[EM] the "meaning" of a vote (or lack thereof)

[Warren Smith]

Kenneth Arrow has worried that range-voting-type "score" votes might have no or
unclear-to-Arrow "meaning."  In contrast, he considers rank-ordering-style
votes to have a clear meaning.
Nic Tideman has also expressed similar worries in email, but now about
the "lack of meaning" of an approval-style vote.
In contrast, I think Tideman regards a plurality-style "name one
candidate then shut up"
vote as having a clear meaning.

E.g. "what does a score of 6.5 mean, as opposed to a score of 6.1, on
some ballot?"

But the Bayesian view is: whether or not Arrow or Tideman or somebody
has a more-or-less muddled mental notion of the "meaning" of a ballot,
is irrelevant.   The only genuinely meaningful thing is "who won the
election?"
All meaning of any ballot therefore derives purely from the rules
for mathematically obtaining the election-winner from the ballots.

For a simple example of how ballots have no inherent meaning without
voting system rules, consider plurality and AntiPlurality voting in which
the meanings of a  "name one candidate" ballot are pretty much opposite
(plurality: most-named candidate wins;
AntiPlurality: least-named candidate wins).

Let us now enquire more deeply about ballot "meaning."  In a non-monotone voting
system like Instant Runoff,  your vote A>B>C can cause A to lose, whereas
your vote B>C>A would have caused A to win.   Would Arrow be right if
he said IRV is wonderful
because "A>B>C" has a "clear meaning"?  Or would a Bayesian be right
in saying this
example indicates the "meaning" Arrow had in mind, was not valid?  Indeed the
Gibbard-Satterthwaite theorem
   http://rangevoting.org/GibbSat.html
shows that in essentially ANY rank-order ballot system and also in the
plurality and
AntiPlurality systems with "name one candidate" ballots -- i.e. exactly
the systems Arrow & Tideman thinks "have meaning" -- there ALWAYS
exist elections
in which some voter's vote of A>B>C will cause a worse election winner
(for the A>B>C
notion of "better" and "worse") than some different
dishonestly-ordered vote would
have caused.  (And with Plurality and AntiPlurality, "dishonestly" ranking
your non-favorite candidate top or your really-not-worst candidate
bottom, can be the only way
for you to get an improved election result.)
In such an election, what is the "clear meaning" of an A>B>C rank-order vote?

Gibbard identified/invented exactly two rank-order ballot systems in
which honest and strategic
voting were the same thing (this required him to employ
non-determinism), but stated
that both of his systems were not good enough for practical use since they
"leave too much to chance."

In contrast, consider the "double range voting" system invented by
F.Simmons and Warren D. Smith
   http://rangevoting.org/PuzzRevealU2.html

This system (or others of the Simmons class) ARE good enough for
practical use if any
rank-order system is (since it leaves only an arbitrarily small amount
of the deciding to chance,
and deviates from your favorite system in an arbitrarily small way).

In this voting system, each ballot contains a part on which the voter
is urged to
provide her honest scores (on, say, an 0-to-9 range) for each
candidate.  In this system,
ONLY voting on this ballot portion in a unique honest manner is strategically
best.  Any deviation from perfect honesty (or omision of information)
is a strictly worse voting strategy.
   That is, if your expected utility if A wins is 6.5 and your
expected utility for B
winning is 6.1 on an 0-to-9 scale (defining the utility scale so
you've rated the
best available candidate 9 and the worst 0)
then you MUST score A=6.5 and B=6.1 EXACTLY, otherwise you are guaranteed
to get in expectation a worse-utility election result.

So contrary to assertions by the likes of Arrow that utility is "unmeasurable"
or that range votes "lack meaning" it seems to me that we have a very
clear, totally unique,
not changeable by one iota, meaning for the scores 6.5 and 6.1 deriving
wholy from the procedure the voting system uses to determine the
winner from the votes.
This is wholy unlike EVERY allegedly-practical rank-order voting system.

So Arrow, and Tideman (and anybody else) are simply wrong if they
assert scoring-style votes
are inherently less-meaningful than rank-ordering-style or
name-one-candiate-style votes.

So now Arrow might perhaps riposte that to HIM, deep in the recesses
of his brain,
rank-order votes have more meaning, even though every voting system he
and his colleagues have ever considered for practical
use, disagrees with his private meaning in at least some situations,
and even though (therefore)
the true meaning of your vote really also depends on how the OTHER
voters are voting, not just
on the candidates and your evaluation of them in your private brain.
(I would have to then counter-riposte: who cares?)

Another riposte might be that under the assumption there are a large number
of other voters all of whom vote TOTALLY RANDOMLY and independently,
your vote can have a clear meaning, and in some rank-order systems (e.g. Borda)
this meaning coincides with "honest ordering."
I would then riposte that (a) that assumption is false, and (b) under
the same assumption
approval voting also has a "clear meaning" (namely: you should
"approve" candidates
above mean utility for you).

Does our argument tell us that score-style votes inherently have MORE meaning
than rank-order style votes?  (Exactly contrary to Arrow?!?)  Well... not
necessarily.   Yes, score-style votes certainly inherently convey more
information
than rank-ordering-style votes (strengths of preference as well as preferences).
And I would claim that if they were employed for the honest-part of
double range voting
ballots, they inherently have more meaning.  But if employed for plain
range voting, then it is posible
to construct 4-candidate election situations in which it is strategically best
for a range voter to misorder, so we run into the same
Gibbard-Satterthwaite-style issues (albeit only with 4, not 3, candidates).

All this analysis really tells us is the Bayesian view is correct.
And certainly that any dismissal
of range- or approval-style voting on the grounds of their claimed
"inherent lack of meaning",
is hogwash.

-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)