[EM] Automated Approval Winner

[fsimmons at pcc.edu]

----- Original Message -----
From: 
Date: Monday, July 25, 2011 3:51 pm
Subject: Automated Approval Winner
To: election-methods at lists.electorama.com,

> Here's an example of a monotone method for converting ranked 
> ballots into approval ballots 
> automatically:
> 
> x: A>B>C
> y: B>C>A
> z: C>A>B
> 
> First we convert to range ballots using first place numbers 
> cumulatively:
> x: A(x+y+z), B(y+z),C(z)
> y: B(x+y+z), C(x+z), A(x)
> z: C(x+y+z), A(x+y), B(y)
> 
> Now we normalize the ratings:
> 
> x: A(1), B(y/(x+y)), C(0)
> y: B(1), C(z/(y+z)), A(0)
> z: C(1), A(x/(x+z)), B(0)
> 
> For fun we calculate the range totals for the respective 
> candidates: 
> 
> x +z*x/(x+z), y+x*y/(x+y), and z+y*z/(y+z), respectively. 
> 
> These values simplify to
> 
> x*(x+2*z)/(x+z), y*(y+2*z)/(y+x), and z*(z+2*y)/(z+y) , respectively.
> 
> The largest of these values determines the range winner.
> 
> Now we begin the conversion of range to approval. First we find 
> the range total for each ballot:
> 
> For each of the ballots from the first faction the range total 
> is (x+2*y)/(x+y).
> For each of the ballots from the next faction the range total is 
> (y+2*z)/(y+z).For each of the ballots from the last faction the 
> range total is (z+2*x)/(z+x).
> 
> These totals tell how many candidates each of the respective 
> ballots should approve.
> 
> Since these numbers are all greater than one, each ballot should 
> approve at least one candidate. The 
> remaining fractions y/(x+y), z/(y+z), and x/(z+x), respectively, 
> tell us what fraction of each faction 
> should approve two candidates. So the approvals are
> 
> x*y/(x+y) : AB, x*x/(x+y): A only
> y*z/(y+z): BC, y*y/(y+z) : B only
> x*z/(x+z): CA, z*z/(x+z): C only
> 
> The respective approval totals are
> x*(x+2*z)/(x+z), y*(y+2*z)/(y+x), and z*(z+2*y)/(z+y) ,
> which are identical to the respective range totals (in this 
> example of a three faction Condorcet cycle, but 
> not in general).

Actually, as long as there are only three candidates, this way of automating approval will always give the 
same approval totals as the range totals (from this way of converting rankings to ratings).

> 
> I'll give the proof of monotonicity in another message.
> 
> Forest