[EM] Optimal Approval strategy

[Richard Moore]

Joe Weinstein wrote:
> Optimal Approval Strategy
> 
> One instrumentally-driven strategy for you as a voter in an Approval 
> election is to vote for each candidate whose utility for you is at least 
> the election's prior utility (i.e. expected utility without your 
> participation).   A couple weeks ago I termed this strategy the 
> 'Rational Strategy'.
> 
> A few days later, as an aside within a long post, Richard Moore noted - 
> very tactfully, not mentioning me or taking me to task in any way - that 
> he had some time before posted an example showing the non-optimality of 
> this 'Rational Strategy'.
> 
> To my frustration, for his example's posting, Richard gave no operative 
> reference - time or place or name.  Maybe this was for the better:  I 
> was forced to devise my own example.  In generic form, this example  
> readily illustrates implications about Approval strategy which likely 
> are well known to experts, including various on this list.

My example can be found in the list's archives at

http://groups.yahoo.com/group/election-methods-list/message/9380

At first I thought that the above-expected outcome strategy would prove
to be equivalent to the delta-P (i.e., Pij) strategy, but this was naive.
When I tried to work out the equivalence, I found out that it held true
*only* if the negative delta-Pij values for i != j (which must exactly
offset the positive value of delta-Pjj, for a given j) were distributed
among the "i" candidates in a manner proportional to those candidates'
win probabilities. If they are distributed disproportionately (a likely
case), then it is possible that the two strategies will give different
results (somewhat less likely, since it depends on how disproportionate).
In an extreme case, such as my example, vote-skipping strategies may
come into play. As Mike pointed out, it is already known that the optimum
strategy can, in rare cases, involve vote-skipping.

Actually taking into account disproportionate delta-Pij values would
require a way of modeling the correlations between votes for various
pairings of candidates: Do A voters tend to vote for or against B, for or
against C, and so forth. I think in general this will be so difficult that
most strategies in the real world will treat these correlations as if we
have zero information about them; i.e., we will ignore the correlations
and just base strategy on the single-candidate probabilities.

Incidentally I do like what Joe calls the Rational Strategy for a couple
of practical reasons.

First of all, the Rational Strategy is actually very intuitive: However
badly you expect the election to turn out, vote for candidate X if you
believe that with that candidate, you will do better than that expectation.
If you are sure that the winner will be Bush or Gore, and you think you
can do better than that expectation with Browne or Nader, then vote for
Browne and Nader, along with your favorite of the two front runners. No
need to take Buchanan into account, and no lengthy calculations required.

Second, it utilizes feedback. You can simplify the Rational Strategy into
simply asking, "Whom do I like better than the incumbent?", reflecting
an expectation that the next winner will typically be about as good as
the current incumbent. You might want to add an approval vote for the
incumbent as well, if you approve of the job the incumbent is doing.

The payoff of this is that, if everyone votes this way, then if the
incumbent is not well liked, the problem will quickly correct itself in
the next election. Bad incumbents -- those with low real approval --
don't become entrenched, as they do in the current system. Thus Approval
voting should turn a rotten political system around quickly. The quality
of the outcome (as perceived by the voters) and the voters' expectations
for the next election's outcome will rise hand in hand. Moreover,
incumbents who are highly approved will tend to be included along with
each voter's better choices and have a better chance of staying in
office.

  -- Richard

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