[EM] Approval vs. CR (again)

wclark at xoom.org wclark at xoom.org
Wed Jan 28 18:02:05 PST 2004

Richard Moore wrote:

> You may be reading to much into the statement "CR is strategically
> equivalent to Approval". What this statement means is just the
> following: If S is an optimum strategy for an Approval election, and
> S' is the equivalent CR strategy (in which an S' ballot is one that
> gives 100% rating to all candidates approved in S and 0% rating to all
> candidates disapproved in S), then S' is also an optimum strategy for
> CR, if the probabilistic information used in determining the strategy
> is the same in both cases.

It's that last part, about the probabilistic information being the same in
both cases, that I'd been missing.  Thanks for the clarification.

> Granted it may be possible to have different sets of probabilities in
> a CR election than you would have if the same election were held with
> Approval -- for instance, you might know that members of party X have
> a tendency not to vote all-or-nothing in CR while members of party Y
> have a tendency to vote optimum strategy, and this may give an edge to
> party Y's second choice over party X's second choice. But strategic
> equivalence does not mandate that the resulting strategies be
> identical when the initial probabilities are different. So this does
> not run contrary to the strategic equivalence assertion.

Then I don't see the significance of the strategic equivalence, at all. 
If my strategy is going to be different for the exact same election with
the exact same voter preferences, with the only difference being whether
Approval or CR is used, then I don't see why they should be treated as
being equivalent in any important way.

My main reason for questioning the strategic equivalence between Approval
and CR is that, in my experience, this equivalence has often been used to
suggest that there's no reason to support CR.

As one example, on the electionmethods.org website, it states:

"Cardinal Ratings are strategically equivalent to Approval but more
difficult to implement, hence they are not worth pursuing."
  [ From http://www.electionmethods.org/evaluation.htm ]

I think CR deserves a lot more attention than it's received.  In many
ways, it's a generalization of various other systems.  By restricting the
range of values in one way, CR can be made to simulate Approval.  By
restricting them in another way, it can simulate Borda.  There are entire
classes of variants that, to my knowledge, haven't been explored at all --
because CR is dismissed as being "not worth pursuing."

> Instead of forcing us to try to conceive of such a case, why not give
> an example?

I did, in a previous email.  I'll include it at the end of this reply.  It
makes additional assumptions about how to convert between sincere CR
ballots and sincere Approval ballots (which probably aren't justified) but
the general idea is still clear, I believe.

> The truth of statement B is not dependent on its being equivalent to
> statement A, so you have a straw man argument.

I was making two different points.  The first was that A and B do not mean
the same thing, nor does A imply B simply in virtue of the meaning of the
statements.  You already made it clear that you agree with that point, but
my reason for raising it was that others sometimes seem to think they're
semantically equivalent.

The second point I was trying to argue was that A doesn't imply B at all,
but that was based on my misunderstanding of what "strategic equivalence"
entails.  The argument I was trying to make was based on precisely the
point you clarified in your definition -- that the probabilistic
information on which optimum strategies are based will differ depending on
whether Approval or CR is used in an election.

I was never trying to argue directly from the first point to the second,
so there was no straw man argument in play.  I was simply trying to clear
up some apparent confusion in the first case, and then went on to generate
some of my own in the second.

Thanks for helping to clarify the issue.

-Bill Clark


[Previously posted on Mon, January 19, 2004 19:18]

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:


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

In thinking about this issue more, I'm no longer convinced that the
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

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