# [EM] Easy voting system optimality theorems

Abd ul-Rahman Lomax abd at lomaxdesign.com
Fri Jun 8 15:11:11 PDT 2007

```At 01:01 PM 6/8/2007, Warren Smith wrote:
>A list of 6 easy theorems (but still interesting, I think)
>showing how various voting systems can be regarded as "optimal"
>under appropriate "quality measures" cooked up for the occasion
>(but not too artificial), is
>    http://rangevoting.org/Copt.html

I've written about measures of election quality previously, it being
my opinion that frequently election "criteria" were indirect, that
is, they were about the process rather than the result.

For example, at first glance, the Majority Criterion seems desirable,
that the first preference of a majority should prevail. However, it's
quite clear that there are situations where this is an undesirable
result in a zero-knowledge election.

Here is what I've come to as the core of the problem with respect to
the Majority Criterion. The foundation of MC is the principle of
majority rule. If we have a meeting, and it is moved that a candidate
be elected, and the motion is seconded and, for some reason, there is
no debate, and that candidate is the first preference of the
majority, and the members do not know the opinions of other members,
we can assume that the candidate will be elected. The vote is a
Yes/No vote on the motion.

However, these conditions are quite narrow. In a real democratic
process, there would be debate and the members voting would generally
have some idea of how the electorate feels in general. Further, in
real democratic process, there is opportunity to reconsider a
decision; any member of the majority can move that the motion be reconsidered.

Indeed, the raw democratic process described, followed carefully by
the members, is Condorcet-compliant, because at any time while the
motion is under consideration, any member can move that it be amended
to substitute a different name, and if the majority prefers the new
candidate, it presumably will vote Yes on the amendment. And this
process, iterated, would presumably find the Cordorcet winner, and
probably rapidly. Further, in the process, preference strengths are
expressed, and it is even possible that a Range poll is used to
nominate the candidate.

But election methods collapse the process, in the name of efficiency.

Now, public debate prior to an election and polls both improve the
self-knowledge of the electorate, but not reliably.

Why is this self-knowledge important? My standard example is the
pizza election. Three election methods aficionados are out for pizza,
and they wish to buy a single pizza. The pizzeria is closing, and
they can only choose one kind. They have three choices: Pepperoni,
Mushroom, and Onion. The majority prefers Pepperoni, but Mushroom is
quite acceptable to them; indeed, their preference of the two for
Pepperoni is only slight. One of them, though, cannot eat Pepperoni,
for whatever reason, but Mushroom is good for this voter. What is the
best pizza that they can elect? What method would they choose if they
wish to find the pizza without debate or discussion, simply through ballot?

Only methods which consider preference strength can make the choice
that most of us would consider the best. There is one exception to
that choice being best, which is if the two voters were to refuse to
accept the decline in value, as they perceive it, from Pepperoni to
Mushroom. Given the preferences noted, it would seem churlish,
wouldn't it? Nevertheless, majority rule dictates that the majority
*could* refuse, and could insist upon Pepperoni.

(Let's assume that they have a free pizza coupon, so the question of
who pays for it is moot.)

Nevertheless, the majority will, we submit, generally accept, in a
healthy society, a minor loss in value, in order to provide a large
increase in value for a minority.

The majority criterion does not consider preference strength at all,
so it fails to measure election quality in situations where
preference strength is significant. And preference strength,
particularly if we assume it is sincere, *does* matter.

Thus the Majority Criterion is really judging a process, rather than
the result. Are there measures of election quality that aren't
process-oriented?

Of course there are, and they are commonly used to judge quality in
other situations.

If we poll, after the election, all the voters, and ask them, "How
satisfied are you with the election result, all things considered?",
rate this on a scale of 0 to 10, with zero being least satisfied and
10 being most satisfied, we have a measure of the election quality
with respect to a single voter. And if we sum these ratings over all
voters, we have a measure of the overall election quality, for this
particular election.

If we were to do this with many elections, sufficient to cover the
various scenarios and problems to which a method may be susceptible,
we would have a measure of the method's quality, which could then be
used to compare the method with other methods.

But, of course, this is a post-facto measure, and it may depend on
information not on the ballot. However, if we ask the voters to
estimate their expected satisfaction, on the ballot, with the
election of each candidate, and with proper instructions -- which I
won't detail at this time --, we do have a measure of election
quality that depends on the ballot itself. It is inferior, as such a
measure, to the post-facto measure, but it should be close in results.

There is another aspect to election through deliberative process
which also applies, quite frequently, to election methods as used by
deliberative bodies. That aspect is the practice of the majority
declaring that an election result is accepted. The majority can
refuse to accept election results, no matter what method was used.
If, however, the consent of the majority to the result is built into
the method, it becomes possible to bypass, reasonably, the need for
ratification. And this is the true appeal of the Majority Criterion,
for if the conditions of the majority criterion are met, and that is
the candidate elected, majority rule is satisfied.

But quite possibly at the result of lower overall satisfaction. This
is the dilemma, and real deliberative process does not have this
problem, for, in such process, the majority makes a deliberate and
clear decision. It's acceptance of the election *is* the election, in
fact, what came before is a kind of poll. And recent events make it
clearer than ever how important the form of the poll is. MSNBC has
been holding two different Range 2 polls, one for the Republican
candidates for U.S. President, and another for the Democrats.

(Range 2 means that there are two preference intervals. There are N+1
ratings possible in Range N. This is different from my prior usage,
which always grated with me. Approval is Range 1, according to this
new definition.)

Votes in the poll in question are of the form -, 0, +, with 0 being
the default vote unless changed by the voter to - or +.

public views these candidates than what we see with the more standard
after seeing them, I wonder why we would ever bother with the older
kinds of polls. Obviously, there is even more information that can be
collected, but it is apparent that in order to be able to predict or
judge election quality, we need some kind of preference strength
information, and Range 2 only begins to collect that (by having 2
levels of preference strength rather than one one as in Approval and
only one restricted to one as in Plurality).

These polls show Ron Paul, a libertarian who became a Republican some
years ago, as being way ahead of all other Republican candidates.
Whereas on the Plurality polls, he is down at the 1% or 2% level.
It's not even close, in either method. He is a landslide winner in a
Range poll, and doing so poorly that he might as well drop out before
he wastes even more time and money in the Plurality polls.

On the Democratic side, the Range polls show results that are much
closer to what we'd see from Plurality polls. They are also
different, tobe sure, but not so dramatically.

I watched Fox pundits opine that these polls were distorted by
legions of "internet voters." Duh! They are all internet voters,
obviously! But there is clearly something else going on here. (The
polls restricted voters from voting for more than once by setting a
cookie, I'm sure, so it is very easy to get around the vote once
restriction. But it takes a certain amount of time to vote, and I
rather doubt that anybody set bots to work on voting. Besides, if
MSNBC is watching the IP addresses, it would take a distributed
botnet to get around this, and those are busy sending spam and are
not designed to interface with the MSNBC polls, which could be subtly
altered to defeat automated voting (it would stand out like a sore
thumb to the MSNBC system operators, if they are paying attention.)

Plurality polls are distorted, apparently, by electability
considerations, whereas the Range poll frees voters to express both
value and electability; in particular, there is no harm in rating
your favorite first. Do that in a Plurality poll, and if enough
people do as you, your second choice is left out in the cold and may
not make it to the final rounds.

So, to restate the problem: Satisfying the Majority Criterion
satisfies majority rule, but does not optimize overall satisfaction
under some conditions. Optimizing overall satisfaction necessarily
can violate the Majority Criterion. Does it also violate majority rule?

Not necessarily. Rather, the result of such a method, unless special
constraints apply, does not determine, by itself, how the majority
would vote in a ratification. Essentially, unless the majority
predetermines the outcome in some way for the immediate election, we
do not know if the majority has not consented. Of late, I've been
suggesting that when conditions appear in a Range election where the
first preference of a majority is not elected, a runoff is triggered.
Apparently this is actually rare, it requires special conditions,
specifically where a small preference of a majority is outweighed by
a large preference of a majority.

My opinion is that the result of that runoff would normally be that
the Range winner would prevail, and I've given my reasons on the
Range list. If this is true, it becomes reasonable, if it is
considered that a runoff is too expensive or an onerous burden, to
consider the Range winner as not violating majority rule.

It's a shortcut, like all election methods. *However,* it is possible
to collapse the process, I'm coming to think.

I mentioned that the majority might predetermine the process. Some
might say that the majority approving of the election method by
having passed some Ballot Question in the past, satisfies this, but
that would be equivalent to consider majority rule to be satisfied if
the majority was bound by some prior decision. This is properly "rule
of law," not "majority rule," and I think that it actually isn't
appropriate for elections. If a ballot had on it a question, "Shall
the winner of this election be determined by X procedure," and the
majority vote Yes, then we could say that the majority has consented
to applying this procedure to this specific election. And the rules
could provide that, if majority consent is not apparent from the
ballots, and the Ballot Question failed, there would then be a runoff.

So, to summarize, there are two basic standards for judging election
quality that proceed from independent understandings of what is best.
There is the Majority Criterion, rooted in the concept of majority
rule, and there are measures of election quality that ideally would
be based on post-election polls, but which might depend on ballot
information, if the ballot contains enough information.

The Majority Criterion likewise, to be used as applied directly to
the information on the ballot, requires that the necessary preference
information be collected. I've expressed my view that Approval
satisfies the Majority Criterion, strictly as written, but it must be
acknowledged that it can suppress the majority preference, and,
therefore, whenever there are two candidates with greater than a
majority support, some other indication of preference must exist on
the ballot to allow the majority to express its preference unhindered.

This is done with what I called A+. This is Approval as counted to
determine the Approval winner. But the extra Plus or Favorite
designator would be used to determine if there has been a Majority
Criterion failure, and because the majority is thus unhindered in its
expression of preference, A+ can clearly fail the Criterion. But with
a runoff whenever the Plus indicator shows a pairwise majority
failure, that is, we can determine from the ballot that the majority
preference has been passed over, the combined method satisfies the
Majority Criterion and therefore majority rule.

I find the interplay between Measure A, on Smith's page, which is a
Plurality measure, and measures based on Voter Satisfaction or, what
is the equivalent, Social Utility or Bayesian Regret, fascinating.

To me, there are basically two measures that make sense, and the best
method maximizes both. The "majority rule" measure, which is related
to the Plurality measure, is fundamental to democracy; this is a
binary, pass/fail measure. "Would the electorate accept the result of
this election?" The presumption is that if the electorate does NOT
accept the result, we have an explicit rejection of the candidate
being offered for ratification by the majority, and in normal
process, presenting that question again would be suppressed (subject
to more specific rules about the reconsideration of motions), and
then there would be further process to determine the winner, possibly
the same election all over again, which would be best but expensive,
or a ratification based on the next winner in line.

This concept of ratification is probably going to meet with
resistance. It flies in the face, it would seem, of the very purpose
of election methods, which is to bypass the supposedly cumbersome
deliberative process. However, reducing the process to only one step,
when there are multiple choices (more than Yes/No) leads to all the
various nasty phenomena that can make it so difficult to determine
the best winner. I'd suggest we need to consider a return to the
democratic roots of elections and *then* attempt to match the best
features of these roots, while improving efficiency. A two stage
rarely, it can be expected, lead to a second stage. And a third stage
would be even more rare, etc. The result is that so many election
criteria violations are fixed, except for ones that are truly
rootless, they are purely about the method and process, not about the
quality of the result.

There are those in the Range camp who object to the introduction of
majority rule. These would impose some apparently higher social
utility winner on the electorate, contrary to the consent of a
majority, or, perhaps, they would wish to avoid presenting such a
question to the majority, believing that the majority would
automatically choose to maximize its own utility at the expense of a
minority. This, I'll note, represents a fundamental distrust of democracy.

It is quite possible to argue that the runoff or ratification stage
is unnecessary, that Range results will be "close enough." And I'd
agree that this is a reasonable position. But the problem is that
there are possible anomalies in how the voter ratings were expressed,
for Range is subject to strategic distortion. (It never encourages
rank reversal, which is why pairwise analysis of Range ballots could
be so interesting. They could be expected to be sincere votes. My
view is that to win the Range election is more important than winning
the pairwise election; only if the electorate has views that are
nailed down could it be different. And such an electorate is in deep
trouble for other reasons. Imposing the Range winner on this
electorate, even though it might seem to maximize satisfaction, could
seriously backfire if the majority rejects it instead of tolerating it.

(It seems appropriate to present a scenario where a Range winner
would sensibly be rejected by the electorate. Let's imagine that the
expressed utilities represent real gain or loss. Now, if the
utilities are expressed on a relative scale, so that each voter looks
at the gain or loss of each candidate, relative to the range of gains
and losses over the candidate set, with the "votes" being normalized
so that they span the full range on the ballot, the actual value of a
preference strength unit for each voter is different. It could happen
that there is a net loss to society if the Range result is accepted,
compared to, say, the preference winner. Somehow there must be some
means to judge the overall result.

Hence the process I am coming to. An initial election which maximizes
total voter satisfaction, with the ballots analyzed for majority
preference (or, alternatively, explicit approval, using an approval
cutoff perhaps), and if the latter does not indicate majority
consent, a runoff or ratification. The runoff is more efficient, the
ratification is more explicit.

Now, the kicker: if the preference votes are used to create an
electoral college, as with Asset Voting, the runoff can be submitted
to that college. The implications are vast, so I will not address
them today. I'm only noting that if an Electoral College exists, the
process can become fully deliberative; it becomes as practical, with
appropriate rules, as it is in small societies. An Asset electoral
college is totally open and therefore accurately representative, for
there is no coercion of voters to pick a popular candidate. Every
recipient of a vote can recast that vote, effectively as a proxy for
the voter. These voters need not be candidates, in fact. And you
could write your own name in, if you want to vote in public, subsequently.

The ultimate measure of election quality is its acceptance by the
electorate when the question is presented. The acceptance vote is the
measure, and a reasonable system would reject any election that is
rejected by the majority. (This is a version of Measure D, the one
maximized by Approval, but "approval" is not inferred, it is explicit.)

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