[EM] Better Than Expectation Approval Voting (2nd try readable format)

[fsimmons at pcc.edu]

Mike's exposition of basic Approval and Range strategy as variations on the theme of Better Than 
Expectation strategy was very interesting and valuable, including the recommendation of introducintg 
Approval after score or grade voting, which are much more familiar to most people.
 
That was probably the most important part of his message, but I want to make a few more remarks about 
Approval strategy.
 
1.  When there are candidates between the two front runners and you are not sure where to draw the 
approval line, put it adjacent to the candidate with the greatest likelihood of winning.  In other words put your 
approval cutoff adjacent to the candidate most likely to win on the side of the candidate next most likely to 
win.  This is what Rob LeGrand calls "strategy A."
 
2.  Suppose that order is easier than ratings for you.  Joe Weinstein's idea is to approve candidate X if and 
only if it is more likely that the winner will be someone that you rank behind candidate X than someone that 
you rank ahead of candidate X.  Note that when there are two obvious frontrunners Joe's strategy reduces to 
Rob's strategy A.
 
3.  Suppose that on principle someone would never use approval strategy on a score/grade/range ballot, but 
is forced to use an approval ballot anyway.  How could they vote as close as possible to their scruples?  
For example suppose that you would give candidate X a score of 37 percent on a high resolution score 
ballot, but are forced to vote approval style.  In this case you can have a random number generator pick a 
number between zero and 100.  If the random number is less than 37, then approve the candidate, 
otherwise do not.  If all like minded voters used this same strategy, 37 percent of them would approve 
candidate X, and the result would be the same as if all of them had voted 37 on a scale from zero to one 
hundred.
 
Now for the interesting part:  if you use this strategy on your approval ballot, the expected number of 
candidates that you would approve is simply the sum of the probabilities of your approving the individual 
candiates, i.e. the total score of all the candidates on your score ballot divided by the maximum possible 
score (100 in the example).  Suppose that there are n candidates, and that the expected number that you 
will approve is k.  Then instead of going through the random number rigamarole, just approve your top k 
candidates.
 
Forest