[EM] Arrow's theorem and cardinal voting systems

Forest Simmons fsimmons at pcc.edu
Mon Jan 13 15:32:35 PST 2020


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

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

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

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.

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