Brief Descriptions of Voting Methods, Part 1

[Bruce Anderson]

On May 27,  5:00pm, Steve Eppley wrote:
> Subject: Re: Attachment CRITTBL1
> Bruce, you've provided a table of some ranked ballot voting methods
> which meet or fail to meet 7 criteria.  Quite a few of the methods
> meet all 7 criteria, including Smith//Condorcet.  
>
> I presume that you're planning to post definitions of the methods 
> mentioned in the table.  I'm looking forward to that, because many of 
> the methods' names are unfamiliar to me and because I'm curious about 
> how Condorcet is defined (and by whom).
>
>-- End of excerpt from Steve Eppley

Quick-and-very-rough outlines of the definitions of the ranked-ballot voting 
methods listed on my table of methods vs. criteria follow. 

A.  Elemental Direct and Sequential Deletion Methods:

Anderson:  selects the candidate(s) with the least pairwise losses, with ties in 
least losses being broken by most pairwise wins -- I am indebted to Mike for 
this simplification of my original, more complex definition.  Anderson reduces 
to Copeland when there are no pairwise ties.  

(Approval is not a ranked-ballot voting method.)

Arrow-Raynaud:  is sequential deletion by the minimum of the row maximums of 
r'(i,j) = r(i,j,1/2).

Borda:  is equivalent to selecting the candidate(s) with the maximum sum over j
of r'(i,j).

Let b(i,k) be the number of voters who rank candidate i uniquely in kth place 
plus 1/m of the number of voters who rank exactly m-1 other candidate(s) as 
being tied with candidate i strictly behind exactly k-h other candidates for 
some h from 1 through m.  Let n be the total number of candidates.  Then Borda 
is also equivalent to selecting the candidate(s) with the maximum sum over k of 
(n-k)b(i,k).

Bucklin:
Let K(i) be the smallest value of k such that 
b(i,1)+...+b(i,k) > n/2.
Then I define Bucklin to select the candidate(s) that minimize K(i), with ties 
being broken by maximum b(i,1)+...+b(i,k), then by maximum b(i,1)+...+b(i,k-1), 
and so on.

Condorcet:  selects the candidate(s) with the maximum of the row minimums of 
r'(i,j).  This definition is not the "official" EM definition, but (according to 
what I have read) it is as consistent with Condorcet's writings as is the EM 
definition, and it greatly facilitates proofs.  I am careful to construct 
counter-examples that are valid for both definitions.

/Condorcet/1a/:  is sequential deletion by the maximum of the row minimums of 
r'(i,j). The /1a/ means (in my notation) to only delete 1 candidate at a time, 
and to continue until only one candidate is left.

Coombs:  is sequential deletion by most last-place rankings, i.e., by the 
maximum of b(i,n).  Coombs continues only until there is a majority winner (or 
until b(i,k) = b(i',k') for all remaining i and i' and all relevant k and k'.

Copeland:  selects the candidate(s) that maximize (pairwise wins minus pairwise 
losses), which is equivalent to selecting the candidate(s) that maximize
2(pairwise wins) + (pairwise ties), which is equivalent to selecting the 
candidate(s) with the best pairwise won-lost record, counting pairwise ties as 
half a win and half a loss.

To be continued.

Bruce