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

[Joe Weinstein]

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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