[EM] Some more ideas

[fsimmons at pcc.edu]

I got a lot of insightful feedback from Jameson, Kristofer, Kevin and others on the distance based ideas, 
Yee diagrams, Centrism, etc.   I'll reinforce some of them ini a later message, but for lack of time I just 
want to get a couple of other ideas on the record for now.

Mike Ossipoff used to say that Approval couold be considered as "plurality done right."

In the same vein we could say that Range can be considered as "Borda done right."

Dodgson is clone dependent for the same reason that Borda is, namely the natural spacing of clones 
(i.e. their closeness) is not reflected in rankings as like it is in ratings.

So a range ballot variant of Dodgson could be considered as "Dodgson done right."

The other problem of Dodgson, its computational difficulty, has been dealt with by approximate 
algorithms elsewhere, but that doesn't do any practical good as long as the clone dependence is 
tolerated.

Bart Ingles used to always point out that if you have a large enough electorate you can do Range with 
Approval ballots. Not only that, but with sufficiently large electorate, you can do range with arbitrarily 
great resolution with only approval style ballots:

You give each person access to a random number generator that gives uniformly distributed decimal 
values between zero and 100 percent.  If the person wants to rate a candidate at 37.549%, she samples 
the random number generator, and if the sampled value is less than 37.549%, she approves the 
candidate, else not.

The "law of large numbers" that makes quantuum effects on the macro scale (like your car "tunneling" 
out of the garage unexpectedly) guarantees that there is no appreciable difference between this 
approach and the full range ballot approach when the electorate is large enough.  "Enough" depends on 
the confidence that you demand, but after about ten thousand voters, other types of random errors errors 
swamp the difference.

Getting back to the clone difficulties with methods like Dodgson and Corda based on rankings instead of 
ratings, I have an idea for directly converting rankings into ratings that removes the clone dependence.  
Once this conversion is made, the Range version can be carried out:

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.  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.  Then the 
rating of Y on ballot B' is just the ratio  S(Y)/SB.

Note that any method based on this conversion automatically satisfies Independence From Pareto 
Dominated Alternatives.

Note how the rankings

70  A>B>C>D
30  B>C>D>A

are converted to the ratings

70 A(100), B(30), C(0)=D(0)
30 B(100), A(0)=C(0)=D(0) .

One more idea I have explained elsewhere: an idea of amalgamation of factions to make a method based 
on Range Ballots summable by precincts. Warren Smith's webpage on that.

Combined with the above ideas, we have powerful tools for overcoming the defects various valuable 
methods..