Arrow's axioms (was Re: [EM] Re: [Fwd: Election-methods digest, Vol 1 #525 - 9 msgs])

[Steve Eppley]

Ken Johnson wrote:
-snip-
> > From: "wclark at xoom.org" <wclark at xoom.org> ...
> >> But why did Arrow stipulate #1?  (rank method)
-snip-
> Based on the preceding discussions, I infer the following: 
> (1) Arrow's theorem is based on an unjustified and
> (according to the theorem's conclusion) indefensible
> bias in favor of ranked methods. 
-snip-

I consider Arrow's axioms justifiable.  In the decades 
leading up to Arrow's theorem, economists and social 
scientists had struggled in vain to find a good way to 
compare different individuals' utility differences (known 
in the literature as the problem of "interpersonal 
comparison of utilities") in order to be able to calculate 
which outcome is most utilitarian.  That is, they were 
interested in being able to sum for each alternative the 
utility of that alternative for each voter, so they could 
elect the alternative with the greatest sum.  By Arrow's 
time, they'd learned that, lacking mind-reading 
technologies, they couldn't elicit cardinal utilities that 
could be compared between individuals, for instance to 
compare the utility difference between your "100" candidate 
and your "0" candidate to the utility difference between my 
"100" candidate and my "0" candidate.  Simply summing our 
reported numbers, which don't have units (such as dollars) 
attached, would not help them find which alternative had 
the greatest utility.  If each voter is constrained to 
assign numbers within a given range, such as 0 to 100, then 
the sum would not be the utilitarian sum.  Maybe these sums 
aren't worthless, but they need careful scrutiny.

Also, as you know, asking each voter to freely assign 
numbers within some range would create a strong incentive 
for individual voters to exaggerate, so that in the long 
run the information elicited from the voters by a cardinal 
utility method would be no greater than the information 
that can be elicited by Approval.  

In the worst case, the socially responsible voters would 
fail to exaggerate and the selfish voters would exaggerate. 
I consider this case extremely disturbing.

Arrow also reasoned that the information about the voters' 
preferences that can be elicited by Approval is far less 
than the information that can be elicited by letting each 
voter express an ordering of the alternatives.  That makes 
sense to me, and I further believe that the best methods of 
tallying preference orders will lead to better outcomes for 
society than if Approval is used, over the long run.  I'm 
perfectly willing to trade complete satisfaction of Arrow's 
"choice consistency" axiom (while satisfying all the other 
Arrow axioms) for outcomes that, over the long run, are 
better for society. (For a description of Arrow's theorem 
in the framework where choice consistency is one of the 
axioms, justifications of each axiom, and a simple proof of 
the theorem, follow the link to Arrow's theorem at my 
website at "http://www.alumni.caltech.edu/~seppley".  Note 
that "choice consistency" was Arrow's name for this axiom, 
not mine.)

The best justification for requiring satisfaction of choice 
consistency is not, in my opinion, the aesthetic value of 
consistency.  It's the thorny problem that arises, if 
choice consistency is not satisfied, about deciding which 
candidates to nominate.  For a current example, look at the 
Democrats deciding their presidential nominee(s).  In 
principle, they could nominate more than one candidate, but 
the manner in which plurality rule fails choice consistency 
gives each party a strong incentive to nominate at most one 
per office (and sometimes zero, to be socially responsible 
by avoiding nominating a spoiler that makes the outcome 
worse). (Should I take a moment to argue that Nader should 
have competed in the Democrat primaries, where he would 
have been able to fully participate in the debates, rather 
than run in the general election?  This isn't the old days 
when parties didn't offer primary elections to choose their 
nominees, leaving Progressive candidates with no 
alternative but to run as third party candidates.  Nader 
argues his right to run, when the debate is really about 
whether it's socially responsible for him to run in the 
general election.)

Approval only satisfies Arrow's choice consistency in a 
narrow technical sense, by boldly assuming voters' 
decisions about which candidates to "approve" or 
"disapprove" are independent of the set of nominees.  But 
this assumption is indefensible.  Society has a lot of 
experience with a voting method that is very similar to 
Approval, also called Approval, which asks each voter to 
vote "yes" or "no" on ballot propositions. (Continuing the 
status quo is implicitly one of the alternatives.)  When 
propositions conflict so that at most one of them may pass, 
which is analogous to a single-winner election with more 
than 2 candidates, enough voters tend to vote "no" on 
compromises (to avoid defeating preferred alternatives, or 
to express their preference for preferred alternatives) 
that the status quo can often win even when the compromise 
is more popular.  As a result, the conventional wisdom is 
to place only one proposition on the ballot, rather than 
let the voters decide between 2 or more conflicting 
propositions.  That smells like the two-party system, with 
each party nominating only one candidate per office.  

Regardless of the voting method, rational choice theory 
models each voter's sincere preferences as being consistent 
with some ordering of the alternatives. (Even if her 
preferences are cardinal utilities.  There is plenty of 
empirical evidence regarding observations of individual 
choices from a varying set of alternatives that demonstrate 
individual choice consistency, which means each individual 
chooses as if her preferences were consistent with an 
ordering.  The evidence isn't perfect, since in complex 
situations, such as choosing a preferred lottery, it's 
common for individuals to employ simplifying strategies 
that can lead to minor inconsistencies.)  Consider the 
following scenario, in which the voters' sincere 
preferences are represented as orderings:

   40%: A>B>C
   33%: B>C>A
   27%: C>A>B

You'll recognize the majority cycle, which demonstrates the 
choice inconsistency of every voting method that reduces to 
majority rule when only 2 candidates compete.  If we make 
the assumption that, given only 2 candidates, Approval will 
behave like majority rule (because voters will approve one 
and disapprove the other to avoid wasting their votes), 
then Approval shows the same choice inconsistency: the 
alternative that wins when all 3 candidates compete, given 
those cyclic preferences, will lose if a particular losing 
alternative is dropped from the set of nominees.  For 
example, suppose the Approval winner is A when all 3 
compete.  If only A & C compete, C would win (since we're 
assuming that most of the 60% who prefer C over A would 
approve C but not A) but choice consistency demands that A 
still win.  Thus the problem of which candidates to 
nominate does not go away under Approval, and the 
conventional wisdom based on experience with "yes/no" 
voting on propositions is to nominate only one proposition 
to compete against the status quo.  So I'm skeptical about 
the value of Approval as a reform, and I don't accept the 
claim that the only Arrow axiom it violates is universal 
domain (which requires every ordering be an admissible 
vote).  I'd rather use a voting method that encourages 
competition to be the best compromise, so voters can rank 
less-corrupt compromise candidates over more-corrupt 
compromise candidates.  And I think it's dangerous to argue 
for a weak reform now and then later come back and say "we 
were only kidding, here's something that will really work."

Furthermore, I'd rather use a voting method that also 
allows each voter to express her preferences for favorite 
alternatives by ranking them over compromise alternatives. 
Unless we require that everyone vote (as Australia does), 
it stands to reason that voters will be more motivated to 
vote if they can rank their favorites over compromise 
alternatives, which I believe will be useful for reducing 
the bias against people who are "too busy" on election day 
to vote, at least for the people who are only slightly too 
busy.

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)