[EM] Craig's strategy puzzle

[Bart Ingles]

I've been meaning to try a mathematical solution to Craig Layton's
approval voting strategy puzzle from a few months back (Feb 11 2001, to
be exact):

> This question is open to all the strategically minded posters; Say with the
> above utilities (100, 25, 20, 0 - assign A,B,C,D respectively).  One opinion
> poll shows (and, being an approval election, these are approval polls so
> they show the predicted winner), that A will get 36 percent of the vote, B
> will get 40, C will get 45 and D will get 44.  You know that there is some
> error in opinion polls; say that you are reasonably sure, based on the
> poll's claims, that there is a 90% chance that the opinion poll will be
> within 5% of the actual vote (that is, 90% chance that A will get between 31
> and 41%), and that it is more likely to be closer to the poll than further
> away (more likely to be 36% than 41%, but we don't know how much more
> likely).  How should you vote?

I dug out an old statistics text and found that if there is a 90% chance
that a candidate's vote total will fall with a 10-percentage point
range, then +/- 5 percentage points must equal +/- 1.65 standard
deviations.  In other words, one percentage point is equal to 0.33 SD.

With this I was able to fill in a pairwise probability matrix, as
follows:

                 Pairwise
Pair             Loser -->
Winner
 |          A      B      C      D
 |    A    -    .255   .068   .093
 V    B  .745     -    .205   .255
      C  .932   .795     -    .566
      D  .907   .745   .434     -


In other words, A has a .255 probability of defeating B, etc.

At this point I became lost in the conditional probability calculations,
and was unable to find the actual win probability of the four
candidates.  This may or may not come back to me, but in the meantime if
anyone wants to post either the formula or the results, feel free.

But as a practical matter, this isn't really necessary in order to know
how to vote.  It's clear that approving C provides a several-fold
increase in the likelihood of defeating D, more than making up for the
utility compromise.  Whether to also approve B, or to just "skip vote"
for A and C, is probably worth a coin-toss.

Bart