[EM] Cardinal Ratings vs. Approval Voting (vs. IRV)

Bart Ingles bartman at netgate.net
Sun Jan 18 03:30:02 PST 2004

Bill Lewis Clark wrote:
> I want to call into question the claim that Cardinal Ratings (CR) is
> strategically equivalent to Approval Voting (AV.)  In particular,
> I'd like to question this claim in light of some of the AV vs. IRV
> points raised on the Fairvote website (among other places.)
> >From http://www.fairvote.org/irv/approval.htm :
> (Claim 1:)
> * Approval voting does not solve the spoiler problem. Voting for
> your second choice candidate can in some cases lead to the defeat of
> your favorite candidate.

But only if your second choice candidate actually wins.  And this can
only happen if the two candidates both manage to beat all others.

Sincerely ranking your top two choices in IRV can sometimes lead to the
defeat of _both_ your first and second choices.  And judging from
Merrill's computer models, this is at least as common with IRV as the
situation you mention under approval.

Which is worse, electing your second choice, or your last choice?

> (This problem is less severe than in
> plurality voting, but instant runoff voting does a better job of
> addressing the spoiler problem.)  Campaigns would urge – quietly at
> least – their supports to "bullet" vote for their candidate only,
> and approval voting would thus tend to revert back to plurality
> voting.  Approval voting is unlikely to work in practice as it is
> supposed to work in theory.

Actually I would expect campaigns to publicly urge supporters to bullet
vote.  I just don't expect voters to do as they're told.

> Suppose voter preferences are as follows:
> A>B>C, approve AB -- 30%
> A>B>C, approve A  -- 21%
> C>B>A, approve BC -- 25%
> C>B>A, approve C  -- 24%
> Under AV, B wins the election with 55% approval (compared to 51% for
> A and 49% for C.)  However, look how things change if a Cardinal
> Ratings system is used (with a 0-2 scale.)
> A>B>C, A:2, B:1, C:0 -- 30%
> A>B>C, A:2, B:0, C:0 -- 21%
> C>B>A, A:0, B:1, C:2 -- 25%
> C>B>A, A:0, B:0, C:2 -- 24%
> Here, A wins with an average rating of 1.02 (compared to 0.55 for B,
> and 0.98 for C.)

If these are the same voters, then the top example is not a reasonable
depiction of approval voting strategy (assuming the Cardinal Ratings
example shows sincere ratings).  Since the first and third groups of
voters rate B exactly midway between the other two, it's not reasonable
to assume they all approve B.  More likely, half of each group view B as
slightly better-than-average, the other half slightly worse.  Or else
they choose randomly, with the same effect.

So a more likely approval outcome would be:
A  -- 15%
AB -- 15%
A  -- 21%
C  -- 12.5%
CB -- 12.5%
C  -- 24%
A = 51%
B = 27.5%
C = 49%

With Cardinal ratings, if voters in the first and third groups actually
rate B as either slightly above or below 1.0, then optimal strategy for
these voters would be to rate all candidates either 2 or 0, as follows:

A:2, B:2, C:0 -- 15%
A:2, B:0, C:0 -- 15%
A:2, B:0, C:0 -- 21%
C:2, B:2, A:0 -- 12.5%
C:2, B:0, A:0 -- 12.5%
C:2, B:0, A:0 -- 24%

Avg rating: A = 1.02, B = 0.55, C = 0.98, or the same ratios as for
approval voting or for sincere Cardinal ratings.

> (Claim 2:)
> * Approval voting forces voters to cast equally weighted votes for
> candidates they approve of.  Voters cannot indicate a strong
> preference for one candidate and a weak preference for another.
> Voters in fact almost always will have different degrees of support
> for different candidates.

You can in effect give partial votes by voting randomly.  In other
words, if you want to give B a half-approval, you can toss a coin and
approve B only if the coin toss is "heads".  Assuming other voters do
likewise, the outcome is the same as for Cardinal Ratings.

Since weak cardinal ratings (or partial approvals) are never optimal
strategy, it's probably better not to entice voters into using them. 
Since tossing coins requires extra effort, approval voting encourages
voters to make the best use of their vote.

> (Claim 3:)
> Approval voting would challenge our notions of majority rule:
> Adoption of approval voting could cause the defeat of a candidate
> who was the favorite candidate of 51% of voters. If this result were
> to happen the system would likely be repealed.

Not likely, since if there were data available to show a candidate as
being so popular as to be the favorite of 51% of the voters, those
voters would likely bullet vote.

> The example already provided addresses this claim, as well.
> Basically, I don't understand why CR and AV were ever considered
> strategically equivalent in the first place.  They're obviously not,
> at least from my perspective.

Best strategy with CR is to give each candidate either the minimum or
maximum rating, making the ballot equivalent to an approval ballot.

> Of course, I haven't (yet) read Samuel Merrill's _Making
> Multicandidate Elections More Democratic_ (which I understand to be
> the canonical text on this matter) so perhaps I'm just talking out
> the wrong orifice.

Seriously, I suggest you read it before investing too much energy in any
voting system.  In particular, pay attention to the effect that
ideologically similar candidates have on the outcome of a race.  With
Plurality, top-two runoff, and IRV, having multiple ideoligically
similar candidates in a race tends to bias the outcome towards a
dissimilar candidate.  This is essentially the spoiler effect, which
tends to encourage candidates, backers, and voters to support one of the
top two "lesser evil" candidates.

Bart Ingles

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