[EM] Power of votes with approval

[Bart Ingles]

Stephane, you wrote:
> For Approval, this is how I would do, I am not sure it is optimal.
> 1) I would cut all candidates into two equal groups the ones I like, the
> ones I do not.
> Without poll information, I believe it is the vote that would optimize
> my voting power...

You seem to be equating 'voting power' with 'number of distinctions
drawn between approved and unapproved candidates'.  This might be true
when averaged across the universe of election possibilities, but not
necessarily for a given election unless the mean utility for all
candidates just happens to coincide with the median utility from your
point of view.

A better definition of voting power might be "the ability to influence
the outcome of an election in a preferred direction."  Voting for
exactly half of the candidates may not do this, even in the absence of
polling information.

Suppose your sincere preferences are A>B>C>D>E>F, and your sincere
cardinal ratings (as an approximation of utility) for these candidates
are 100, 8, 6, 4, 2, and 0 respectively.  The average rating for these
candidates is 20.  Since only candidate A is rated above the average,
you should vote only for A.

Using the strategy you proposed above would require you to vote ABC,
which would merely increase the likelihood of electing someone you
strongly dislike (B or C).  In other words, the expected outcome without
your vote is 20.  Voting for candidates rated less than 20 can only
worsen the outcome.

Similarly, if your sincere cardinal ratings for A>B>C>D>E>F are 100, 98,
96, 94, 92, 0, you should vote for ABCDE, since only candidate F is
worse than the average rating of 80.


> 2) If I can get some poll information I judge reliable enough:
> out of my desired candidates, I would vote for the one I prefer the most
> and would approve too all my desired candidates that obtain a better
> position
> according to the poll.
> Example: sincere preferences A>B>C>D | E>F>G>H
> No poll: vote ABCD.
> The poll: D>H>B>A>C>G>F>E. I would vote ABD.

Again, optimal strategy would depend on your sincere ratings for the
candidates, and would require more polling information than how the
candidates were ranked in the poll.

For example, suppose your ratings for the candidates are 100, 98, 12,
10, 8, 4, 2, 0, and the polls (in addition to the order shown above)
also show that D, H, and B are in a fairly close 3-way contest, well
ahead of the remaining candidates.  In that case, a vote for AB would be
optimal.

A simple, close-to-optimal strategy would be to first identify two or
more frontrunners, if possible.  I would then determine the average
rating of these frontrunners (36 being the average for D, H, and B in
the above example) and vote only for candidates whom I rate higher than
that average.

Bart

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