# [EM] Sunday reply to Bill Lewis Clark

Bill Lewis Clark wclark at xoom.org
Mon Jan 19 17:17:47 PST 2004

```> *IF* those sincere votes suffered from a sort of "roundoff error"
> during conversion from CR/RV to Approval, this might distort the
> overall distribution of voter preferences such that Approval and
> CR/RV would require different optimal strategies.

Here's a concrete example, which might make things clearer:

Suppose sincere voter ratings are as follows:

(I)    A:10    B:1     C:2     D:0     46%
(II)   A:1     B:9     C:10    D:0     28%
(III)  A:0     B:2     C:1     D:10    26%

Assuming 100% of voters are strategists, these are the optimal strategies:

(I)    A:10    B:0     C:10    D:0
(II)   A:0     B:10    C:10    D:0
(III)  A:0     B:10    C:0     D:10

Assuming 67% of voters are strategists, these are the optimal strategies:

(I)    A:10    B:0     C:10    D:0
(II)   A:0     B:10    C:10    D:0
(III)  A:0     B:10    C:0     D:10

(Yes, they're the same in this case.)

Now, let's convert the example to Approval -- and assume that 100% of
Group I will approve A, 100% of Group II will approve BC, and 100% of
Group III will approve D.  This assumption is improper, but I'm making it
here anyway in order to show how the rest of my reasoning went through.

The new sincere voter approvals are as follows:

(I)    A:1     B:0     C:0     D:0     46%
(II)   A:0     B:1     C:1     D:0     28%
(III)  A:0     B:0     C:0     D:1     26%

Now, assuming 100% of voters are strategists:

(I)    A:1     B:0     C:1     D:0
(II)   A:0     B:1     C:1     D:0
(III)  A:0     B:1     C:0     D:1

This matches up with the optimal strategies for CR/RV, when all voters are
strategic voters (with a rating of 10 corresponding to approval, and 0
with disapproval.)

HOWEVER, look what happens to the optimal strategy when 67% of the voters
are strategists:

(I)    A:1     B:0     C:0     D:0
(II)   A:0     B:1     C:1     D:0
(III)  A:0     B:1     C:1     D:1

The situation is analogous to my earlier example -- the support that
Groups I&II give to B aren't enough to overtake A's lead, and so Group I
has no reason to support C.

Thus, Group I has a different optimal strategy, depending on whether
Approval or CR/RV is used.  Thus, Approval and CR/RV are not strategically
equivalent -- *IF* a significant portion of the voters are sincere, and
*IF* the (unwarranted) assumption regarding conversions from ratings to
approvals is allowed.

Anyway, that's pretty much what I was thinking.  I've left a good bit of
the calculations out of this email, but if you'd like to check my work on
your own, I've put a copy of the spreadsheet I used to construct this
example at the following location:

http://strategy.xoom.org/strategy.xls

(The green fields on the spreadsheet are the ones you can tinker with,
everything else is derived from those cells.)

probabilistic model is entirely realistic.  I doubt that in a real-world
situation, 10% of voters who sincerely rated a candidate as a 1-out-of-10
would sincerely approve of that candidate under Approval -- and I'm
*extremely* doubtful that 10% of those who *did* approve such a candidate
would sincerely reject a candidate they rated as a 9-out-of-10 (yet this
is what the probabilistic model would have us believe.)

However, I don't think I'm warranted in making my original assumption,
either.  I think the truth lies somewhere in the middle.  Exactly *where*
in the middle is what will determine the viability of my argument (or, to
put it another way, my argument makes the assumptions underlying the
strategic equivalence between Approval and CR/RV more explicit.)

-Bill Clark

--
Dennis Kucinich for President in 2004
http://www.kucinich.us/

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