[EM] 0-info approval voting, repeated polling, and adjusting priors

[Simmons, Forest]

As usual, Jobst has given us lots of food for thought.
 
First I would like to compare Joe Weinstein's approval strategy with its marginal version based on tie probabilities.
 
The marginal version is to approve alternative X iff X is more likely to tie for first place with an alternative that you prefer X to, than with an alternative that you prefer over X.
 
Weinstein's own suggestion was to approve X iff an alternative worse than X is more likely to win than a better alternative.
 
It seems to me that the marginal version is the approval strategy that maximizes your probability of making your approval ballot pivotal (in the desired direction).  In other words, it is the max "voting power" approval strategy.  It is the one that minimizes the probability that given an election where your ballot could have made a difference, it either didn't make a difference or it made it in the wrong direction.  In other words, it minimizes the probability of voter regret, given the opportunity of being tie maker or tie breaker.
 
But most elections are not that close, which is why I prefer to approve precisely the candidates that I believe deserve my support.  And if I like X better than Y and think that Y deserves my support, then I think that X does also, so an approval cutoff is all I need to decide, once I have ranked the alternatives.  I vote to support, not under the illusion that my vote will be the tie breaker.
 
Furthermore, it seems to me that Weinstein's original suggestion is more robust than its marginal version.  By "robust" I mean it works well even in less than ideal conditions.
 
Next, what do we mean by probability of winning?
 
Suppose that voter preferences are known precisely:
 
40 A>B>C
35 B>C>A
25 C>A>B
 
What is the probability that A will win?
 
That depends on the method.
 
But what if the method itself incorporates (prior) winning probabilities as part of the method?
 
Where do we get those probabilities?
 
They could be subjective, but is their a better way?
 
What if we take random samples (with replacement) of size ten, say, from the ballot set and see how a bootstrap version of the method works on those sets.  Then we get the win and tie probabilities empirically.  The results might be mathematically predictable, so that the actual samples wouldn't have to be taken, etc.
 
Gotta Go,
 
FWS
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