[EM] difficulty of interpersonal comparisons in utility (fwd)

[Forest Simmons]

On Fri, 21 May 2004, James Green-Armytage wrote:

>
> Gervase asked for comments on this paragraph:
> >"The concept of cardinal utility suffers from the absence of an objective
> >measure of utility when comparing the utility gained from consumption of
> >a
> >particular good by one individual as opposed to another individual.  For
> >this reason, neoclassical economics abandoned utility as a foundation for
> >the analysis of economic behaviour, in favour of an analysis based upon
> >preferences [i.e. rankings]."
>
> 	I think that the key point in this paragraph is "comparing the utility
> gained ... by one individual as opposed to another individual".
> 	It is not hard for me to say "I prefer Nader to Gore, and Gore to Bush,"
> and for Joe Schmo to say "I prefer Bush to Gore, and Gore to Nader." But
> it is harder for me to have a basis to say something like "I prefer Gore
> to Bush *more* than Joe Schmo prefers Bush to Gore. It may be true, but it
> is hard to find out whether or not it's true, especially on a large scale.
> You can ask people how much they care, but if saying that they don't feel
> strongly means that their vote is simply reduced to a fractional value, it
> seems unlikely that people will make such an admission.
> 	However, I may point out that it might be somewhat easier to say that "I
> prefer Gore to Bush more than *I* prefer Nader to Gore." Thus, I think
> that it is more possible to prioritize your own preferences in comparison
> to one another than to prioritize your preferences in comparison to the
> preferences of another person.
>

Part of the beauty of Declared Strategy Voting (DSV) based on CR style
ballots (with Approval outputs) derives from the idea that you have
expressed above (all the CR comparisons are within ballots, as opposed to
between ballots).  In fact, there is no need to normalize the CR ballots,
since once the probabilities have been calculated, the "above expected CR"
strategy is applied to each ballot, and normalization doesn't change which
candidates are above expected CR on any given ballot.

My current favorite DSV method goes like this:

Ballots are Cardinal Ratings on any scale.

For each i between one and the number of candidates, let p(i) be the
probability that candidate i would be the [your favorite deterministic
method] winner for a set of one thousand randomly chosen ballots from the
ballot set.

Then for each CR ballot in the entire ballot set calculate the expectation
E given by the sum over i of p(i)*r(i), where r(i) is the ballot's rating
of candidate i.

If r(i) is greater than E, then that ballot approves candidate i.

If r(i) is less than E, then that ballot fails to approve candidate i.

In the unlikely case that r(i)=E, then that ballot gives half approval to
candidate i, unless r(i) is one of the extreme ratings on the ballot, in
which case the wishes of the ballot owner are respected (top rating ->
approval, bottom rating -> lack of approval).

The DSV approval scores for all of the ballots are summed to find the DSV
winner.

[End of description of method]

Note that once the ballots have been collected [and your favorite
deterministic method has been specified], then for each i, the probability
p(i) is a well defined number.  So the method I have defined, though
probabilistic in spirit, is nevertheless as deterministic as Plurality.

In fact, the Plurality winner can be described as the candidate most
likely to be chosen by random ballot (once the ballots have been
collected).

However, some well defined probabilities can be hard to calculate from
first principles. Sometimes it is easier to use very precise estimates of
those probabilities obtained by experiment. This is the essence of the
famous Montecarlo method. The values of the p(i)'s can be estimated to any
desired number of decimal places with any desired degree of confidence by
sufficient experimentation.  In our context of elections it would be easy
to make the reliability of the (Montecarlo) probability estimates far
greater than the reliability of most other aspects of the election
process.

On another vein of this thread concerning normalization of CR ballots:

Long ago Richard Moore and I pointed out that when CR is used additively,
the only way to avoid incentive for distortion of voter CR is to have a
zero information election AND to normalize via the L_2 norm (i.e. the root
mean square norm), so that each ballot (after normalization) is on the
boundary of a round ball centered at the origin of Euclidean N space,
where N is the number of candidates.

If some other norm is used, then there will be corners or other bulges in
the boundary the feasible region (i.e. the unit ball determined by the
norm in question), so there will be incentive (even under zero info
conditions) to distort ratings towards those bulges in order to maximize
the effect of one's ballot.

Forest

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