[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.
> 

One of the main applications of converting ranked ballots into range ballots via the technique described 
above is to be able to see what (plausibly) would have happened if an actual election had been carried 
out with RRV instead of STV.

Of course, the results would vary depending on the probability distribution p that was used in the ballot 
conversions.  If the comparison is repeated using a variety of distributions (preferably from monotonic, 
clone free lotteries, like the ones I have suggested) then you can get a good idea of how RRV might have 
changed the result.

One distribution that would almost be good enough to use for the conversion would be the STV 
distribution itself: give each of the n STV winners a fraction 1/(n+1) of the probability.and distribute the 
remaining probability according to random favorite in order to better separate the resulting candidate 
ratings.