Arrow's theorem and cardinal voting systems
On Thu, Jan 9, 2020 at 11:46 PM robert bristow-johnson wrote:
"strong preference" vs. "weak preference" implies a Score ballot.
Usually, but not necessarily:
https://electowiki.org/wiki/Strong/weak_preference_option
my question that i have asked the Score Voting or Approval Voting advocates
years ago remains: "How much should I score my second choice?"
If asked to rank three ice cream flavors, my preference would be Strawberry
Chocolate > Garlic.
If then asked to choose between:
- Chocolate
- A mystery box with a 75% chance of containing Strawberry and a 25%
chance of containing Garlic
I would choose #1, which shows that:
A. My preference for Chocolate > Garlic is significantly stronger than my
preference for Strawberry > Chocolate.
B. If voting honestly, I should give Chocolate at least a 4 out of 5 on a
Score ballot.
The odds can then be varied, to narrow in on a more precise rating, which
is essentially what we all do internally when we rate a movie or restaurant
or product or student or respond to a Likert scale survey, etc.
Of course, this is imprecise, but so is forcing voters to rank many
candidates when they are indifferent between some of them.
If Vanilla and French Vanilla were both on the same ballot, I would be
indifferent between them. Forcing me to choose between them and then
arbitrarily assigning the same weight to this very weak preference that was
applied to my Chocolate > Garlic preference would be rather undemocratic,
no?
http://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html
Organisms don't have ordered lists of equal-strength preferences in their
brains. They have fuzzy estimates of utility that they then convert to
rankings when necessary.
"The majority judgement experiment proves that the model on which the
theory of social choice and voting is based is simply not true: voters do
not have preference lists of candidates in their minds. Moreover, forcing
voters to establish preference lists only leads to inconsistencies,
impossibilities and incompatibilities."
https://hal.archives-ouvertes.fr/hal-00243076/document#page=40
that tactical question faces the voter in a Score or Approval election the
second he/she steps into the voting booth. but not so for the ordinal
Ranked ballot.
From what I've been told (though I haven't read and understood it myself),
Gibbard's theorem proves that ALL voting systems require voters to make
tactical decisions, no matter whether they are ranked or rated or otherwise.
On Fri, Jan 10, 2020 at 2:02 AM Rob Lanphier wrote:
that cardinal voting systems are provably free of any
sort of impossibility paradox.
I've never heard anyone claim that they are. The claim is simply that
Arrow's theorem, in particular, doesn't apply to cardinal systems, which
Arrow seems to agree with in that interview (and I don't know why that
would be a big deal or why he shouldn't be trusted to interpret his own
theorem). Satterthwaite's also doesn't. It's Gibbard's theorem,
specifically, that applies to the general case of all conceivable voting
systems: https://politics.stackexchange.com/a/14245/10373
On 10/01/2020 12.41, Steve Eppley wrote:
For the criterion that matters most to me, I don't have a rigorous definition. Here's a non-rigorous definition: The voting method should give candidates who want to win a strong incentive to take positions that the voters themselves would collectively choose given a well-functioning direct democracy... even on issues that most voters don't care strongly about. Here's how I relate that to voting methods like Maximize Affirmed Majorities (MAM), which facilitate competition, count all pairwise majorities, and pay attention to the sizes of the majorities: Suppose candidate Alice wants to win, and is considering taking position p on some issue. Although she knows a majority of the voters prefer alternative q over p, her wealthy campaign donors favor p and most voters care more about other issues. Given a voting method like MAM, the risk to Alice is that by advocating p, she would create an opportunity for another candidate Bob to enter the race, take position q and copy
Alice's positions on all other issues. The larger the majority who prefer q over p, the larger the majority who would tend to rank Bob over Alice. Defeating Alice. A deterrent against taking unpopular positions to benefit donors.
How about this? If you clone A into A1 (Bob) and A2 (Alice), and A1 is
ranked above A2 on more ballots than A2 is ranked above A1, then if the
original winner was A, the new winner should be A1.
That most voters care about other issues than p vs q means that Alice
and Bob should be near-clones, since "Alice but with q" is a slight
improvement to "Alice with p", but not enough of an improvement that
some other candidate is ranked between A1 and A2.
If voters care more about q vs p, then A1 and A2 will no longer be
near-clones, but hopefully the method should generalize robustly from
the clone case so that it follows the spirit of the criterion.
On 1/11/2020 6:48 AM, Kristofer Munsterhjelm wrote:
On 10/01/2020 12.41, Steve Eppley wrote:
For the criterion that matters most to me, I don't have a rigorous definition. Here's a non-rigorous definition: The voting method should give candidates who want to win a strong incentive to take positions that the voters themselves would collectively choose given a well-functioning direct democracy... even on issues that most voters don't care strongly about. Here's how I relate that to voting methods like Maximize Affirmed Majorities (MAM), which facilitate competition, count all pairwise majorities, and pay attention to the sizes of the majorities: Suppose candidate Alice wants to win, and is considering taking position p on some issue. Although she knows a majority of the voters prefer alternative q over p, her wealthy campaign donors favor p and most voters care more about other issues. Given a voting method like MAM, the risk to Alice is that by advocating p, she would create an opportunity for another candidate Bob to enter the race, take position q and copy
Alice's positions on all other issues. The larger the majority who prefer q over p, the larger the majority who would tend to rank Bob over Alice. Defeating Alice. A deterrent against taking unpopular positions to benefit donors.
How about this? If you clone A into A1 (Bob) and A2 (Alice), and A1 is
ranked above A2 on more ballots than A2 is ranked above A1, then if the
original winner was A, the new winner should be A1.
That most voters care about other issues than p vs q means that Alice
and Bob should be near-clones, since "Alice but with q" is a slight
improvement to "Alice with p", but not enough of an improvement that
some other candidate is ranked between A1 and A2.
If voters care more about q vs p, then A1 and A2 will no longer be
near-clones, but hopefully the method should generalize robustly from
the clone case so that it follows the spirit of the criterion.
If by "How about this?" you're suggesting satisfaction of that clone criterion ("... the new winner should be A1") implies satisfaction of my non-rigorous criterion ("create a strong incentive to take positions the voters would choose"), I don't see why that would be so.
Instant Runoff would elect A1 in that clone criterion's scenario, yes? But Instant Runoff doesn't create the incentive. To the contrary, Instant Runoff defeats candidates who advocate compromises that voters would collectively choose, and makes those candidates & positions appear unpopular. Instant Runoff rewards extremists the same way Plurality Rule does, because it counts at most one of the majorities, which can be a coalition of minorities. For example, a minority who want abortion banned, plus a minority who want immigrants deported, plus a minority who want guns unregulated, plus a minority who want capital gains taxes slashed, etc, can together add up to a majority.
--Steve
On 11/01/2020 17.46, Steve Eppley wrote:
On 1/11/2020 6:48 AM, Kristofer Munsterhjelm wrote:
On 10/01/2020 12.41, Steve Eppley wrote:
For the criterion that matters most to me, I don't have a
rigorous definition. Here's a non-rigorous definition: The
voting method should give candidates who want to win a strong
incentive to take positions that the voters themselves would
collectively choose given a well-functioning direct democracy...
even on issues that most voters don't care strongly about.
Here's how I relate that to voting methods like Maximize Affirmed
Majorities (MAM), which facilitate competition, count all
pairwise majorities, and pay attention to the sizes of the
majorities: Suppose candidate Alice wants to win, and is considering
taking position p on some issue. Although she knows a majority of the
voters prefer alternative q over p, her wealthy campaign donors favor p
and most voters care more about other issues. Given a voting method
like MAM, the risk to Alice is that by advocating p, she would create an
opportunity for another candidate Bob to enter the race, take position q
and copy Alice's positions on all other issues. The larger the
majority who prefer q over p, the larger the majority who would
tend to rank Bob over Alice. Defeating Alice. A deterrent
against taking unpopular positions to benefit donors.
How about this? If you clone A into A1 (Bob) and A2 (Alice), and A1 is
ranked above A2 on more ballots than A2 is ranked above A1, then if the
original winner was A, the new winner should be A1.
That most voters care about other issues than p vs q means that Alice
and Bob should be near-clones, since "Alice but with q" is a slight
improvement to "Alice with p", but not enough of an improvement that
some other candidate is ranked between A1 and A2.
If voters care more about q vs p, then A1 and A2 will no longer be
near-clones, but hopefully the method should generalize robustly from
the clone case so that it follows the spirit of the criterion.
If by "How about this?" you're suggesting satisfaction of that clone
criterion ("... the new winner should be A1") implies satisfaction of my
non-rigorous criterion ("create a strong incentive to take positions the
voters would choose"), I don't see why that would be so.
Instant Runoff would elect A1 in that clone criterion's scenario, yes?
I don't think it would in every such scenario. Consider this election pair:
Before cloning:
110: A
100: X>A
100: Y>A
X and Y are eliminated and then A wins.
After cloning:
110: A2>A1
100: X>A1>A2
100: Y>A1>A2
First A1 is eliminated, and then X and Y are eliminated, and then A2
wins. But A1 is the CW and beats A2 pairwise 200-110.
If the q-preferring majority ranks A1 and A2 low enough, then IRV may
exclude A1 before it gets to determine who should win of A1 and A2. It's
the usual center squeeze.
Does that make the clone criterion more suited to your purposes, or
would it have to be stronger? I suppose the clone criterion is a sort of
local optimum criterion (if Alice exists, then Bob can copy all of
Alice's positions except the one a majority dislikes, and overtake
Alice), while your non-rigorous criterion is a global optimum criterion.
(In passing, I think I see that LIAA + clone independence implies this
clone criterion, as well.)
On 1/11/2020 2:42 PM, Kristofer Munsterhjelm wrote:
On 11/01/2020 17.46, Steve Eppley wrote:
On 1/11/2020 6:48 AM, Kristofer Munsterhjelm wrote:
On 10/01/2020 12.41, Steve Eppley wrote:
For the criterion that matters most to me, I don't have a
rigorous definition. Here's a non-rigorous definition: The
voting method should give candidates who want to win a strong
incentive to take positions that the voters themselves would
collectively choose given a well-functioning direct democracy...
even on issues that most voters don't care strongly about.
Here's how I relate that to voting methods like Maximize Affirmed
Majorities (MAM), which facilitate competition, count all
pairwise majorities, and pay attention to the sizes of the
majorities: Suppose candidate Alice wants to win, and is considering
taking position p on some issue. Although she knows a majority of the
voters prefer alternative q over p, her wealthy campaign donors favor p
and most voters care more about other issues. Given a voting method
like MAM, the risk to Alice is that by advocating p, she would create an
opportunity for another candidate Bob to enter the race, take position q
and copy Alice's positions on all other issues. The larger the
majority who prefer q over p, the larger the majority who would
tend to rank Bob over Alice. Defeating Alice. A deterrent
against taking unpopular positions to benefit donors.
How about this? If you clone A into A1 (Bob) and A2 (Alice), and A1 is
ranked above A2 on more ballots than A2 is ranked above A1, then if the
original winner was A, the new winner should be A1.
That most voters care about other issues than p vs q means that Alice
and Bob should be near-clones, since "Alice but with q" is a slight
improvement to "Alice with p", but not enough of an improvement that
some other candidate is ranked between A1 and A2.
If voters care more about q vs p, then A1 and A2 will no longer be
near-clones, but hopefully the method should generalize robustly from
the clone case so that it follows the spirit of the criterion.
If by "How about this?" you're suggesting satisfaction of that clone
criterion ("... the new winner should be A1") implies satisfaction of my
non-rigorous criterion ("create a strong incentive to take positions the
voters would choose"), I don't see why that would be so.
Instant Runoff would elect A1 in that clone criterion's scenario, yes?
I don't think it would in every such scenario. Consider this election pair:
Before cloning:
110: A
100: X>A
100: Y>A
X and Y are eliminated and then A wins.
After cloning:
110: A2>A1
100: X>A1>A2
100: Y>A1>A2
First A1 is eliminated, and then X and Y are eliminated, and then A2
wins. But A1 is the CW and beats A2 pairwise 200-110.
If the q-preferring majority ranks A1 and A2 low enough, then IRV may
exclude A1 before it gets to determine who should win of A1 and A2. It's
the usual center squeeze.
Does that make the clone criterion more suited to your purposes, or
would it have to be stronger? I suppose the clone criterion is a sort of
local optimum criterion (if Alice exists, then Bob can copy all of
Alice's positions except the one a majority dislikes, and overtake
Alice), while your non-rigorous criterion is a global optimum criterion.
(In passing, I think I see that LIAA + clone independence implies this
clone criterion, as well.)
You're right that Instant Runoff fails "clone A1 should win."
I don't know whether its satisfaction implies satisfaction of "the incentive to take positions the voters would choose." My election method analysis skills are very rusty.
I don't recall LIAA. I assume you mean LIIA (Local Independence of Irrelevant Alternatives, promoted by Peyton Young).
There appears to be a flaw in that clone criterion. Suppose 3 clones majority cycle: Bob > Alice > Charlie > Bob. The premise of the "clone A1 should win" criterion could hold: In the "original" scenario where Bob doesn't run, Alice wins. We don't have enough information to show that Bob will win if Bob runs too. Alice could still win if the Bob>Alice majority is the smallest of the three cyclic majorities. (When I described my thinking about the incentive in MAM, I wrote: "The larger the majority who prefer q over p, the larger the majority who would tend to rank Bob over Alice.") But that clone failure isn't necessarily a failure of the voting method to create the strong incentive. My hunch is that typically, candidates like Alice won't be able to rely on a Bob>Alice majority being the smallest in a cycle, when taking positions on issues. The chance that Bob>Alice won't be smallest in a cycle is a risk to be avoided, all else being equal.
Thanks for spending time on this. I hope you can continue.
--Steve
Hi Jim,
Thanks for the thoughtful response! I want to respond to everyone,
but I suppose I'll just circle back on respond to yours now. More
inline...
On Thu, Jan 9, 2020 at 8:12 PM Faran, James jjfaran@buffalo.edu wrote:
About Score voting failing Unrestricted Domain:
Part of the confusion of those advocating score and you is not a confusion on anyone's part, but rather a difference in what each considers a preference. (It's not possible to have a good reasoned argument until both sides agree on what the words mean.) Score voters would say
A:100; B:95; C:0
and
A:100; B:5; C:0
are different preferences, but you seem to say that these are both A>B>C and so are the same.
Hmm, that seems to be an interesting way of putting it. I agree that
this difference is at the heart of the matter. It seems that Arrow's
proof also relies on these being the same, and I believe the criterion
stipulates that they must be the same. That doesn't mean Arrow's
theorem doesn't apply to systems that allow for this extra
information; it just means that those systems don't meet that
criterion.
I would say the Electowiki page on Unrestricted Domain needs to be edited to include both possibilities, but I can't be bothered.
Well, I'm not going to edit it for you ;-P
I suppose it would be good to cover many of the points of this
discussion on that page, but I'd prefer to come to a shared
understanding before I make any edits.
You also seem to think that most voters would not be able to understand that sort of nuance. You may be right there, especially in today's political climate (especially in the United States?), where there are two sides and the other side is always demonized.
Well, it's not the first time the United States had problems with
partisanship causing things to get personal:
- https://en.wikipedia.org/wiki/George_Washington%27s_Farewell_Address
- https://en.wikipedia.org/wiki/Burr%E2%80%93Hamilton_duel
- https://en.wikipedia.org/wiki/Petticoat_affair
- https://en.wikipedia.org/wiki/Caning_of_Charles_Sumner
- https://en.wikipedia.org/wiki/American_Civil_War
I don't think the problem is today's political climate (here in the
USA or elsewhere). The problem is with getting enough people to agree
that the system is fair.
Note that any new voting system will almost always try to be replaced by the loser under the new system. ("The current government is illegitimate! If it wasn't for the biased voting system we would have won!" -- cf. the recently revived call for the elimination of the U. S. Electoral College after Mr. Trump won with a minority of the popular vote.) If the winner can't keep support, the losing side will be able to push through a change.
However, a question: If we had the following score ballots:
9000: A:100; B:95; C:0
1000: B:100; C:85; A:0
giving A a score of 900,000 and B a score of 955,000, hence a victory for B, would there really be enough antipathy to B to cause outrage? All the A voters seemed to think B was pretty good.
I think the answer would be "yes", once the people who were passionate
about A discovered that A was preferred to B 9:1 by 9,000 out of the
10,000 voters, and that it was only those people who rated crackpot C
as "85" who swung the election. As others have pointed out on the
list, it's probably not a good idea for a voting system to push these
mathematical nuance problems onto voters.
Of course (see above), the losing side could always complain.
Well, sure, but when the losing side has a point, that's a problem for
the advocates for the electoral system in question.
Anyone wedded to Condorcet winners would be outraged. And, of course, no real world election would end up like this. Score may be a little too ripe for manipulation. Gibbard-Satterthwaite, anyone?
Like I said in my original email, I think the great thing about these
impossibility theorems (like Arrow's and Gibbard-Satterthwaite) is
that they demonstrate cases where tradeoffs will be necessary. Your
exaggerated example is helpful in coming to a shared understanding of
a edge case in the system that could also be manifested in a more
realistic example.
Rob
On 11/01/2020 06.24, fdpk69p6uq@snkmail.com wrote:
On Thu, Jan 9, 2020 at 11:46 PM robert bristow-johnson wrote:
my question that i have asked the Score Voting or Approval Voting
advocates years ago remains: "How much should I score my second choice?"
If asked to rank three ice cream flavors, my preference would be
Strawberry > Chocolate > Garlic.
If then asked to choose between:
- Chocolate
- A mystery box with a 75% chance of containing Strawberry and a 25%
chance of containing Garlic
I would choose #1, which shows that:
A. My preference for Chocolate > Garlic is significantly stronger than
my preference for Strawberry > Chocolate.
B. If voting honestly, I should give Chocolate at least a 4 out of 5 on
a Score ballot.
The odds can then be varied, to narrow in on a more precise rating,
which is essentially what we all do internally when we rate a movie or
restaurant or product or student or respond to a Likert scale survey, etc.
Of course, this is imprecise, but so is forcing voters to rank many
candidates when they are indifferent between some of them.
That sounds like it's still a normalized ballot, so that "honest Range"
with this calibration would fail Steve Eppley's independence. It doesn't
solve the problem of Range having many sincere ways to express the same
ballot, either, though it does narrow down which ballots are honest/sincere.
Suppose we have an ice cream election and (for the sake of the argument)
you're exactly indifferent between a 75:25 lottery between Strawberry
and Garlic, and a certain choice of Chocolate. (Suppose also that you're
risk neutral, because risk aversion is not the point.)
Then what we know is that
0.75 * utility(Strawberry) + 0.25 * utility(Garlic) = utility(Chocolate)
This is a linear equation with three unknowns. We need two more to
unambiguously determine the values. A standard zero and a standard unit
will do.
A standard unit is reasonable to have, because multiplying every unknown
by some constant preserves all the lottery-based equations, so someone
who likes exaggerating his scale (e.g. by saying "OMG, this is the best
thing ever!" every time he sees something good) will have more influence
than someone who likes to be economical with his values.
But it's hardly clear how to find that standard zero and unit. For
e.g. a pizza election, you could say the standard zero is no pizza at
all and the standard unit is a Margherita (say). But for a political
election? And strictly speaking, the scale would have to be unbounded so
that it can both accommodate people who don't particularly like pizza
and people who have lived their whole lives for the purpose of getting a
pizza.
So the point is that the lack of a single reference honest ballot for
Range is a due to cardinal utilities being very hard to calibrate
between people. And if you can't say what the one honest ballot is, then
there will still be ambiguity in any cardinal system as to what
constitutes sincerity, and how you should rate your choices.
You may try to define a single honest ballot for a semi-cardinal method
that automatically normalizes the endpoints to max and min value, so
that the two unknowns are given and the voter can just use the lottery
method to fill in the remaining data. But if you do so, then a
two-candidate election becomes a majority election and you're back to
Eppley's independence failure example. So in some sense, the
impossibility is "tight" - if you want IIA, the ballots must be
independently calibrated. If you let them be relatively calibrated even
a little, IIA goes away.
That doesn't mean that rankings are better than ratings, period. But a
ranked ballot makes it possible to have a single honest ballot without
needing to standardize it -- at the expense of ranked methods failing
IIA. And the ambiguity of rated ballots makes honesty and strategy blur
together. Since no election method can know if you've chosen the right
scale, it seems like honesty and strategy will always be blurred
somewhat together, no matter the cardinal method.
If Vanilla and French Vanilla were both on the same ballot, I would be
indifferent between them. Forcing me to choose between them and then
arbitrarily assigning the same weight to this very weak preference that
was applied to my Chocolate > Garlic preference would be rather
undemocratic, no?
http://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html
To some degree, what I said above also holds for truncation and
equal-rank, but there it doesn't seem to be as serious a problem. It
would be interesting to find out why.
Perhaps truncation and equal rank are convenience features, so the voter
says "determining who wins of these is worth less to me than the effort
it is to rank those candidates, so I'll let someone else decide". That
might be a decision that a voter can do without needing to do any
absolute calibration. But if so, the problem with cardinal ballots is
then not that there are many honest ballots as such, but rather that
the voter is required to make a strategic effort.
Organisms don't have ordered lists of equal-strength preferences in
their brains. They have fuzzy estimates of utility that they then
convert to rankings when necessary.
(I have a suspicion that what we really have are utility vectors, and
what we call "utility" is more like a norm of these. But I have no
proof, and it's sort of besides the point.)
that tactical question faces the voter in a Score or Approval
election the second he/she steps into the voting booth. but not so
for the ordinal Ranked ballot.
From what I've been told (though I haven't read and understood it
myself), Gibbard's theorem proves that ALL voting systems require voters
to make tactical decisions, no matter whether they are ranked or rated
or otherwise.
All deterministic ones, to be precise :-) And Gibbard doesn't say voters
need to do it - it only says that a voter who wants to maximize the
impact of his vote needs to do so.
On 12/01/2020 21.59, Steve Eppley wrote:
On 1/11/2020 2:42 PM, Kristofer Munsterhjelm wrote:
I don't think it would in every such scenario. Consider this election pair:
Before cloning:
110: A
100: X>A
100: Y>A
X and Y are eliminated and then A wins.
After cloning:
110: A2>A1
100: X>A1>A2
100: Y>A1>A2
First A1 is eliminated, and then X and Y are eliminated, and then A2
wins. But A1 is the CW and beats A2 pairwise 200-110.
If the q-preferring majority ranks A1 and A2 low enough, then IRV may
exclude A1 before it gets to determine who should win of A1 and A2. It's
the usual center squeeze.
Does that make the clone criterion more suited to your purposes, or
would it have to be stronger? I suppose the clone criterion is a sort of
local optimum criterion (if Alice exists, then Bob can copy all of
Alice's positions except the one a majority dislikes, and overtake
Alice), while your non-rigorous criterion is a global optimum criterion.
(In passing, I think I see that LIAA + clone independence implies this
clone criterion, as well.)
You're right that Instant Runoff fails "clone A1 should win."
I don't know whether its satisfaction implies satisfaction of "the
incentive to take positions the voters would choose." My election
method analysis skills are very rusty.
I don't recall LIAA. I assume you mean LIIA (Local Independence of
Irrelevant Alternatives, promoted by Peyton Young).
Yes, that should have been LIIA.
There appears to be a flaw in that clone criterion. Suppose 3
clones majority cycle: Bob > Alice > Charlie > Bob. The premise of
the "clone A1 should win" criterion could hold: In the "original"
scenario where Bob doesn't run, Alice wins. We don't have enough
information to show that Bob will win if Bob runs too. Alice could
still win if the Bob>Alice majority is the smallest of the three
cyclic majorities. (When I described my thinking about the incentive
in MAM, I wrote: "The larger the majority who prefer q over p, the
larger the majority who would tend to rank Bob over Alice.") But
that clone failure isn't necessarily a failure of the voting method
to create the strong incentive. My hunch is that typically,
candidates like Alice won't be able to rely on a Bob>Alice majority
being the smallest in a cycle, when taking positions on issues. The
chance that Bob>Alice won't be smallest in a cycle is a risk to be
avoided, all else being equal.
I'm not entirely sure what you mean. Do you mean that even if "A1 should
win" happens to be necessary, it isn't sufficient; or that even if it
happens to be sufficient, it isn't necessary?
I think you're saying it's not sufficient, because there could already
be a clone of Alice, and then when Bob enters, he could have the
smallest of the cyclic majorities and create a cycle, and he won't win.
More generally, we can say that he creates a three-cycle and the cloning
comes out so that, according to the cycle-resolution mechanism of the
method in question, he doesn't win even though he's in the Smith set.
But then it would seem that no matter what method you have, it's
possible to construct the cloning so that the right clone loses.
If that's right, then there has to be some kind of additional structure
that makes it possible for the method to distinguish the right clone
from the other clones. In the original example, that is that Bob copies
all of Alice's positions except the disliked one, where he does better
according to a majority. For the three clones to create a cycle, there
has to be some set of properties so that a majority prefers Bob's to
Alice's, Alice's to Charlie's, and Charlie's to Bob's. But then,
wouldn't Bob have to differ from Alice by more than one property?
Thanks for spending time on this. I hope you can continue.
--Steve
Election-Methods mailing list - see https://electorama.com/em for list info
On 1/22/2020 6:50 PM, Kristofer Munsterhjelm wrote:
On 12/01/2020 21.59, Steve Eppley wrote:
On 1/11/2020 2:42 PM, Kristofer Munsterhjelm wrote:
I don't think it would in every such scenario. Consider this election pair:
Before cloning:
110: A
100: X>A
100: Y>A
X and Y are eliminated and then A wins.
After cloning:
110: A2>A1
100: X>A1>A2
100: Y>A1>A2
First A1 is eliminated, and then X and Y are eliminated, and then A2
wins. But A1 is the CW and beats A2 pairwise 200-110.
If the q-preferring majority ranks A1 and A2 low enough, then IRV may
exclude A1 before it gets to determine who should win of A1 and A2. It's
the usual center squeeze.
Does that make the clone criterion more suited to your purposes, or
would it have to be stronger? I suppose the clone criterion is a sort of
local optimum criterion (if Alice exists, then Bob can copy all of
Alice's positions except the one a majority dislikes, and overtake
Alice), while your non-rigorous criterion is a global optimum criterion.
(In passing, I think I see that LIAA + clone independence implies this
clone criterion, as well.)
You're right that Instant Runoff fails "clone A1 should win."
I don't know whether its satisfaction implies satisfaction of "the
incentive to take positions the voters would choose." My election
method analysis skills are very rusty.
I don't recall LIAA. I assume you mean LIIA (Local Independence of
Irrelevant Alternatives, promoted by Peyton Young).
Yes, that should have been LIIA.
There appears to be a flaw in that clone criterion. Suppose 3
clones majority cycle: Bob > Alice > Charlie > Bob. The premise of
the "clone A1 should win" criterion could hold: In the "original"
scenario where Bob doesn't run, Alice wins. We don't have enough
information to show that Bob will win if Bob runs too. Alice could
still win if the Bob>Alice majority is the smallest of the three
cyclic majorities. (When I described my thinking about the incentive
in MAM, I wrote: "The larger the majority who prefer q over p, the
larger the majority who would tend to rank Bob over Alice.") But
that clone failure isn't necessarily a failure of the voting method
to create the strong incentive. My hunch is that typically,
candidates like Alice won't be able to rely on a Bob>Alice majority
being the smallest in a cycle, when taking positions on issues. The
chance that Bob>Alice won't be smallest in a cycle is a risk to be
avoided, all else being equal.
I'm not entirely sure what you mean. Do you mean that even if "A1 should
win" happens to be necessary, it isn't sufficient; or that even if it
happens to be sufficient, it isn't necessary?
I think you're saying it's not sufficient, because there could already
be a clone of Alice, and then when Bob enters, he could have the
smallest of the cyclic majorities and create a cycle, and he won't win.
More generally, we can say that he creates a three-cycle and the cloning
comes out so that, according to the cycle-resolution mechanism of the
method in question, he doesn't win even though he's in the Smith set.
But then it would seem that no matter what method you have, it's
possible to construct the cloning so that the right clone loses.
If that's right, then there has to be some kind of additional structure
that makes it possible for the method to distinguish the right clone
from the other clones. In the original example, that is that Bob copies
all of Alice's positions except the disliked one, where he does better
according to a majority. For the three clones to create a cycle, there
has to be some set of properties so that a majority prefers Bob's to
Alice's, Alice's to Charlie's, and Charlie's to Bob's. But then,
wouldn't Bob have to differ from Alice by more than one property?
Where you ask whether I meant "isn't sufficient" or "isn't necessary," what I meant is that the clone criterion you proposed, "A1 must win," doesn't clearly distinguish between voting methods that create the desired incentive and voting methods that don't. (Neither does the "weaker" criterion "Alice must not win.") Alice can still win even though clone Bob is ranked over Alice by a majority. There may be voting methods that fail the clone criterion yet create the desired incentive anyway. The clone criterion might not be necessary.
And perhaps some voting methods that satisfy the clone criterion may fail to create the desired incentive in more general, non-clone cases. The clone criterion might not be sufficient. A "clone recognition filter" could be tacked onto a bad voting method, contrived to recognize when the "A1 must win" premise exists in the votes and guarantee the defeat of A2 in the rare case when the premise holds, yet not create the desired incentive in other cases. For example, the voting method "if clone A2 must lose then elect clone A1; else elect the Instant Runoff winner" would satisfy whatever clone criterion you like that implies A2 must lose, but fail to create the desired incentive.
When you say "there has to be some kind of additional structure that makes it possible for the method to distinguish the right clone" it looks like you're thinking specifically about clone criteria, and possibly not generally enough to cover non-clone cases too. Although you may be right that any rigorous criterion that distinguishes voting methods that create the desired incentive must be some kind of clone criterion, I think that's just speculation. The ability to contrive a "clone filter" in the previous paragraph suggests the speculation is wrong.
Regarding your question about whether Bob would need to differ from Alice by more than one property (policy) for there to be a "Bob>Alice>Charlie>Bob" majority cycle (in which the Bob>Alice majority might unfortunately be the smallest majority), the answer is No, Bob can differ from Alice on only one policy. Suppose issue I1 is abortion and issue I2 is taxes. Suppose that on abortion, a majority prefer policy aMaj over policy aMin. Suppose that on taxes, a majority prefer tMaj over tMin. Suppose Charlie takes positions aMaj and tMaj (consistent with my goal that the voting method should create an incentive to take majority-preferred positions). Suppose Alice takes positions aMin and tMin, after cleverly calculating that the minority who prefer aMin and the minority who prefer tMin are "single issue voters" and thus will together comprise a majority coalition who prefer policy pair aMin&tMin over policy pair aMaj&tMaj. Bob has two obvious options, either of which
produces the majority cycle: (1) Bob can take positions aMaj and tMin, or (2) Bob can takes positions aMin and tMaj. In either option, Bob differs from Alice by only one property. The option that's better for Bob depends on which issue majority is smaller (given a voting method that pays attention to the sizes of the pairwise majorities).
There may of course be other reasonably simple options that are even better for Bob, if I1 or I2 isn't a dichotomous issue. For example, suppose policy aMaj is "no restrictions whatsoever on a woman's right to have an abortion" and aMin is "completely ban all abortions." Bob could consider a range of abortion policies between those two extremes... for example "no restrictions during the first 6 months after conception, and after 6 months allow abortion only for medical need." (Note: issues are rarely one-dimensional. Where I use one-dimensional terms like "between" I'm simplifying for the sake of discussion.)
--Steve