Forest Simmons fsimmons at pcc.edu
Tue Apr 17 13:07:32 PDT 2001

```Martin, your idea for the "in between" approved and unapproved is
similar to mine in the "Five Slot" method. That method could be summarized
in our notation as  A > B >> C >> D > F .

As long as there are candidates in all three of the coarse categories,
those in the C category should get 1/2 an approval point. As soon as the
{A,B} or {D,F} category is depleted in a runoff, then any candidate in the
C category would start to receive a one or zero, respectively.

Here's what I wrote, which is similar but not exactly the same:

"If the number of slots between the first and last non-empty slots is odd,
then that cutoff will fall smack on top of one of the slots. That cutoff
slot counts as the average of the two extremes in the Approval
possibilities, i.e. if Approved=1 and Disapproved=0, then the candidates
in the cutoff slot get 1/2 point each . If Approved=1 and Disapproved=-1
(as in Demorep) then the current approval cutoff slot counts as zero in
the current round of the Approval runoff."

Now, to change the subject slightly, here's an example that Craig Layton
sent me in his critique of Approval Completed Condorcet. There are several
interesting things about it.  The first is that our notation is catching
on!

2  A >> B > C
49 B >> C > A
49 C > A >> B

Although Craig didn't say so, these have to be zero info sincere
preferences, otherwise in this close of a contest the A faction would
extend approval to B to try to keep C from winning, and the B faction
would extend approval to C to keep A from winning the Approval vote.

So let's just pretend these are sincere zero info preferences. Although
utilities are not given, it is very likely that C is the highest utility
candidate, as can be seen by comparing the two biggest factions.

Dyadic Approval and Universal Approval both give the same pairwise
results:

(Double all the numbers for Universal Approval.)
A beats B 51 to 49
B beats C 50 to 49
C beats A 49 to 2.

Marginal methods give the win to C.  "Winning Vote" methods give the win
to A.  In this example, it is clear to me that marginal considerations
make better sense.

Note that C beats A on 98% of the ballots, that A comes in last place on
49% of the ballots, and that C comes in last on only 2% of the ballots.
Although utilities could be cooked up that would give A more average
utility than C they would have to give A 97% utility or so in the C
faction to pull it off.

It seems to me that it would be extremely unlikely to have a C > A >> B
faction that large with that high of utility for A without also having a
sizable faction of the persuasion A > B >> C.

So it seems to me that this example adds credence to the quality of
Universal Approval.

As far as Approval Completed Condorcet goes, it would give the wrong
answer A in this zero info election, but would give the right answer C in
the more likely partial info environment, since the strategic adjustments
would obviously lead to  B > C >> A for most if not all of the B faction,
yielding close to 98 percent approval for C.

Forest

On Tue, 17 Apr 2001, Martin Harper wrote:

> I was considering using a Dyadic Approval vote for this poll, primarily to get a
> feel of what it'd be like to vote with it unaided by fancy computer graphics and
> the like. Anywho - the following occured to me:
>
> Suppose I want to vote  A > B > C >> D  - such that my approval vote would be
> A,B,C, and I believe that B is equidistant in utility between A and C. Or,
> equivalently, that I know that A>B>C, but have no info to guide me on whether B
> is closer to A or C - whether it is above or below the mean of A,B,C.
>
> This ought to be weighted to be conceptually equivalent to half a vote for
> A>B>>C>>>D and half a vote for A>>B>C>>>D. In Universal Approval, Simulated
> Approval Runoff, and the like this means that in an {A,B,C} approval election we
> give a point to A, no points to C, and half a point to B.
>
> A similar generalisation allows votes like A>B>C>D>>E. This is all very nice,
> because increased flexibility for the voter means that the voter doesn't have to
> do arbitrary tie-breaking and other nasties - but it makes the interface to
> create the votes directly look even more complicated. But hey, that can be dealt
> with eventually... :-)
>
>
>

```