[EM] Anyone got a good analysis on limitations of approval andrange voting?

[Abd ul-Rahman Lomax]

At 12:22 PM 11/8/2009, Terry Bouricius wrote:
>A somewhat more accessible (and available online for free) analysis of
>strategic vulnerability of various methods is in this doctoral paper by
>James Green-Armytage ("Strategic voting and Strategic Nomination:
>Comparing seven election methods"). He found that Range and Approval were
>just about the worst in terms of manipulability.
>http://econ.ucsb.edu/graduate/PhDResearch/electionstrategy10b.pdf

There is a great deal of confusion on the topic of "strategic 
voting," based on a shift in the definition that took place after 
Brams suggested Approval Voting as "strategy-free." Originally, 
strategic voting referred to expressing a preference contrary to 
one's sincere preference, in order to improve, in the eyes of the 
voter, the probability of a better outcome. In the context of ranked 
methods, the meaning was clear, and, for the most part, students of 
voting systems neglected the implications of equal ranking and the 
function of compromise in decision-making.

In that context, there is something offensive about strategic voting, 
it seems "dishonest." Yet strategic voting is how voters attempt to 
remedy defects in voting systems. The problem, if any, is in the 
voting system, not in the strategic voting itself. In a properly 
designed decision-making system, what we call "strategic voting" may 
facilitate a final decision or may postpone it, but would never harm 
the value of the outcome, unless it was a poor strategy that harms 
the outcome from the point of view of the voter, i.e., the voter 
would have been better off voting sincerely.

It's clear that in the real world, people have strong preferences and 
weak preferences. When preference is weak, the voter may rationally 
decide to equally rank voters; I know that in real elections, I 
sometimes have difficulty deciding which of two candidates to vote 
for; an equal ranking there would be "fully sincere," as an accurate 
expression of my preferences, and for me to prefer one over the other 
is actually insincere. Thus pure ranked voting systems force a kind 
of insincerity; equal ranking systems still allow the option of the 
expression of strict preference, so they only increase the options 
for expression open to the voter.

However, what about the situation where a voter does have a 
preference between two candidates, but opts to equally rank them? In 
a Range system of adequate resolution, a "fully sincere" vote should 
be possible, without strategic harm to the voter's interests, but 
this needs definition, which is elusive. Practical Range systems, 
with limited resolution, resemble Approval voting, where preference 
strength below a certain level results in equal ranking. And 
"preference strength" is not independent of the voter's judgment of 
the strategic situation.

I might prefer, say, Jan Kok for President, and I could write his 
name on the ballot, but there is a problem. I would prefer him, 
perhaps, to any candidate actually printed on the ballot, but if I 
rank him above one of the printed names, I'm almost certainly going 
to waste my vote. (Students of voting systems, again, have neglected 
the influence and implications of write-in votes, for, where write-in 
votes are possible, we could argue that most voters vote 
"insincerely" or "strategically." They do not write in their true 
favorite because it would waste their vote, thus they rank one of the 
possible winners above their favorite, which meets the classical 
definition of strategic voting.

I vote strategically so as to exercise maximum power from my right to 
vote, and this is, in fact, what we want voters to do, and what we 
should expect them to do. Voting is a method of making collective 
decisions, and collective decisions must necessarily involve, most of 
the time, some level of compromise, and compromise is "strategic," 
wherein I approve an outcome even though it is not my first preference.

The last classical neglect has been in the implications of majority 
approval or disapproval of an outcome. The study of voting systems 
has been focused on improving methods for making decisions from a 
single ballot. Major respect has been given to the Majority 
Criterion, because of basic democratic traditions, but there is no 
way to extract a majority-approved winner from a single ballot that 
is guaranteed to work, because the reality may be that there is no 
candidate approved by a majority. However, this problem was resolved 
long ago as to practical function: repeated balloting is used. Under 
Robert's Rules of Order, no election is valid unless approved by a 
majority of voters, and Robert's Rules does not allow restricting 
candidacies, i.e., the common runoff voting where the only eligible 
candidates in the runoff election are the top two from the primary, 
violates Robert's Rules, hence special bylaws are required if it is 
decided to implement this. However, a reasonable and not uncommon 
compromise is to print the names of the two two on a ballot, but 
allow write-in votes. Still, under the Rules, a true majority of 
votes cast is required. Thus the end result of a standard election 
process under Robert's Rules must necessarily satisfy the Majority 
Criterion, if applied to votes as expressed. But it may violate the 
Majority Criterion as to unexpressed preferences.

Nevertheless, if voter expression was not constrained in the process, 
the *purpose* of the Majority Criterion is satisfied if the majority 
approves the outcome, which is required. That is, the majority 
approves of the selection of other than its first preference, 
presumably in order to satisfy a wider goal, and, further, we may 
presume that the unexpressed preference of the majority is comparatively weak.

Now, to James Armytage-Green, from the paper cited:

>In this paper, I will attempt to compare voting methods with regard 
>to how often they
>are vulnerable to strategy. In order to make a fair comparison, it 
>seems necessary to seek an
>environment where sincere preferences are entirely transparent; 
>thus, I use simulations.
>Broadly speaking, strategic behavior in elections can be divided 
>into two categories: strategic
>voting and strategic nomination. Strategic voting, of course, is 
>expressing a preference
>(through voting) that differs from ones sincere preference. 
>Strategic nomination can be
>thought of as adding or subtracting a non-winning candidate in order 
>to change the election's
>outcome.

James proceeds to use an issue space distance simulation to predict 
votes, but he reduces votes to pure preference. With respect to 
Approval Voting, he determines an average issue space distance for 
the variation between the voter's position and the candidate's 
position, and then assumes a "sincere approval vote" based on whether 
or not the candidate is closer to the voter's position than the 
average or not. If closer, "approve," if more distance, do not 
approve. Thus his simulation method guarantees vulnerability to 
"strategic nomination," since the addition of a new candidate to the 
space can drastically alter the "average," but in real voting, with 
write-ins possible, a practically unlimited number of candidates are 
possible. In reality, voters will weight the average based on the 
probability of success for each candidate, thus drastically limiting 
the effect of irrelevant candidacies. By definition, if they are 
irrelevant, they have probability zero of election, and the voter will so vote.

Range voters, as well, will factor in probability of election when 
determining Range ratings. In a good Range system, the voter may vote 
a weighted score, and it has been shown that this is a unique 
counterexample to Arrow's theorem, as modified to allow score voting. 
But, of course, this is "strategic voting," since the voter alters 
the "sincere preferences." But, note, the alteration never results in 
an "insincere vote," where the voter reverses preference. It is only 
by looking at preference strength that we can even detect the "insincerity."

In the discussion of voting systems, too often, basic common sense, 
how people *really* make decisions, has been lost. I use the pizza 
election as an example. Three persons want to select a single pizza 
to buy; let's say they are buying a frozen pizza, and they can only 
buy one. They have three choices: Cheese, Mushroom, and Pepperoni. 
Two of the three prefer Pepperoni. One is Jewish and prefers 
Mushroom. The Majority Criterion appears to require that Pepperoni 
win, but in real choices in functional human groupings, Pepperoni may 
be the worst outcome. Let's propose sincere Score ratings. And there 
are four possibilities, the three pizzas and No choice -- which then 
requires discussion and further process, or they all go hungry.

Let's assume these normalized sincere Scores, listing the voters' 
ratings after each option:

No Choice: 0,0,0    They are hungry.
Cheese:    8,8,8
Mushroom:  9,9,10
Pepperoni: 10,10,0

Score voting tells us that the best choice for the group is Mushroom. 
Further, from the votes, we can predict the likely result of a 
ratification vote: Mushroom would be explicitly approved, and in that 
vote, the Majority Criterion's purpose is satisfied.

Our three friends won't bother with the ratification vote at all, 
rather, they will effectively ratify it by contributing to the price 
of the pizza.

This example shows the defect in the Majority Criterion: it 
completely neglects preference strength. If the preference of the 
majority is weak, the majority may well prefer to set it aside to 
gain a more widely acceptable outcome. In this example, in fact, the 
criterion which is sought to be satisfied is Unanimity. Unanimous 
approval may be preferable to Majority preference, and societies 
which respect this will tend to be more successful, because they will 
be more unified. Unanimous approval may not be attainable, there are 
practical limits, obviously, but the desirability of broad public 
support of election outcomes should not be neglected.

In real elections, we should be so lucky as to see violations of the 
Majority criterion from Approval and Range. These methods only 
violate the majority criterion in order to make a better choice. They 
have only done so if the majority, in voting, made compromises based 
on an umderstanding of the necessary compromises. So, for example, 
with the pizza election, suppose they used Approval Voting. I would 
guess from the range scores that the votes would have been No Choice 
0, Cheese 3, Mushroom 3, Pepperoni, 2. The friends would look at each 
other, one of them, preferring Mushroom would say, "I suggest 
Mushroom," and the others would immediately agree, and Mushroom would 
be the unanimous choice.

Two-round voting. James describes it as:

>Two round runoff: Each voter chooses one candidate. The top two 
>vote-getters then compete
>in a runoff election. After plurality, this is the next most 
>widely-used single-winner election
>method.

He statement about two-round runoff is misleading. The most widely 
used election method is plurality voting, majority of ballots cast 
required to determine an outcome, election repeated until this 
condition obtains. In each election, the voters decide whether 
sticking with their favorite is more important, or determining an outcome.

Score voting merely improves the efficiency of this process; but 
something must be added to score voting to make it work: 
specification of an approval cutoff. When the simplest form of score 
voting is used, Approval, the cutoff is explicit. Approval Voting 
theorists have often assumed a series of elections where voters 
increasingly compromise until a majority-approved choice appears. 
This is somewhat simulated even by single ballot systems by the 
nomination and election process, including public knowledge of the 
popularity of candidates. Call that a "virtual ballot."

With Range or Score Voting, approval cutoff must be specified; 
otherwise Range can elect a candidate opposed by a majority. Range 
theorists have neglected this problem.

James, in his consideration of 2-round runoff, neglects the effect of 
the second election process, and his description of two-round is 
incomplete. Sometimes write-in votes are allowed, so the second 
ballot is not, in fact, restricted to the top two, and a recent 
mayoral election in Long Beach, California, a write-in had a 
plurality in the primary, but not a majority, and so a runoff was 
held. Because the write-in was the incumbent mayor, prohibited by 
term limit legislation from appearing on the ballot, the runoff 
ballot had only one name on it. In the runoff, the incumbent gained a 
plurality, nearly a majority, there was another write-in candidacy.

Two-round voting with write-ins allowed is quite close to standard 
iterative democratic process, and it is obvious that it could be 
improved. Approval and Range voting (with explicit approval cutoff) 
make iterative process more efficient, making it more likely that a 
majority-approved winner will appear in the first round, and it would 
be rare that majority approval does not appear in the second round. 
Both methods avoid the spoiler effect of additional write-in 
candidacies in the second round.

It's remarkable that a method as advanced as Two-Round Approval could 
be implemented simply by Counting All the Votes in Two-Round. And 
there is a method which is even more compatible with what people want 
in voting systems:

Bucklin. With a majority requirement. Use it in both rounds. In the 
second round, it would probably be quite sufficient to have two 
ranks: Preferred and Accepted. (The third rank is no-vote, or 
Rejected). In the first round, two ranks might be adequate, the same, 
or a third rank could be used.

In analyzing Two-Round, not only does James neglect write-ins, but he 
assumes that preferences remain identical in the second round. This, 
as well, assumes the same voter set. But, in fact, in real elections, 
it's a different voter set. And that brings in the way in which 
Two-Round simulates Range. Voter turnout in a special election, if 
the runoff is a special election, is heavily affected by preference 
strength. If a voter has no preference between the candidates, or if 
the preference is sufficiently weak, the voter is not motivated to 
turn out. Low turnout in runoff elections is not an indication that 
the process is flawed. Rather, it could be a sign that the voters 
equally approve of either candidate. Or, unfortunately, that they 
equally disapprove.

Bucklin has some distinct advantages: It was widely used in the U.S. 
at one time, and it was popular. The reasons why it passed from use 
are unclear. A probable reason would be, in some cases, low usage of 
the alternate votes, which is the same reason that IRV was abandoned 
where it was previously used, in some cases. In other cases, there 
either were explicit political reasons or political reasons were 
hidden underneath spurious legal arguments.

Bucklin allows a fully sincere first preference vote. In the original 
form, say, as used in Duluth, ballots would be voided due to 
overvoting in first and second preference, but approval-style voting 
was allowed in the third rank. Hence with Bucklin one could, in fact, 
vote, Anybody But Bush, while still expressing first and second 
preferences. I would modify this by allowing all votes to be counted, 
which allows a much more expressive vote. It is possible to 
incorporate higher resolution in each round, i.e, instead of being 
Bucklin/Approval, it would be Bucklin/Range/Approval.

But one step at a time. Start with Count All the Votes. Leave 
everything else the same. Really, this is such an obvious and simple 
and *harmless* reform that it should have been done long ago. Most 
voters won't use it. Which is fine. We would expect, in a first 
round, fully sincere voting, where the existence of any significant 
preference strength would cause bullet voting for the favorite. In 
most elections there are two major candidates and nobody else has a 
serious chance of winning. Even plurality usually picks a majority 
winner. Approval will have no effect on this, and the possibility 
that two candidates get majority approval is low; if it happens, the 
outcome is, almost by definition, harmless. Some will raise the 
Majority Criterion bugaboo, for it could then be true that the 
majority preference was passed over for one more widely approved. As 
I wrote, we should be so lucky. It will be rare, and it can easily be 
argued that the Approval winner was the best choice. That would only 
not be true if there was widespread *and ignorant* multiple approval, 
perhaps due to the appearance of some seriously poor candidate who 
was then given a chance of winning in the estimation of the voters. 
Bucklin would fix this problem, easily.

No voting reform, however, should be considered as a replacement for 
runoff voting, which, especially if write-ins are allowed, is so far 
advanced because of the single iteration, that using some more 
advanced single-round systems, such as IRV, is a step backwards. IRV 
is sold as "guaranteeing majority choices," but that's highly 
deceptive. Robert's Rules does suggest, as a possibility, sequential 
elimination preferential voting, but that's in a context where a true 
majority still must be found; they are suggesting it as a way of 
increasing the possibility of finding a majority over simple 
plurality. However, they explicitly comment that if voters do not 
rank all the candidates, there might be majority failure "and the 
election will have to be repeated." What the editors of Robert's 
Rules probably did not know is that in nonpartisan elections, IRV 
closely tracks the performance of plurality. Bucklin does much 
better, because Bucklin does count all the votes, hence it is more 
likely to find majority approval, and it will not pass over a 
compromise winner, as IRV can easily do -- and Robert's Rules notes 
this as a flaw in their suggested form of Preferential Voting.

James Armytage-Green's shallow analysis of Two Round Runoff, and 
similar analyses by other voting systems experts, has resulted in 
inadequate defense of Two Round. Robert's Rules also notes as a 
problem with Preferential Voting that voters are deprived of the 
benefit of voters being able to consider the results of the first 
round in how they vote in the second. Voting systems analysts have 
typically neglected this very important characteristic. It makes a 
difference in real elections.

James goes on to consider how each method is affected by "strategic 
voting," which he apparently defines as a vote different from what 
his simulation considered "sincere," even though, with Approval and 
Range, his "sincere" votes were still modified, in certain fixed 
ways, from absolutes, and by "strategic nomination," which then 
assumes modification of the "sincere votes" based on how this affects 
his simulation, even if it would not affect real-world voting, so: 
garbage in, garbage out.

It's simplest to look at Approval. He defines the "sincere approval 
vote" as requiring that the voter vote for any candidate whose issue 
space distance is less than the average. Basically, this is a stupid 
strategy that nobody will follow, and that Approval advocates have 
never recommended. Because the vote for a candidate then depends on 
the candidate set, it's obviously going to be vulnerable to 
"strategic nomination." In real Approval voting, there will be far 
less effect, because the real Approval voting strategy, followed by 
nearly all voters, will be "Approve any candidate who is better than 
your expected outcome for the election." Or, alternatively, the 
simplest strategy: "Approve the best of the top two and then any 
candidate you prefer to that one." Most voters, in fact, practically 
by definition, will only vote for one. But we won't know for sure 
until we have systems being used. It's possible that support for 
third parties is wider than we realize, and it would show up in 
Approval, and even more in Bucklin. Or Range, of course. (And IRV 
does accomplish this discovery as well).

James is trying to judge vulnerability to "strategic voting" by 
defining some "sincere Approval vote." So he has to use some 
objective standard. But he picked one that is itself a strategy, 
simply a poor one. The real problem: the idea that "strategic voting" 
is a Bad Thing, to be avoided. This is true, in my opinion, with the 
original meaning of strategic voting: preference reversal. But equal 
ranking is not preference reversal, it is, instead, a decision to not 
express a preference, it is an abstention from a pairwise election or 
a set of such elections. Range voting is even more flexible, if the 
Range resolution is sufficient or if the method allows preference 
expression with equal ranking. In that case, preference may still be 
expressed with a strategically optimal vote.

(It's been assumed that a strategically optimal vote in Range 
involves equal ranking, but, in fact, that is only true when the 
probability of election of the favorite is reduced to zero. Since 
each one of us is a sample human being, it could never be assumed 
that the probability of our favorite winning is zero, for it is 
possible that others think like us. It could be, however, rationally 
reduced to a very low level. In order to define strategically optimal 
votes, one would vote Von Neumann-Morganstern utilities, which are 
normalized absolute utilities modified by probabilities of relevance. 
We use these kinds of utilities routinely in making choices, we don't 
allocate limited resources to pursuing impossible alternatives. When 
there are only two choices, in our perception, we don't allow our 
"internal strategic nomination" to significantly modify our choice 
between the realistic alternatives, by causing us to spend our single 
plurality vote on our favorite write-in, and, while we may rue it, 
most of us also don't vote for no-hope third party candidates, same reason.)

See http://en.wikipedia.org/wiki/Expected_utility_hypothesis
http://www.rangevoting.org/DhillonM.html
http://ideas.repec.org/a/ecm/emetrp/v67y1999i3p471-498.html, note 
that there is a copy of the Dhillon-Mertens paper linked from the 
RangeVoting page above. Abstract of the Dhillon-Mertens paper:

>In a framework of preferences over lotteries, the authors show that 
>an axiom system consisting of weakened versions of Arrow's axioms 
>has a unique solution, 'relative utilitarianism.' This consists of 
>first normalizing individual von Neumann-Morgenstern utilities 
>between zero and one and then summing them. The weakening consists 
>chiefly in removing from IIA the requirement that social preferences 
>be insensitive to variations in the intensity of preferences. The 
>authors also show the resulting axiom system to be in a strong sense 
>independent.

Notice that the requirement of Arrow that "social preferences be 
insensitive to variations in the intensity of preferences" was 
preposterous. Arrow apparently insisted on this because he believed 
that it was impossible to come up with any objective measure of 
preference intensity; however, that was simply his opinion and 
certainly isn't true where there is a cost to voting. And there is a 
cost associated with voting in almost every context. If votes were 
bets of money, for example, as a tax rate to be paid on, say, net 
worth, we would vote our expected utilities, as modified. Warren 
Smith correctly, however, generates, by simulation, absolute 
utilities, and then his simulation engine can follow various voting 
strategies and see how voting systems behave. There is no superior 
approach that has been proposed, to my knowledge.

What opposition to Smith's approach boils down to is: I don't like 
the results, and there are flaws in the approach, therefore I'm right.

If you *must* make a decision by a single ballot, with no other 
input, Range is optimal. But even Smith's simulations find that Two 
Round Range is superior to single round Range. The restriction to 
single ballot is one of the most damaging assumptions of voting 
systems analysis, reducing all methods to being some form of 
plurality election. We need a broader approach, and the field is wide 
open, there has been so little work done in it that, if I had time, 
I'd be able to write several papers that would be likely to be 
accepted under peer review. Students, notice! Fame awaits you, if not 
fortune. The papers I would write at this time:

Performance of Instant Runoff Voting in Nonpartisan Elections, 
Compared to Top Two Runoff.
The Concepts of Strategic Voting and Strategic Nomination as 
Misapplied to Approval and Range Voting.
The Harm of Later No Harm.
Election Reform Long Overdue: Count All the Votes.
Modern Neglect of Two Top Runoff as a Major Election Reform.
The History of Bucklin Voting in the United States.