[EM] Some more ideas

[fsimmons at pcc.edu]

> > For each candidate X, let p(X) be the probability that X would 
> > be chosen by random ballot, i.e. in the 
> > case of of no "equal first rankings" it is just the percentage 
> > of ballots on which X is ranked above all other 
> > candidates.
> > 
> > Convert ranked ballot B to a range ballot B' as follows:
> > 
> > Let SB be the sum of all p(X) such that X is ranked above 
> bottom 
> > on B. 
> 
> Bottom means truncated. So SB is the sum of p(X) for ranked X.
> 
> >For each candidate Y, form the 
> > sum S(Y) of all p(X) such that X is ranked above bottom but 
> > lower or equal to Y on ballot B. 
> 
> So S(Y) is the sum of p(X) for the X that are ranked equal to 
> or below Y, but not truncated.
> 
> >Then the rating of Y on ballot B' is just the ratio S(Y)/SB.
> > 

With the bottom=truncated interpretation, a more apt probability distribution (instead of the random 
favorite lottery distribution) would be the distribution corresponding to the "random ranked lottery:" The 
random ranked lottery works like this:

Initialize a set variable V with the entire set of candidates. Then while the number of members of V is 
greater than one, draw ballots at random until some ballot B has non-empty intersection with V.  
Replace V with the intersection of B and V.  EndWhile.

Note that we don't actually carry out the lottery; we just use its probability distribution.  This keeps the 
method deterministic.

When I ask a student for an example of a random variable, sometimes they will say, "How about the 
probability that a tossed fair coin comes up heads?"  The answer is 50%, which is not a random variable.

On the otherhand, if the probabilities are hard to calculate exactly, we can do a MonteCarlo experiment 
to determine the distribution to arbitrary degree of accuracy, i.e. so accurate that the possible error 
could not possibly change the result of the election based on that distribution.