[EM] How to convert a set of CR ballots to a set of Approval ballots

[Forest Simmons]

On Fri, 1 Aug 2003, [iso-8859-1] Kevin Venzke wrote:

>  --- Forest Simmons <fsimmons at pcc.edu> a écrit :
> > Kevin, I think that the refined method can be slightly improved as
> > outlined below.
>
> I implemented the (original) refined method and it seems to be a great
> improvement, although I haven't done a side-by-side comparison.  (I was
> judging based on how long I had to wait for a disagreement.  The display
> is pretty slow and I'm not using an efficient sort method.)
>
> [...]
> > In our refined method we amalgamated adjacent slots with minimal combined
> > viability.
> >
> > A slightly simpler and (I believe) improved rule would be to amalgamate a
> > slot of minimal viability with the less viable of its two neighbors (or
> > with its only neighbor if it is one of the two extreme slots).

It turns out that this rule isn't quite right either.

Now (August 2, 2003) I believe that the rule should go like this:

The adjacent pair of slots whose maximum viability is minimal is the pair
to merge.  If two pairs have this MinMax viability, break the tie by
merging the pair (among the tied) whose second highest viability is
minimal.

An example of Kevin's shows the deficiency of the previous rule:

9  C D B A
10 C A B D
13 B D A C

The starting viabilities are the Borda Scores 33, 58, 57, and 44,
respectively for A, B, C, and D.

According to the previous rule the pair BA is the one to merge in the top
faction (9 C D B A -> 9 C D BA).

The new rule yields the correct merger of the pair CD.

The old rule leads to a win by A, which is both the Borda and Condorcet
loser.

The correct rule leads to a win by C, the first choice majority winner.

Note the line up of viabilities in the top faction: 57,44,58,33

The first pair (57,44) has a max viability of 57.

The other two pairs both have max viability of 58.

Since we are trying to shrink the weak links, we should merge pair where
the max viability is weakest, i.e. the first pair.

Suppose in a longer example a ballot has the following lineup of
viabilities at some stage:

1,10,6,8,7,9,3.

The pairs (6,8) and (8,7) both have the same max viability of 8, so we
note that 6 is less than 7 and merge the (6,8) pair. [All other pairs have
max viability greater than 8.]

In this example all of the previous rules give the wrong merger.

However, in the case of three slots this rule agrees with all of the
previous rules.

>
> > I would like to call this method Max Power Cardinal Ratings (MPCR) since
> > its hueristic is to convert the CR ballot into an Approval ballot with the
> > maximum likelihood of being positively pivotal, i.e. maximizing the voting
> > power of the ballot.
>
> So it's doing Joe Weinstein's strategy, right?  I sort of figured it must be
> doing something like that, because when you amalgamate ratings based on
> viability, you kind of make a mess of what those ratings were supposed to
> stand for.
>
> I wonder if anyone would object for this reason, since Weinstein's strategy
> isn't the optimal Approval strategy.  (It only considers relative utility and
> the odds.  It cares that you gain, but not by how much.)
>

Yes, Joe Weinstein's strategy is what I had in mind.

There are two reasons for this.

(1) For Joe's strategy you don't need to know the utilities, only the
ordering, and as the procedure progresses the CR slots lose their
proportionality to the utilities, even if they started out proportional.

(2) Optimizing expected outcome pays off in the long run, but in the short
run (one election) you really want to maximize the chance that your vote
will make a positive difference.

To see this suppose that you must choose between a 50% percent chance of
winning a thousand dollars and a 1% chance of winning sixty thousand
dollars.  If you had the opportunity to play this game an hundred times,
you would choose the higher expectation choice every time.  But if it were
a one time opportunity, and you desperately needed the thousand dollars to
make ends meet, you would probably take the 50% chance.


One more related point.  CR ballots are not really needed for this method,
but they are the most convenient way to get ranked preference ballots that
allow equal rankings at any level, not just the bottom level (via
truncation).

Furthermore, a method that depends only on the orderings is easier to
compare with other methods that depend only on ordinal ballots, i.e.
Condorcet, Borda, IRV, etc.

Forest