[Election-Methods] Improved Approval Runoff

[Abd ul-Rahman Lomax]

At 04:58 PM 8/20/2007, Peter Barath wrote:
> >Sure. That's been proposed many times. However,
> >it's not a very good method. First of all, it is
> >blatantly obvious, if you care to look, that the
> >Condorcet winner is sometimes *not* the best
> >winner, by far.
>
>I guess this is an unjust blame because this thing
>affect all voting methods.

No. It *particularly* affects ranked methods, because ranked methods 
obscure preference strength. While there may be methods which promote 
the expression of absolute utilities in the votes (one possibility is 
mentioned below by Mr. Barath), if we set these aside, even Range 
methods may fail to accurately aggregate utilities because of 
normalization, resolution, or, yes, issues of strategy.

Assuming that the Condorcet winner in an election is, 
unconditionally, the best winner, is blatantly an error, because it 
is easy to construct scenarios where reasonable people will agree 
that a different winner is better, and deliberative process would be 
almost certain to choose that winner. Election methods are shortcuts, 
which reduce what may be considered impossibly tedious or complex 
deliberative process to a matter of counting and analyzing ballots, 
but it is practically inherent that the shortcut introduces flaws; 
this does indeed affect Range as well as ranked methods. But not as 
badly, not *nearly* as badly.

Most analysts here comment that, with "sincere voters," Range is the 
best method; they then, often, go on to claim that, however, because 
of the issue of strategic voting, Range is impractical or dangerous 
or whatever. Yet I have never seen a scenario which actually shows 
this, and, at the same time, shows votes that make any sense, that 
real voters would be at all likely to cast. The basis of the claim is 
often that voters will essentially disable themselves by voting weak 
votes against smart strategic voters who vote strong votes. However, 
if I vote a weak vote, it means that my preference is weak, and I 
have little ground to complain if, therefore, someone who expresses 
strong preference prevails. I allowed that by my vote. And, I claim, 
we should take the votes as writ.

In any case, I find the question interesting, "What is the ideal 
method with sincere voters?" It is obvious that Condorcet methods 
*fail* rather badly with sincere voters, when they trip over the 
matter of preference strength.

In a ranked method, where A>B>C is expressed, no information at all 
is provided about preference strength. If we assume that A>>B>>C is 
impossible, an assumption that Range generally makes (you only have 
one vote to express, so you cannot express full vote strength in the 
AB pair and at the same time full vote strength in the BC pair), we 
can look at the three rough possibilities: A>>B>C, A>B>C, A>B>>C, 
plus a ranked method that does not allow equal ranking may also have, 
as actual preferences of the voter, A=B>>C, and A>>B=C. (I'm 
neglecting the weak A=B>C and A>B=C; in the language of Range, I'm 
normalizing.)

Those votes are really quite different in meaning and value. I may be 
able to discern a preference and therefore express it in a ranked 
method, but this preference may be insignificant compared to the 
preference I have for both these candidates as compared to all the 
others. Yet any ranked method will treat this maximally weak 
preference -- it may actually be no preference if the method forces 
ranking and does not allow equal ranking -- as quite the same as a 
life-or-death, full vote preference.

Blatantly, this causes, under some conditions, poor results. The 
Condorcet Criterion sounds good, it would seem obvious that a proper 
election winner, if the election allows full ranking, should not lose 
the pairwise contest with any other candidate, i.e., if the election 
were immediately held, at the outset, only between these two, vote 
for one, we would think that the ideal winner would not lose. But 
that is only true if we neglect preference strength.

In real elections, the effect I'm talking about is more rare than we 
might expect because if, in fact, voters have a weak preference, they 
might not even bother to vote, depending on who the frontrunners are.

>  Even in a two-candidate
>contest where every considerable method becomes
>Plurality, it's possible that the minority has
>stronger preference, so the winner is not the
>social optimum.

In public elections under present conditions, absolutely, no method 
will choose the true SU winner unless somehow absolute utilities are 
expressed. It does happen, sometimes, and sometimes we forget that 
election methods are general and public elections are not the only 
application. Further, in some public elections, the context is *not* 
the highly competitive, polarized situation we commonly think of with 
regard to elections.

Normalization with two candidates only obscures the preference 
strength, causing all preferences to become equal. However, it is a 
serious error to apply this fact to elections with more than two; as 
the number of candidates increases and the range of candidates 
becomes broader, normalization has less and less impact, and votes 
will tend more toward becoming a relatively accurate expression of utilities.

Note that the common characterization of bullet voting as "strategic" 
and "not sincere" is, arguably, an error. Bullet voting in Range can 
be sincere; what is overlooked is that there is no defined sincere 
mapping of internal utilities to Range Votes. We use simulated 
internal utilities and map them to Range Votes with various 
assumptions, but it is not at all a clear matter to term a mapping of 
some finite preference to a vote equality as "insincere." We would 
never term, in plurality, a vote as "insincere" because it ranks more 
than one candidate bottom equal; in this case it would be forced by 
the method. But what if the method were Approval? Obviously, with 
Approval one must equal rank somewhere. In Range, though, I have 
defined "truncation" to refer to the process whereby an internal 
utility scale is mapped to the Range Vote scale such that some finite 
spread in the internal utilities, a subset of the full possible 
utilities, is mapped to the Range Votes, and some of the candidates 
are thus off-scale. Or even all of them are off-scale, and are thus 
rated either top or bottom. Even if there are preferences between 
them. Quite simply, there is no defined "sincere" rating for a candidate.

We often talk about Range strategy, i.e., about methods of mapping 
utilities or expected satisfaction to Range Votes. Suppose a voter 
were to think of the absolute best possible candidate as 100%, I call 
this the Messiah vote. And the absolute worst, 0%, I call the 
Antichrist vote. (I use these terms not to propose some religious 
meaning, only as references to an extreme polarity. I once did this 
and someone objected that perhaps someone doesn't like the Messiah. 
That is a contradiction, because in this context, "Messiah" only 
means the absolute best possible candidate under any conditions.)

In a sense, this scale is the same for all voters. I call it the 
"first normalization," though, because it is not really the same, we 
are assuming that all people have the same complete range of 
preference, or, more accurately, we want to treat their voting as if 
they do. These are still not absolute utilities, but we may choose to 
treat them as such.

(And this is what the simulators do, they assume a commensurable 
internal scale. This is *not* the Range Votes, necessarily, but if 
voters simply vote these full-scale utilities, Range does the best 
possible job of maximizing overall voter satisfaction. it will choose 
the right pizza, if any method can.)

Take those same utilities and use them to rank candidates, and you 
often get the same result, but also you can get a worse result. It happens.

But we do not expect people to vote absolute utilities, and those are 
the ones that would be most deserving of the term "fully sincere 
votes." Obviously, in most elections, the range of candidates present 
in the election is not that full range, we do not have both the 
Messiah and the Antichrist running, except for a very small minority 
of mentally ill voters. (or that terminally rare election where they 
actually are running, which we will neglect. If that is the election, 
we have more to worry about than election methods!)

If I vote these fully sincere utilities in a real election, then, I 
will be casting a weak vote. Indeed, if it were Range 100, I might be 
voting max 51 and min 49, casting, really, only 1/50th of a vote. Or 
something like that, maybe more, maybe less, but never the full Range.

But if someone comes to me and says, we have these three choices, I 
*don't* think of the full range of possible candidates. Rather, 
first, and most easily, I rank them. And I can readily discern 
variations in preference strength. I can thus construct a set of 
ratings without even looking at the Antichrist and Messiah. I would 
choose my favorite and max rate. I would choose my least favorite and 
min rate. And the other, where would I put the other? It might seem 
that somewhere in the middle would be "sincere," unless I could 
discern no difference between the middle one and the ones at the ends.

But consider this. I mentioned that someone came to me and asked me 
to choose among three. In most elections, the *real* choice is 
between two. So what do I do? I do just what was described above. I 
normalize to the set of *real* candidates. So I max rate the favored 
frontrunner and min rate the least-favored. And where the others go 
is, to me, fairly inconsequential *unless* my favorite is among the 
others. In any case, my vote would probably be to rate the others in 
the middle unless they were better than the max frontrunner or worse 
than the min frontrunner. Or equal to them.

This is a different mapping, but really quite the same principle as 
the first mapping where the best of all candidates fixes the top end 
of the scale, and the worst fixes the bottom end. Instead of that 
mapping, the best frontrunner and worst frontrunner are used.

All of these are sincere, in the ordinary meaning. Generally, equal 
ranking when there is some preference between the candidates is *not* 
insincere. I would not call it "fully sincere" either, since some 
preference strength is concealed in the process.

In any case, I see utterly no harm in voters choosing how to vote. In 
Range, I would treat their votes as sincere expressions of their preferences.

For example, there are three candidates. As to performance in office, 
a voter considers A to be the best, C to be the worst, and B is right 
in the middle. And then there is D, who is almost as good as A, in 
the view of the voter. The voter votes, in high-res Range, A and D 
max, B midrange, and C min. Is this a sincere vote? Well, trick 
question. The voter is D's mother. She is not going to vote against 
D, period. Indeed, this gives her a strong preference for D. Does 
this mean that her sincere vote would be D max and all the rest min? Maybe.

It's up to the voter. We see scenarios where supposedly a voter has 
"sincere preferences" of A=B, nearly, but bullet votes for A (there 
are other candidates as well, of lower preference), and supposedly 
this is "strategic," which is a term which conceals the motive and 
true preferences. A voter votes "strategically" because the voter 
hopes to gain value by doing so. We *want* voters to use the system 
to maximize value! If the voter has sufficient preference for A to 
cause the voter to bullet vote for A, this is a contradiction with 
the assumption that the preference is weak. The voter wants the 
favorite to win!

Range is not vulnerable to Favorite Betrayal, unless one makes the 
strange condition that the Favorite is betrayed by voting the 
Favorite equal with another. The application of the term "strategic 
voting," and the approbation that accompanies it, is problematic when 
applied to equal ranking or rating. More properly, and more 
offensive, is preference reversal, which Range never rewards. 
Essentially, there is no strategic voting pattern which is not a 
monotonic mapping of the internal utilities to the Range Vote, but it 
is possible that what I called the "magnification" is different for 
different parts of the scale. And it is possible that there is 
truncation, such that the extremes are collapsed, which is equivalent 
to a compressed expression of utilities.

>(The only defense against this is the money voting, the
>Clarke-tax, which is - I think - treated also a little
>unjustly. At least the theoretical honor should be given
>for showing the possibility of strategy-freeness.
>And who knows, one day it can be proven even practical
>in some circumstances.)

This, of course, contradicts what was said at first, but no problem....

"Strategy-free" is, in my view, not a proper goal, unless it means 
that the method is free of incentive for preference reversal. That is 
laudable. Generally, Condorcet methods are not strategy-free by this 
definition, though certainly it is possible that strategic voting 
could be difficult to pull off.