[EM] IRV ballot pile count (proof of closed form)

[Abd ul-Rahman Lomax]

At 11:55 PM 2/11/2010, Dave Ketchum wrote:
>We all get careless and stumble, sooner or later!
>
>But I choke on two details here:
>
>You misuse the label "plurality" - having only the ability to vote for
>1 even though, for many races most intelligent voters will find there
>is only one candidate deserving approval.
>      Even Approval has more power, letting voters vote for more than
>one, though unable to differentiate.
>      Condorcet is another important step up, letting voters vote for
>more than one while indicating which they like best.
>
>Forcing voters to act as if they wanted to vote for more than they
>wish to is a step backward, and should not pretend to be an asset for
>a method.

I'm not following Mr. Ketchum's arguments here. But "plurality" was 
used in a very ordinary sense. Any method which elects without a vote 
of a majority of those who cast non-blank ballots in an election is 
an election by plurality, using the definitions of Robert's Rules 
(and of most parliamentary procedure manuals, I believe, if not all). 
There is room for interpretation on whether or not a non-blank ballot 
that does not contain a legal vote should be included in the basis 
for majority, but no room for excluding from the basis those who do 
cast a valid vote, but for a candidate that is, say, later eliminated 
due to low vote count.

Hence almost all voting systems that have been considered, absent 
vote coercion (as with mandatory full ranking or penalization of 
partial ranking, as happens with some versions of Borda Count), are 
"plurality methods," including Approval and Range and, the point 
here, Condorcet methods.

I did incorrectly state the case at first, by showing lower rankings 
that did add additional votes for other candidates by A. The example 
was clearer with all bullet votes. What this points out is that a 
ranking of, say, A>B>C>D>D>F>G>H is, from this point of view, a vote 
for G over H. Should this be considered an "approval" of G? The voter 
has expressed that, in an election between G and H, the voter would 
prefer H, though, in fact, in a deep ranking like that, this is 
probably noise for the most part. (Robson Rotation is, in fact, used 
to eliminate some of this noise by averaging it out so that, at 
least, it is not produced by ballot position.)

>"Majority" is a word whose merits need more serious thought - see an
>earlier post from today.
>
>Ditto "runoffs".

Indeed. Voting systems theory, early on, focused on attempts to find 
the ideal single-ballot system, from various perspectives. While this 
is a theoretically interesting question, it essentially misled the 
entire field when applied to real election reform, ignoring the most 
widely used voting reform, top two runoff, as if it were merely a 
more expensive and cumbersome version of Sri Lankan Contingent vote. 
Or batch-elimination IRV, same thing. It isn't. It produces different 
results than IRV, in about one-third of runoffs in nonpartisan 
elections. (Probably in partisan elections, it produces roughly the 
same results.)

In addition, this approach ignored the *universally used* direct 
democratic method, repeated balloting, with no decision being made 
without a majority of those voting supporting it. None. No exceptions.

Ignoring explicit voter approval, then, is one of the widespread 
systemic errors. Another one, arising early on, was the assumption 
that pure preference profiles were adequate to understand how voting 
systems would amalgamate votes and produce a useful social ordering, 
when, in fact, any sane method of studying how voting systems work 
would realize that a strong preference is different from a weak or 
barely detectable one, not to mention an indistiguishable one that is 
forced by a voting system to be crammed into one of A>B or B>A, with 
no allowance for A=B. And real, human, social decision-making 
systems, outside of voting, do consider preference strength, very much.

And any system that attempts to maximize benefit to a society based 
on preference profiles would have to take preference strength into 
account. That it may be difficult to do this, that it may be 
difficult to determine commensurability, does not change this. What 
we can see through the device of assuming absolute utilities for 
voters in simulated elections is that the Condorcet Criterion and the 
Majority Criterion, for similar reasons, can require preposterous 
results, in situations where, with a single ballot and no other 
amalgamation method operating, will produce a result that will later, 
if it's tested, be *universally rejected.* I'd call that a Bad 
Decision. And any system which considers preference strength, that 
allows the expression of it and then uses that information for other 
than simply resolving a Condorcet cycle, *must* fail the Condorcet Criterion.

Originally, my assumption, as with many students of this field, was 
that the Condorcet Criterion was the King of Criteria. Well, the King 
has been dethroned. It's a good and useful criterion, it has a place. 
But applied rigidly, it is quite possibly harmful.

I've argued that if ballots show a Condorcet winner, an election 
should not resolve in favor of another candidate, except possibly in 
situations where a legitimate cause of Condorcet failure can clearly 
be identified and it can be known that voters would then reject the 
Condorcet winner, knowing the results of the first election. Other 
than that possibility, I would always want to see Condorcet failure 
submitted again to the voters for review and possible confirmation or 
rejection of the failure. Rejection of the failure would probably 
mean election of the Condorcet winner, and, as long as the voters can 
do this, it must be said that the *overall method* satisfies the 
Criterion. It only looks like it didn't by not immediately jumping 
for the Condorcet winner in a primary. The later ballot completely 
supercedes the former, which is totally standard democratic process.