[EM] Arrow's theorem and cardinal voting systems

Forest Simmons fsimmons at pcc.edu
Wed Jan 15 12:57:19 PST 2020

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> 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. 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
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