# [EM] Hybrid Beats-All/Approval v. Straight Approval

Joe Weinstein jweins123 at hotmail.com
Sun Oct 21 02:13:02 PDT 2001

```Hybrid Beats-All/Approval  v.  Straight Approval.

Forest Simmons has asked (Archive message 8222, 'Re: two bit ratings', 11
Oct 2001) whether the public might be easier persuaded to go with a hybrid
method - Beats-All/Approval - rather than straight Approval.  In the hybrid
method, the Beats-All candidate wins if such candidate exists, and otherwise
the Approval winner wins.

I don't know about the public, but for me straight Approval is clearly
superior in result, as well as appealingly simpler to describe.  For
instance, consider just the following simple examples involving two, three
or four candidates.  (Apologies if these examples unwittingly resemble
others posted lately to this list.)

Each line describes a voter bloc:  the left-hand number is the percentage of
voters in the bloc, >> divides approved candidates from non-approved
candidates, and a notation X=Y signifies that within the given bloc equal
numbers of voters take X>Y and Y>X.

Example 1.

55   B>A >>
45   A >> B

Example 2.

55   B>A >> C
45   A>C >> B

Example 3.

12   B>A >> C1=C2
08   A >> C1=C2>B
20   C1 >> B>C2>A
20   C1>A >>B> C2
20   C2 >> B>C1>A
20   C2>A >> B>C1

In all examples, A is the Approval winner and B is the Beats-All candidate.

In both Examples 1 and 2, A has 100% approval,  and B has just 55% approval.
In Example 3, A is the only candidate with majority (> 50%) approval, and
indeed has 52% approval, whereas B is the least approved candidate, with 12%
approval.

To be sure, in Examples 1 and 3, A is beaten pairwise by every other
candidate X!! (More voters at least slightly prefer X to A than do A to X.)

Nonetheless, in all three cases A would be the most accepted candidate, more
than any other candidate X, basically because more voters materially prefer
A over X - i.e. approve A and do not approve X - than materially prefer X
over A.

Given suitably assumptions, we can argue that average satisfaction with A is
higher than for any other candidate.  Most notably and simply, suppose each
voter rates each approved candidate at roughly 100 (%) (totally
satisfactory) and each non-approved candidate at roughly 0 (totally
unsatisfactory).  Then averaged over all voters, approval ratings (percent)
for each candidate represent average voter satisfaction.   (Actual values
are A=100 and B=55 in Examples 1 and 2, and A=60 and B=12 in Example 3.)

Some people might object to Example 1 because each voter in the larger voter
bloc there fails to take advantage of the election method to make the
biggest possible strategic distinction among candidates - namely approval
for at least one v. disapproval for at least one.  In Examples 2 and 3,
however, every voter does take this opportunity.

Example 2 illustrates the most basic scenario requiring more than just two
possible grades which (as Forest has noted) IRV supporters demand the
election method be able to register.

In effect, the simplest available election method to handle this scenario
would be ‘Three-Slot Approval' with three allowed grades:  ‘strong
approval', ‘weak approval', and  ‘disapproval'.  Three-Slot Approval would
work better than - and be as easily explained as - the Beats-All/Approval
hybrid.

Joe Weinstein
Bixby Knolls, Long Beach CA USA

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