[EM] Approval Strategy

[Forest Simmons]

Basic Approval Strategies:

1. Given a list L of winning probabilities for the various alternatives, 
you should approve an alternative A if and only if it is more likely that 
the winner will be worse than A than that it will be better than A.

That's the recommendation when the alternatives are ranked, but not rated.

If the alternatives are rated, then you can do better:

2. First calculate the expected winner rating E = Sum over A of P(A)*R(A), 
where P(A) and R(A), respectively, are the winning probability and rating 
for candidate A.

Then approve all candidates rated above E.


Where do we get the rankings or ratings? From the ballots. In other words, 
I'm thinking of Declared Strategy Voting where the voters fill out 
rankings or (preferably) ratings and we use one of the two basic approval 
strategies to calculate their approval cutoffs automatically.

Where do we get the list L of probabilities?

That's the most interesting part.

If you are a Bayesian Statistician, you start by assuming that all of the 
winning probabilities are equal, then you find the winner (based on these 
prior probabilities and a random sample of the ballots) and use Bayes 
formula to update the probabilities, and then repeat until the 
probabilities converge (if possible).

That's too messy, especially when there are other more elegant ways of 
coming up with probabilities.

Chris Benham's Weighted Median Approval uses (symmetrically completed) 
ranked ballots, and then for winning probabilities uses the probability of 
winning under random ballot.

I emphasize that Chris' method (WMA) is perfectly deterministic, but his 
"weights" can be interpreted as probabilities.

This suggests that the list L could be the probabilities generated by any 
lottery method.

In other words, for each "lottery method" there corresponds a 
deterministic method that uses the lottery probabilities as winning 
probabilities in one of the two basic Approval strategies, depending on 
the ballot type.

I have more to say about this, but I want to let folks digest this much 
first.

In summary, good lotteries are good sources of winning probabilities for 
use in the two basic Approval strategies.  This gives a standard way of 
converting random methods into deterministic methods for use in cases when 
randomness is not considered essential for fairness or for thwarting 
manipulators.

Forest