[EM] Some more ideas

[fsimmons at pcc.edu]

> 
> 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 range variant of Dodgson is to elect the candidate that requires the least total change in its ballot 
ratings to turn it into a CW.

> 
> 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. 

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

I'm afraid that this claim is wrong.  However, it does change Borda, which is highly dependent on Pareto 
Dominated Alternatives into a method that is Independent of 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) .

With the truncation clarification, since A is not truncated this chages to

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

Note that with the truncation clarification

70 A>B
30 B>A

which converrts to

70  A(100), B(30)
30  B(100), A(70)

is different than

70 A
30 B

which converts to

70 A(100), B(0)
30 B(100), A(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.

To amalgamate factions with range ballots create a faction for candidte X by taking a weighted average of 
all of the ballots that rate X at 100%.  The weight for a ballot is the reciprocal of the number of 
candidates that are rated at 100% on that ballot.  The weights of the amalgamated factions are the 
respective totals of the weights that went into their creation.

This amalgamation is summable over precincts, so it can convert non-summable methods based on 
range or approval ballots (which are range ballots on a scale of zero to one)  into summable methods.

With large elections the resulting amalgamated faction ballots will be the same whether or not the range 
voters voted actual high resolution range ballots or they voted approval ballots with the assistance of 
random number generators as explained above.

This opens up the possibility of using approval ballots for any method that can be based on range 
ballots, including Dodgson, as explained above; just apply Dodgson to the amalgamated faction ballots.

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