[EM] Arrow's theorem and cardinal voting systems
Richard Lung
voting at ukscientists.com
Mon Jan 20 10:49:14 PST 2020
Thankyou Forest Simmons. It is appreciated.
The link is:
https://www.smashwords.com/books/view/997528
Richard Lung.
On 19/01/2020 22:30, Forest Simmons wrote:
> Fascinating! We need people like you to knock us out of complacency
> ... out of the comfortable ruts that we inhabit.
>
> I look forward to following the link you gave me!
>
> On Thu, Jan 16, 2020 at 10:25 AM Richard Lung <voting at ukscientists.com
> <mailto:voting at ukscientists.com>> wrote:
>
> This is obviously a very learned summary that would require
> considerable study to do justice to, much more than my old head is
> capable of, quite apart from severe personal distresses, that have
> been over-whelming our lives.
> Yet, just the other day, after a half century of amateur study, to
> my surprise, my naive physicists mathematical text crawled across
> the finishing line of publication. Only the last chapter is of
> direct interest to electoral mathematicians. It contains an
> explanation of how to conduct a two-dimensional election: FAB STV 2D.
> The two dimensions are Representation and Arbitration. The latter
> graphs at 90 degrees (neutrally) to the former, and the count is
> conducted without disturbing the normal one-dimensional count that
> is FAB STV. But taken together as a complex variable the count is
> according to the rules of complex variables.
>
> from
> Richard Lung.
>
>
> On 15/01/2020 20:57, Forest Simmons wrote:
>> Just a couple of additional thoughts:
>>
>> Besides Arrow and Gibbard-Sattherwaite we have lots of criterion
>> incompatibility results from Woodall, and defensive strategy
>> criteria from Mike Ossipoff, Steve Eppley, et. al.. In
>> particular we never worried about Later No Help and later No Harm
>> until Woodall came along. Venzke and Benham picked up the torch
>> and brought Woodall into the EM Listserv discussion.
>>
>> Many EM contributors have clarified which combinations of various
>> criteria of more practical than academic stripe are compatible or
>> not: Participation, FBC, Precinct Summability, Chicken, etc.
>>
>> In particular, we now know through the work of Ossipoff, Venzke,
>> Benham, and others that the Chicken Defense and Burial Defense
>> (against CW burial) are incompatible in the presence of Plurality
>> and the the FBC, unless we allow an explicit approval cutoff or
>> some other strategic switch on the ballots. Standard ordinal
>> ballots are not adequate for this even when truncation and equal
>> rankings (including equal top) are allowed. A non-standard
>> ballot that allows us to get compatibility to all of these except
>> the CC is MDDA(sc) which is Majority Defeat Disqualification
>> Approval with symmetric completion below the approval cutoff.
>> This method also satisfies other basic criteria such as
>> Participation, Clone Independence, Mono-Raise, Mono-Add, and
>> IDPA, for example.
>>
>> In the context of the current discussion, the approval cutoff or
>> some equivalent strategic switch is essential for the
>> compatibility of chicken resistance and burial resistance.. No
>> strategy, no compatibility. So basically there is no decent
>> method that is resistant to both Burial of the CW and Chicken
>> offensives. (IRV is chicken resistant and has a form of burial
>> resistance, but routinely buries the CW unless voters
>> strategically betray their favorite to save the CW. Furthermore,
>> it fails mono-raise.)
>>
>> Before collaborative efforts of EM List members there was no
>> known clone independent, monotonic method for electing from the
>> uncovered set. The closest thing was Copeland, which is clone
>> dependent.
>>
>> Again the main point is that Arrow, and Gibbard-Satterthwaite are
>> not the "end of history" for election methods, just like the
>> collapse of the USSR was not the end of history as Fukuyama once
>> proclaimed or Thatcher's famous TINA "there is no alternative"
>> (to capitalism). Arrow and G-S give very valuable insights and
>> help us avoid cul-de-sacs, but they are not the last word in
>> election methods progress. The "end of history" and TINA slogans
>> are an excuse for giving up prematurely for lack of imagination.
>> We cannot allow Arrow and G-S to become excuses for lack of
>> imagination in Election Methods. What if Yee had given up before
>> inventing the beautiful Yee diagrams that constitute an
>> Electo-Kaleidoscope for the study of election methods analogous
>> to the telesope and the electron microscope in astronomy as
>> instruments in other branches of knowledge?
>>
>> On Mon, Jan 13, 2020 at 3:32 PM Forest Simmons <fsimmons at pcc.edu
>> <mailto:fsimmons at pcc.edu>> wrote:
>>
>> Rob,
>>
>> Thanks for starting this great thread!
>>
>> The "no perfect car" analogy is good. More definite is the
>> "no 100 percent efficient internal combustion engine" analogy
>> that follows from the second law of thermodynamics. It
>> applies to all kinds of engines, but that doesn't mean that
>> internal combustion is as good as it gets.
>>
>> If Gibbard-Satterthwaite tells us that we cannot have all of
>> the nice properties we want in one election method, that
>> doesn't mean that one method is as good as the next.
>>
>> It follows from Arrow that we cannot have the Majority
>> Criterion and the IIAC at the same time, but there are many
>> decent methods (like River) that do satisfy the MC, and a
>> bunch of other nice properties, like Monotonicity, Clone
>> Independence, the Condorcet Criterion, and Independence from
>> Pareto Dominated Alternatives, as well as the basic
>> Neutrality and Anonymity fairness criteria.
>>
>> The way to think of Arrow's "Dictator" theorem is that it is
>> extremely hard to get a rankings based method with even
>> minimal decency conditions (like non-dictatorship) without
>> scuttling the IIAC.
>>
>> In other words, no decent ordinal based method can satisfy
>> the IIAC, which is the same point of view that Toby and
>> Eppley expressed. It comes down to the mere existence of a
>> Condorcet Cycle. Here's the subtle part that most people
>> don't understand. Condorcet Cycles can exist in the
>> preference schedules of an election even if the election
>> method makes no mention of Condorcet, for example even in
>> IRV/Hare/STV/RCV elections:
>>
>> 45 A>B>C
>> 20 B>C>A
>> 35 C>A>B
>>
>> There exists a majority preference cycle A>B>C>A even though
>> it causes no problem for IRV, since B is eliminated and then
>> C is the majority winner between the two remaining candidates.
>>
>> Now let's check the IIAC. Suppose that A, one of the losers
>> withdraws from the race. Then the winner changes from C to
>> B, since B beats C by a majority. This shows that IRV does
>> not satisfy the IIAC, because removing a loser from the
>> ballot changes the winner.
>>
>> But this is not just a problem for IRV, it's a problem for
>> any method that respects the Majority Criterion; if the
>> method makes A the winner, then removing B changes the
>> winner. If it makes B the winner, then removing C changes
>> the winner. If it makes C the winner, then (as we saw in the
>> case of IRV above) removing A changes the winner. to B.
>>
>> So Arrow's "paradox" can be considered as forcing us to
>> realize that the IIAC is not a realistic possibility in the
>> presence of ordinal ballots because such ballots allow us to
>> detect oairwise (head-to-head) preferences, and when it comes
>> down to a single pair of candidates the Majority Criterion
>> says the pairwise winner must be chosen,
>>
>> However, as someone mentioned, Approval Voting avoids this
>> "paradox" once the ballots have been submitted, since the
>> Approval winner A is always the "ballot CW," and in two
>> different ways:(1) For any other candidate X, candidate A
>> will be rated above X on more ballots than not, and (2) A's
>> approval score will be higher than the sore of any other
>> candidate. From either point of view, if we remove a loser Y
>> from the ballots, then A will still be the winner according
>> to the same ballots with Y crossed out.
>>
>> That's at the ballot level. But if Y withdrew before the
>> ballots were filled out, it could change the winner, because
>> if Y were the only approved candidate for a certain voter
>> before the withdrawal, that voter might decide to lower her
>> personal approval cutoff before submitting her ballot. Or
>> she could raise the cutoff if Y had been the only disapproved
>> candidate.
>>
>> As others have mentioned in this discussion, Approval Voting
>> externalizes the problem of the IIAC from being a decision
>> problem for the method itself to a strategical decision
>> problem for the voter. A voter might think of that as an
>> unfair burden.
>>
>> One answer to this problem could be DSV (Designated Strategy
>> Voting): You submit your sincere ratings, and the DSV machine
>> applies a strategy of your choice or a default strategy to
>> transform the ballots into approval style ballots. Rob
>> LeGrand explored some of the possibilities and limitations of
>> this approach in his master's thesis. He doesn't claim to
>> have exhausted the possibilities. (I also have some ideas in
>> this vein that still need exploring.)
>>
>> What constitutes a "sincere rating." One approach to that
>> has already been mentioned in the ice-cream flavor context in
>> this thread. Another is to use as a rating for candidate X
>> your subjective probability that on a typical issue of any
>> significance candidate X would support the same side you support.
>>
>> It's not just Approval that requires some hard thinking in
>> conjunction with filling out the ballots. Ranking many
>> candidates (think about the number of candidates in the
>> election that propelled Schwarznegger into office) may be
>> just as burdensome as trying to decide exactly which
>> candidates to mark as approved. In Australia you can get
>> around this difficulty by copying "candidate cards" or by
>> voting the party line. Presumably these experts are
>> reflecting state of the art strategy in their rankings ...
>> the strategy that is indispensable for optimum results
>> according to Gibbard-Satterthwaite. This is not just a
>> problem of Approval, though it may seem worse in Approval.
>> In actuality, aoproval and score/range are the only commonly
>> used methods where optimal strategy never requires you to
>> "betray " your favorite.
>>
>> To cut the Gordian knot of this complexity Charles Dodgson
>> (aka Lewis Carroll) suggested what we now call Asset Voting.
>> Each Voter delegates her vote to the candidate she trusts the
>> most to rep[resent her in the decision process. Since
>> write-ins are allowed, she can write in herself if she
>> doesn't trust anybody else to be her proxy. These proxies
>> get together with their "assets" (delegated votes) and
>> choose a winner by use of some version of Robert's Rules of
>> Order.
>>
>> Which criteria are satisfied by this method? Does Gibbard
>> Satthethwaite have anything to say about it? How about
>> Arrow? For that matter does first past the post plurality
>> satisfy the IIAC? (No more or less than Approval in reality.)
>>
>> Let's talk about Gibbard-Satterthwaite. Is there any
>> incentive for a person to delegate as proxy someone other
>> than her favorite?
>>
>> If we are talking representative democracy, then why would
>> you want to delegate your vote to candidate B when candidate
>> A was the one you trusted most to represent you in making
>> important decisions once in office?
>>
>> All of the "problems" with the method are essentially
>> externalized to the deliberations governed by"Robert's Rules
>> of Order" in the smoke filled room.
>>
>> Gibbard-Satterthwaite is taken to say that it is impossible
>> to obtain sincere preferences or sincere utilities from
>> voters in the context of full information (or disinformation)
>> elections. Yet it turns out to be relatively easy; you just
>> need to separate the ballot into two parts. The first part
>> requires strategic voting to pick the two alternatives as
>> finalists. The second part is used solely to choose between
>> these two options. (In the case of cardinal ballots the
>> finalists are lotteries.) A version of the uncertainty
>> principle obtains here: if you use the sincere ballots for
>> any other instrumental purpose than to choose between the two
>> finalists, then you almost certainly destroy their sincerity.
>>
>> However there would be no problem comparing tthe sincere part
>> with the strategical part to get statistics about voters'
>> willingness to vote insincerely in choosing the finalists.
>>
>> Another avenue that has been barely explored is the use of
>> chance to incentivize consensus when there is a potential for it.
>>
>> For example, suppose that preferences are
>>
>> 60 A>C>>>>B
>> 40 B>C>>>>A
>>
>> Under approval voting the A faction has a strong incentive to
>> downgrade C and vote 60 A>>>>>C>B making A the (insincere)
>> approval winner as Gibbard-satterthwaite would predict.
>>
>> However, if the rules said that in the absence of a full
>> consensus approval winner, the winner would be chosen by
>> random ballot, then (assuming rational voters voting in their
>> own interest) C would be the sure outcome; no rational voter
>> in either faction would prefer random ballot expectations
>> over a sure deal on C.
>>
>> Jobst Heitzig is the pioneer in this area.
>>
>> In sum, Arrow, et.al <http://et.al>. should not constitute a
>> nail in the coffin of creative progress in Election Methods.
>> IMHO that is an important message we need to send if we want
>> to attract new talent.
>>
>> Forest
>>
>>
>>
>>
>>
>> ----
>> Election-Methods mailing list - seehttps://electorama.com/em for list info
>
>
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