[EM] Sincere Range and Approval

[Forest Simmons]

I have written before about how to convert sincere ratings into sincere
approval ballots.  This time I want to step back and explain a way to
compute sincere ratings: when the meanings of the sincere ratings are more
evident, then the sincere approvals take on additional meaning, too.


Suppose that by examining the voting record of candidate X you see that on
a random issue of interest to you there is a probability p that she would
vote the same way you would vote if you had the opportunity to vote in the
representative body, i.e. there is a probability p that she would correctly
represent your wishes if she were your representative.


 I will now explain why I consider this probability p to be a natural
choice for your sincere rating of candidate X.


In fact, if every voter V rated candidate X according to the voter’s
subjective
(if not calculated) probability of X correctly representing V on a random
issue of interest, then the sum of these ratings would be the expected
number of voters that would be correctly represented by X on a random issue
of interest.

So with this definition of sincere ratings, the candidate with the highest
sum of ratings is by definition the candidate expected to correctly
represent the greatest number of voters on a random issue of interest:  i.e.
the Range Winner maximizes the expected number of correctly represented
voters (as long as ratings are sincere).


One of the most interesting things about this point of view is that from it
follows a simple definition of sincere approval voting.  We will get to
that definition in two conceptual steps:


(1) An approval voter trying to stochastically mimic her sincere rating p
of candidate X could spin a spinner with fraction p of the circle shaded
green.  If the spinner arrow lands in the green, then the voter approves X.
If all approval voters were to use this strategy, then the expected sum of
approvals for candidate X would be the same as the expected sum of ratings:
i.e. Approval Voting would be statistically equivalent to Range Voting were
all voters to use this spinner strategy.  Both estimate the number of
correctly represented voters.  The main difference is that compared to the
Range Voting estimate, the Approval estimate would have a larger variance
(although both variances would be inversely proportional to the number of
voters).  To significantly reduce this variance in the case of Approval
Voting is the purpose of the second conceptual step.


(2)  The expected number of candidates approved by voter V using the
spinner method is the sum of voter V’s ratings of the candidates.  So
instead of using a spinner for each candidate, voter V should simply
approve her top k candidates and use a q-shaded spinner to decide whether
or not to approve the candidate she ranks (k+1) from the top, where k is
the integer part of the sum of V’s candidate ratings and q is the
fractional part of that sum. This procedure defines what I call “Sincere
Approval.”


One slight change is to allow an affine transformation of the probabilities
to adjust the extremes to zero and one, even if the voter V sincerely
neither completely disagrees nor completely agrees with any candidate.


Another tweak is to allow the voter V to use strategy instead of a spinner
for deciding whether or not to approve the candidate Y that she ranks (k+1).
One such strategy is to approve Y only if voter V’s ballot is more likely
to be pivotal between Y and some candidate that V considers inferior to Y
than some candidate that V likes better than Y.


I will finish by observing that it is well known that a strategically
sophisticated Range Voter can achieve optimal results while using only the
extreme ratings: i.e. Range and Approval are strategically equivalent.  What
I have shown above is that Range and Approval are also statistically
equivalent in the case of sincere voting.  These two facts taken together
suggest that Approval, with its simpler ballots, is an adequate substitute
for Range.  However, we must remember that it is very likely that
psychological considerations outweigh both strategical and statistical
considerations.


Forest
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