A better system than "true preferences"

[Saari at aol.com]

>If you believe a single winner election should be based on the strength
>of each voter's "feelings," rather than on crude rankings, then please
>propose such a system, structured so that each voter would have the
>incentive accurately to report the intensity of those "feelings", rather
>than to exaggerate them so as to maximize the chances of his or her most
>favored candidate.  
>-- Hugh Tobin

OK, I'll bite.  By the way, I should note from the start that the system
described next can be used both for the special-case "single-winner"
situation as well as the more generic "parliamentary procedure" case (e.g. a
group decision for the proposal "Should we do X?")  I have developed a system
which seems to avoid the inevitable and unresolvable tri-lemma between
majority/authority/consensus - by allowing for passing thresholds stricter
than majority but less strict than consensus.

But for now I will just discuss a single-winner election.  I will assume that
all voters are knowledgable and rational (but with different opinions
nonetheless).  The method I am developing does not have a name, but I suppose
that "Bivalent Approval with Gradations" would be pretty accurate.  Voters
would be allowed to vote SUPPORT or OPPOSED (or neither) on every candidate,
and fractional votes are also ok (i.e. 1/2SUPPORT, etc.).  These could be
easily represented numerically by "Any real number between +1.0 and -1.0".  I
often refer to all such systems (where you rate each candidate with any value
within a predefined set) as "rated voting" (in contrast to A>B>C which is
"ranked voting").

Now the scoring/tallying system needs to be defined.  There are lots of
possibilities, including:
-Add up all support, highest score wins (encourages expensive campaigns)
-Add up all opposition, lowest score wins (encourages timid or nonexistent
campaigns)
-Add up all support, subtract all opposition, highest score wins
(spending-neutral??)
-Same, but "scale" opposition votes by a constant real factor N.N before
subtracting.  Could also implemented as "Vote a number between +1.0 and -3.0"

-Add up all support, all up all opposition, divide for a "support/oppose
ratio", highest score wins
-Same, and also require a specified minimum qualifying score else the winner
will be chosen by lot, by Joe, or by another election.

It is clear that the exact choice of rules may well influence the outcome,
and might also influence voter strategy.  But I'll go out on a limb here.  I
actually believe that the arguments I am about to present will hold up pretty
well with ANY of the above scoring/tallying methods.  But for purposes of
discussion and specificity, let us assume that the group has already agreed
to use the following method:

[Add all support votes, add up all oppose votes, compute the "support/oppose
ratio", highest score wins, except that if no candidate gets a support/oppose
ratio of 2:1 or better then the winner will instead be appointed by
so-and-so.]

OK, now to proceed.  Hugh's challenge is for me to show that this system (1)
accurately represents the voters' true feelings, and (2) provides no rational
incentive to exaggerate or distort.

As for (1), I will point out that we have NO defined method in our language
to fully and precisely describe a person's feelings.  Every voting system to
date only allows an approximate representation.  

Does one person's "First-place" or "full-strength" vote really "mean" the
same thing as another person's?  Of course not.  Using money as a crude
yardstick, it could be that one person's top vote was a $1 opinion whereas
another's top vote could be a $1,000 opinion.  Finding some way to allow
people to vote their "true" feelings is a wonderful research goal - I have
some ideas but haven't solved it yet.

So I will not argue that rated votes accurately measures people's feelings.
 Instead, I merely argue that rated votes provide a BETTER APPROXIMATION than
ranked votes.  It is clear that neither system tells you anything meaningful
as to the "real" strength behind a given top-choice (or bottom-choice) vote.
 However, it is clear that a ranking can be unambiguously derived from a set
of rated votes, 
e.g. rated(A=1.0, C=.9, B=-1.0) => ranked(A>C>B).  
It is also clear that the opposite direction doesn't work - a rating CANNOT
be unambiguously derived from a ranked vote.  Thus it is proven that rated
votes contain more raw information than ranked votes.

Now the issue (2) arises - what is the quality of that additional information
in a rated vote?  Is it likely to be accurate, or is it likely to be
exaggerated or distorted?

As to exaggerated, let us first acknowledge that a given voter (assuming that
at least one candidate is at all acceptible) will probably vote a "FULL
SUPPORT" on their top candidate.  We still don't really know if it was a $1
opinion or a $1,000 opinion - so asking whether this vote was "exaggerated"
is really meaningless.  Let's instead just take as a given that each voter
will vote FULL SUPPORT on at least one candidate.  This has no more or less
meaning in and of itself than the fact that in a ranked ballot SOME candidate
gets marked in first place.  (The mere fact that a person ranked some
candidate "first" does not imply an exaggeration in a ranked system.)

Now, would that voter also automatically (i.e. rationally) vote the worst
candidate a FULL OPPOSE?  Perhaps, but perhaps not.  For instance, with the
scoring system defined for use in this example, excessive use of OPPOSED
votes increases the overall likelihood of no candidate winning and thus may
not be an optimal strategy.  (For example, with two evenly matched
candidates, if every voter "rationally" chooses to maximize their influence
by voting FULL SUPPORT for one and FULL OPPOSED for the other, a likely
result is that neither candidate will get the qualifying 2:1 ratio - leaving
the decision to so-and-so which could be perceived as even worse.  Rational
voters may well choose to vote FULL SUPPORT for one and (neither support nor
opposed) for the other.

Finally, the question of "distortion".  Let's assume the rational voter does
vote FULL SUPPORT on their top choice and FULL OPPOSED on their most disliked
choice.  How should he/she vote on the remaining choices?  Let's suppose the
remaining choices are more-or-less "evenly distributed" in that voter's range
of feelings (whatever that means...)

The naive approach would be to simply vote more-or-less in proportion to
their feelings, e.g. A+1.0, B+.9, C+.3, D+0, E-.4, F-1.0 or whatever seems
sensible.

More sophisticated people often conclude that it is more rational to
"extreme-vote", i.e. A+1.0, B+1.0, C+1.0, D?, E-1.0, F-1.0  Such a vote
indeed makes it more likely that one of your "liked" candidates will win.
 But at a cost.  This vote also discards any distinction between your
first-place choice and your second- and third-place choices.  It also
discards any distinction between your two bottom choices.

AFTER THINKING ABOUT THIS LONG AND HARD, I have concluded that the "naive"
vote (distributed approximating your true feelings) IS ACTUALLY THE MOST
RATIONAL CHOICE.  I cannot prove this rigorously - because feelings are still
too fuzzy to handle analytically - but have at least come up with an example
which seems to illustrate the concept.  

Suppose that three friends wish to buy a pie and split it.  There are twenty
choices on the menu.  Each friend has about the same "distribution of
feelings" - each one "likes" roughly half of the choices and "dislikes" the
other half.  There is also a comfortable "spread" between all of the choices,
e.g. a typical "naive" vote might thus be C+1.0, F+.95, Q+.8...A-.9, T-1.0
 So while all three voters have roughly the same "distribution", the actual
flavors liked or disliked is different for each (assume a random
distribution).
Since each friend likes ten of the twenty flavors, we should expect there
will probably turn out to be two or three liked by everybody.

Now the voting.  Two of the three friends choose the "sophisticated" voting
style - voting FULL SUPPORT for ten of the choices and FULL OPPOSED for the
other ten choices.  The third friend chooses the "naive" vote, i.e. A+1.0,
R+.9, F+.8 etc.

What will be the outcome?  Any flavor disliked by anybody will most certainly
lose to any flavor liked by all three friends. The winner will certainly be a
flavor liked by all, assuming there is one which seems likely for this
particular situation. Let's say there were 3 flavors which were liked by all.
 Thus the voting would be:
flavorF: +1.0, +1.0, +.8 winner
flavorQ: +1.0, +1.0, +.6
flavorM: +1.0, +1.0, +.2
[all other flavors had at least one vote of -something]

My conclusion: Regardless of whether the voter cast a "naive" or a
"sophisticated" vote, the winner will be a choice liked by all three.  AND
THE NAIVE VOTER (who spread out the votes according to their feelings) IN
ADDITION got to have THEIR opinion act as a tie-breaker (among the three
flavors liked by all three).

The "sophisticated" voters got more "clout".
The "naive" voter gave up a bit of "clout" but got extra "influence" as a
result.

I conclude that "naive voting" is actually the most rational choice.  I
cannot prove this rigorously, but the above example seems compelling.

I believe (but cannot prove) that whatever behavior is rational in a
small-#-of-voters situation should apply equally well with larger numbers of
voters.  (Pretend you are one of a block of voters, totalling one-third of
the group, who all make the same decision...)

Finally, we must address the issue of "tactical voting" (where advance
knowledge of the likely outcome induces voters to adjust their votes,
producing a distorted result).  This problem shows up very strongly in some
voting systems, such as "vote-for-one".  Advance "hints" as to the likely
matchup (i.e. the "top two contenders" out of a large field) cause more and
more people to switch their single vote from their true favorite to one of
the "top contenders" - since not doing so means their vote has no affect on
the outcome.  The result is a "snowball effect" - the advance "hints" become
a self-fulfilling prophesy.

I claim that rated voting systems are, in general, immune to this problem.
 The reasoning is two-fold:

One, if two candidates are well-liked by most and the likelihood of a close
race between the two is revealed, any tactical votes would mostly appear as
"downgrading" someone's second choice to an artificially-low vote.  If lots
of people do this, the result may well be that neither candidate wins.  So
instead of an accelerating snowball effect, you get more of a damping-out
effect.

Two (and this whole section is tricky) is that tactical votes only work well
when there is RELIABLE information as to the likely outcome.  If the advance
data is inaccurate, then any attempted tactical vote has a good chance to
backfire producing a worse result (in the opinion of the given voter).  So
tactical votes only make good sense if the advance data is accurate.

Now...so suppose some well-studied and accurate "advance poll data" is
released to the public before the election.  Such data will tempt a great
number of voters to tactical vote, and some number may actually do so
(according to their particular preferences and the overall situation).  Now,
EITHER the tactical votes will change the outcome, or they will be too few to
do so.

If they are too few to change the outcome, then it didn't really matter if
they did or not.  No gain, no loss.

But IF the tactical votes actually change the outcome, then this necessarily
means that the original "advance poll data" was in fact NOT reflective of the
actual voting.  And as previously stated, tactical voting based on inaccurate
advance data can easily backfire.  In this particular case (where the outcome
changed), some of the tactical voters will be glad they did, but other
tactical voters will conclude that they made a mistake believing the polls
(and worsened their outcome as a result).

I conclude: Even when the "advance poll data" makes me very tempted to
tactical vote (under this rated voting system), this very fact means I should
not trust the advance poll data (since others may tactical vote).  Thus, as a
fully rational voter I am actually better off overall to IGNORE the
temptation and simply vote my true feelings (to the degree that the voting
system will allow, anyway). 

Well, that's about it.  I've been using this system for awhile now for a
variety of "group decision" situations.  Instead of a majority requirement
for any given group decision, we use a single stricter passing standard
(generally a support/oppose ratio of 3:1, but likely any value between 2:1
and 8:1 gives satisfactory results).  The result is more consensus-like than
simple majority, yet will not devolve into terminal indecision (as consensus
often does) when the group size exceeds 10-20 people. 

We also have some other rules, such as the right of any member to call for a
vote at any time on any subject, which most existing majority systems lack
but which *should* be a fundamental tenet of any rational (non-hierarchical)
group decision system.  And we allow multiple competing, overlapping
proposals as well.  (All proposals are posted on a wall or other medium for
all to see.)  Each proposal has a "deadline" chosen by the proposal author -
may be short or long or whatever.  It sounds chaotic, but it seems to work
(and without needing a chairperson).  The higher passing ratio means that the
good proposals still pass easily but the weaker ones are quite easily shot
down.  The group seems to proceed smoothly as a result.  Testing is ongoing.
 

Mike Saari