[Election-Methods] YN model - simple voting model in which range optimal, others not

[Abd ul-Rahman Lomax]

At 08:30 AM 3/30/2008, Dave Ketchum wrote:
>On Sat, 29 Mar 2008 23:28:22 -0400 Abd ul-Rahman Lomax wrote:
>>At 03:32 PM 3/29/2008, Dave Ketchum wrote:
>>
>>>Some see forest; some see trees; who sees all?
>>
>>Those who see a forest made up of trees.
>>
>>>Looking at the 31 voter Plurality example:
>>>      16 voted for candidates with 3 or more Ys - but NONE for 4 Ys.
>>>      9 voted for candidates with 3 or more Ns - 9 going all the 
>>> way to 4 Ns.
>>>
>>>Seeing that as unbalanced, I inverted it.
>>
>>Which, of course, does nothing. Y and N are arbitrary. In the 
>>example, Y means the majority position. Period.
>
>Except, with complete inversion, N inherits majority position.

(by the way, you meant to write *5* going all the way....)

Yes. All you have done is to change the names of things. N is now the 
majority position. And YYYY wins. Your point?

The point is that Plurality is a method which considers only first 
preference. If voters vote sincerely, all that matters is that first 
preference, and it is entirely possible (and rational, though not 
"probable*) that there are more first preference voters with the 
majority-rejected position than for any other. Start to consider 
mixed sincere/tactical voting and it gets even more possible.

Plurality only works as a voting method when the field is restricted, 
and, even then, as we know, it can fail. With many candidates, it 
thoroughly breaks down. Range works well with many candidates. 
However, "works well" is still relative. Single-ballot election 
systems with many candidates are inherently flawed; natural systems 
would break the decision down and deal with it hierarchically. 
Hierarchical systems may tend toward the Condorcet winner; however, 
in practice, in the biological applications, decisions are weighted 
and passed on, up the hierarchy, as weighted, thus both Condorcet and 
Range aspects are functioning.

(Note that basic democratic process is hierarchical. Decisions are 
broken down into a series of Yes/No votes. "Election methods," which 
break this pattern, are so problematic precisely because they deviate 
from this practice. There still remains the basic Condorcet vs. Range 
problem; but functional systems do incorporate preference strength 
measures and thus move what might otherwise be accidental (or 
designed) victories for particular members of a Condorcet cycle 
toward the Range winner. In pairwise contests, a weak preference 
attracts little campaign funding....)

(My view is that if it is average Range as distinct from sum-of-votes 
Range, it has somewhat similar problems when there are many 
candidates, and the "quorum rule" is an arbitrary fix that likewise 
can break down with many candidates. Unlike basic Range voting, which 
is quite solid and has been extensively studied with simulations (and 
the simulations had no "abstentions" if I am correct), average Range 
has not been studied and is merely an idea that can seem good at 
first sight. It is a radical departure from existing practice, and, 
if included in Range proposals for real elections, will *kill* them. 
It has a place in advisory systems, where the use of average rating 
is then subjected to further judgment, thus covering all the nasty 
possibilities without trying to create a rigid rule.)

>Agreed ANY collection of voters would be possible, but YN offers 
>this collection as demonstrating what to expect from Plurality.

No. That is the basic error Dave made from the beginning. "What to 
expect" from Plurality is a gamut of behavior. The example was 
designed to show a bad element of that gamut. Not to claim that this 
is what to "expect."

With many candidates, though, failure to elect YYYY becomes the norm. 
Election of NNNN would still be rare.

Consider, though, that if on each measure the electorate is divided 
equally (almost) between Y and N, and if these positions are 
uncorrelated, the pattern described becomes, with many voters, about 
as likely as any other pattern. (That's my intuition. It may not be 
exactly true. But for NNNN to have 5 votes more than YYYY is almost 
as likely as for it to have 4 or any other small number. The 
distribution would be relatively flat around that peak.)

>Actually Plurality is neutral as a method - a desirable feature.

Sure. So is flipping a coin. Is that desirable?

Another neutral method, actually better than flipping a coin, is 
Random Ballot, used with a plurality ballot. A ballot is selected at 
randon and the candidate on that ballot wins. It is actually a 
reasonable method. It *usually* selects the best winner. It behaves 
quite like Plurality in certain ways. But, obviously, it can fail 
spectacularly.

The Plurality failure in the example given isn't as spectacular as it 
might seem, because the electorate is actually close to balance. 
Range shows that the balance isn't as close as it might look -- the 
Range margin isn't small -- but from other measures, the result is 
not as bad as it might seem. As much as I detest the result of the 
2000 U.S. Presidential election, all the violations of law and 
precedent, it must also be seen that the electorate was close to 
balance. It's not as extreme as what is theoretically possible. (IRV, 
as an example, can elect a candidate who is opposed by two-thirds of 
the electorate, as shown by the votes, compared to another candidate. 
That's pretty bad. It's also pretty unlikely to approach that 
extreme, but milder examples probably occur (or would occur if 
top-two runoff is replaced with IRV, as is happening in the U.S.)

>Plurality's inadequacy, which causes us to look for better methods, 
>involves limiting approval to a single candidate.  When faced with 
>candidate collections such as Bush/Gore/Nader, voters sometimes wish 
>to approve more than one as better than those they want to reject.

Yes. Approval actually brings us back, closer, to standard 
deliberative process. It holds multiple simultaneous elections. Now, 
suppose that, with Approval, a runoff is held whenever there is 
majority failure, and let's assume that there are two kinds of 
majority failure: no candidate gets a majority, or more than one 
candidate gets a majority. (We should be so lucky as to see the 
latter; in spite of this being a major argument made against Approval 
--- when this happens, it could be that the majority favorite has 
fewer votes than another also-approved candidate, thus the Majority 
Criterion fails --, in real, contested elections, it is quite 
unlikely. It would require a significant number of voters to, say, 
vote for both Gore and Bush. Yeah, right.)

If a runoff is held in either of these two situations, there is then 
the problem of who makes the runoff. The method is quite good with 
simply top-two. However, standard deliberative process, which uses 
repeated balloting to find a majority winner, with *no* candidate 
eliminated (and which can thus go to *many* polls), would indicate 
that this could be restrictive. Still, if we are choosing between two 
candidates, say, both of which were majority approved, the result is 
not going to be truly bad no matter how it turns out.

But Approval is also a blunt instrument, still. The method gains 
flexibility if fractional votes can be cast, thus allowing the voter 
to express true preferences *and* preference strength, and then the 
method can actually maximize what Clay Shentrup first called, as I 
recall, Total Voter Satisfaction. If voters vote sincerely. If they 
don't, the method obviously cannot maximize their overall 
satisfaction; however, it appears that it generally fails to do so in 
a direction *not* favorable to those who exaggerate. Exaggeration 
*may* help such voters elect their favorite, but only when that 
favorite is close to winning anyway, and it can help to defeat the 
second choice of those exaggerating voters, and the punishment can be 
quite a bit more severe than the reward.

When Range allegedly disappoints "sincere voters" due to the "greedy 
tactics" of others, the fact is that the disappointment is small. In 
the examples I've seen, those sincere voters will be, as they 
expressed, quite happy with the outcome, and it would only be an 
insinuating voice that would say, "But, if you had exaggerated like 
they did, you'd have ended up with $1.00 instead of $0.99." Given 
that my loss of $0.01 -- multiplied by all those who voted like me -- 
was more than balanced by a net gain of my neighbors, I'd tell that 
voice to go fly a kite. And if it was a politician saying that, he or 
she would never again have a chance to win my vote.

Plurality works under current conditions, most of the time, because 
generally the candidates have been vetted so that the choice is 
really between two. Duverger's Law. Plurality creates a two-party 
system. When third parties arise, though, Plurality starts to break 
down. Top-two runoff works *much* better than Plurality (for the same 
reasons as IRV, but with an additional factor that actually makes it 
better than IRV. TANSTAAFL, though. Top-two can cost more; it's not 
surprising that a superior election method would cost more. Approval 
gets us closer at no cost, and Approval plus top-two (which would be 
less common, since Approval helps reach a majority) is a good 
compromise. Bucklin, even better, because it does allow expression of 
preferences. (Three, in the Duluth implementation, with multiple 
approvals allowed in the third rank. I would simply allow multiple 
approvals in all ranks. Thus if you are actually happy with two 
candidates as winners, you can vote for both in first rank.)

Bucklin has started to seem to me as even better than Approval for 
U.S. practical implementations, though we can get to Approval with no 
cost at all. Bucklin does require more ballot space, though, like 
Approval, it can be done with all existing equipment. With 
hand-counting, it could also be counted in rounds, like IRV, so it 
does not raise hand-counting costs unless it is needed. (But I'd 
still want to see all the votes counted; the common IRV practice of 
ceasing counting when a winner has been found makes it impossible to 
analyze what happened, and to discover and measure IRV majority 
failure -- where a majority preferred another candidate to the IRV winner.)

Bucklin really deserves more attention. Consider that it was popular 
in the U.S. for a time, and consider that it was working. Arguments 
against it were not based on actual harm that made sense. My theory, 
as yet unsubstantiated, is that it was outlawed or rescinded 
precisely because it was working, and it would have allowed third 
parties to gain a toehold.



>I object to calling this example "likely" - think of such as 16 with 
>3 Ys together with ZERO 4 Ys.

Yes, but. Think of 100016 with 3 Ys together with 100000 with 4 Ys.

That is, add 100000 to all position counts and thus to all vote 
counts. The results are the same. It looks drastic because the 
election has been boiled down to the *margins*.

Note that if positions are random, we would expect 4 occurrences of 
3Ys for every occurrence of 1 Y. It is like the number of occurrences 
of coin toss patterns. If we throw a coin many times, and count the 
occurrence of each sequence of four, we will get these patterns of 
equal probability:

YYYY
YYYN
YYNY
YNYY
NYYY

There are four of the 3-match positions to each one of the 4-match. 
Actually, what is unlikely is that there are so many NNNN positions, 
and, in the expanded situation I mentioned (add 100,000) to all 
votes, that there would be so *many* YYYY positions and NNNN 
positions is the oddity, not that there are so few.

But these were *not* random positions. They were chosen to show a 
behavioral extreme. Yet Range functions fine with that extreme. 
That's the point. Plurality breaks down, IRV performs okay (choosing 
a 3Y candidate in that situation), but Range performs perfectly, 
because it was, in fact, designed to. So does Bucklin, so does 
Approval (probably).

>>However, not only does it seem probable that Dave would not 
>>understand this, I'd guess he is not reading this at all, because 
>>he hasn't responded to any of my explanations, only to Warren. So, 
>>would someone who isn't filtered out by Dave be so kind as to 
>>explain this to him?
>
>Tracing the filtering back to what you sent me on 23/3/08 at 2359, I 
>assume it was unintentional, and will undo it.

I've looked at my email records and find nothing that would explain 
this comment. Perhaps someone can help us out. To my knowledge, I 
sent no comments personally to Mr. Ketchum, but only replied to 
"All." Mr. Ketchum's copy of that mail would have the original 
headers, I assume. If he sends them to me personally, I'll examine them.

>I did not attempt to analyze the quality of the Range analysis, but 
>suspect that, based on what was done to Plurality.

An example was presented where Plurality does maximally poorly, Range 
performs like it always performs, to maximize expressed voter 
satisfaction. Range would perform this way with *all* election 
scenarios. Now, how often would Plurality choose the YYYY winner? 
(i.e., over all possible permutations).

You can estimate it from the relative frequencies. Plurality with 
sincere voting (one of the conditions) will choose the specific 
candidate with the most voters matching that position. As the 
situation is designed, all the specific positions would have the same 
probability. (This is equivalent to saying that every exact sequence 
of coin tosses has the same frequency as every other exact sequence. 
There are four times as many 3Ys as there are 4Ys because the latter 
is an exact sequence and the former is comprised of four different 
exact sequences.)

Thus Plurality would choose YYYY one out of sixteen times.
It would choose one of the 3Y candidates four out of sixteen times.
It would choose one of the 2Y candidates six out of sixteen times.
It would choose one of the 1Y candidates four out of sixteen times.
It would choose NNNN one out of sixteen times.

However, Range would always choose YYYY. It was *designed* to do that.

Always. If voters vote sincerely.