more expensive to convert CAB into BAC, than vice-versa.
Which order is the most expensive to convert into its opposite?
It could be argued the more democratically expensive to reverse an order,
the more stable the order.
Let's try reversing ABC:
The cost is ab'+ac' +bc', or ...
0+(a+b)c'=52×28=1456
Reversing back from CBA to ABC costs ... cb'+ca'+ba'=(c+b)a'=(48+24)76=5472
Reversing BCA:
b(c'+a')+ca'=2400+48×72=5856
Reversing ACB:
ac'+(a+c)b'=28×28=784
It seems that. BCA is the hardest permutation to reverse.
Note that B is not the anti-favorite of any faction and that A is the
antifavorite of every faction except its own.
Also BCA is a beatpath with defeat strengths of B>C=52 to 48 and C>A=48 to
28.
Because the C faction did not explicitly strongly disapprove B, candidate A
ended up with most of the default strong disapproval .... and B with none
of it.
On Sat, Aug 12, 2023, 2:49 PM Kristofer Munsterhjelm km_elmet@t-online.de
wrote:
On 8/8/23 01:17, Forest Simmons wrote:
I think it depends on the person. Myself, I find ranking easier
than rating, because I'm always trying (and failing) to find some
natural calibrated scale when rating, but ranking is easy: just "do
I prefer a world with X to one with Y?". And then if it's below my
JND, equal-rank.
To me "Below my JND" is the same as "infinitely close." And "I strongly
approve X" means I consider X to be infinitely close to my favorite. "I
strongly disapprove Z" means I consider Z to be infinitely close to my
anti-favorite.
Yes, that's an intuitive idea. But after further consideration, I think
it also depends on the effort, which brings a sort of "paradox of
voting" logic to it.
Suppose that I'm an agrarian leftist voter. Suppose that candidate X has
a strong agrarian position while candidate Y has a strong leftist
position. On intial examination, I find that they're about equally good,
so I would rank X and Y equally.
So far so good, "very close".
But let's say I were instead part of a deliberative body (something like
a citizens' assembly) writing a reference on the different candidates'
policies for voters to consult. Then I might investigate the candidates'
past records, the results of policies they supported, and so on, because
the assembly is smaller and the effect of getting it wrong is more
serious (assuming the guide would be used by the voters). And after
careful investigation, I might find out that, in my opinion, X is better
than Y.
Because the stakes are higher, I would make an additional effort to
distinguish X from Y. While complete instrumental rationality is
completely unrealistic for elections (or nobody would vote), there's
some part of it to ranking otherwise very close, or very hard to tell
apart, candidates.
These candidates may not even be an epsilon apart in the limit of time
spent scrutinizing them going to infinity. I just can't determine what
the actual distance is at a glance. So my equal-rank is an expression
that I trust the rest of the electorate enough, and that it would not be
worth it to spend excessive effort trying to determine if X is really
better than Y.
I'm kind of mixing "personal preference" (i.e. what I like the most) and
"best for society" (what candidate would be best for society), but it's
the best I can do at getting at what my intuition says.
Perhaps I would disapprove of the other end of the scale from where
my preferences lie, but if you were to add a (hypothetical)
Stalinist party and a Norwegian NSDAP (to mirror the Stalin and
Hitler example above), then my disapproval thresholds would
probably change so that I would disapprove of those two and approve
of all the democratic parties. >
And what that suggests to me is that when multiparty rule happens
and there's more of a gradual scale, then it gets harder to place
dividing lines [...]
You seem to be forgetting that strong approval and strong disapproval
are optional designations. If you do not feel strongly about approving
or disapproving a candidate, then you cannot honestly use those
designations.
In infinitesimal calculus, you are not required to classify every number
you use as infinitely large, infinitesimal, or neither ... but it is
nice to have those options.
I think what I was trying to say is that it seems on principle very hard
for a method to infer anything consistent from the approval cutoffs, due
to the voters' differences in idea about where they should lie. Like I
said in my quick and dirty STAR post, even if we assume consistent
utilities in a vNM sense, the voter-dependent affine scaling values
makes it very difficult to compare my expression of a cutoff to someone
else's.
With ranking, there's no problem, because the affine transformations are
all monotone.
The relative difficulty in a gradual setting (like multiparty democracy)
also makes sense in that context. Suppose U(v, x) is voter v's utility
if x is elected, and suppose that we have two voters with rating functions:
R(v1, x) = a_1 * U(v1, x) + b_1
R(v2, x) = a_2 * U(v2, x) + b_2
and some very large threshold values A >> B so that a voter v strongly
approves of every candidate for which if R(v, x) >= A, and strongly
disapproves of every candidate for which R(v, x) <= B.
Then if we suppose that the values of a and b are bounded in magnitude,
something kinda like Balinski and Laraki's "common language" idea, then
as long as candidates are easy to tell apart, then you can compare
different voters' below-B/above-A statements. In addition, the voters
can more easily classify them, particularly if the difference between
the sides are clear; if U(v, "my side") >> U(v, "their side"), then the
distinction is natural.
On the other hand, if U is a sliding scale, then either it's very
difficult to say just where the cutoffs should be, or the meaning won't
be preserved.
I have a kind of vague feeling that to the degree the meaning isn't
clear, honest voters will be incentivized to strategize because there
are multiple honest ballots. So the harder it is to understand, the
harder it is to "just stay honest".
Or in the terms of the above: if I'm only willing to disapprove of
totalitarian dictators, and I see that nobody on the ballot is a
totalitarian dictator, then I may start thinking "what other use could I
put this cutoff to?". Which seems to go against the purpose of elections
as information gathering.
I'm repeating myself, but maybe it'll give a better idea of the hard to
express intuitive feeling I have that approval cutoffs are hard and have
very unclear interpretations. Maybe it'll give both of us a better idea,
even!
-km