[EM] Easy voting system optimality theorems

[Abd ul-Rahman Lomax]

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 +.

Reading these polls provides so much more information about how the 
public views these candidates than what we see with the more standard 
Plurality polls (which ask, Who is your favorite candidate?) that, 
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 
election is already accepted; the proposals I've made would only 
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.)