Brief Descriptions of Voting Methods, Part 2

[Bruce Anderson]

A.  Elemental Direct and Sequential Deletion Methods (continued):

/Copeland/2a/:  is sequential deletion by Copeland's score (i.e., by # of 
pairwise wins minus # of pairwise losses, or by any of the equivalent scores 
listed in Part 1).  The /2a/ means (in my notation) to delete all of the 
candidates tied with the lowest Copeland score at the same time (provided this 
does not delete every remaining candidate), and to continue until only 1  
candidate is left (or until all of the remaining candidates have the same 
Copeland score).

Copeland-mod:  selects the candidate(s) with no pairwise losses if any such 
candidate(s) exist(s); otherwise, it selects the candidate(s) with the best 
pairwise won-lost percentage ignoring pairwise ties (as if such ties never 
occurred).  Copeland-mod reduces to Copeland when there are no pairwise ties.

Dodgson:  selects the candidate(s) that minimize the number of (contiguous) pair 
reversals in voters rankings needed to make that candidate not lose any pairwise 
matchup.

Fishburn:  selects the candidate(s) that is (are) not a Fishburn losers.  A 
candidate, say i, is a Fishburn loser if there is some other candidate, say j, 
such that every candidate that pairwise beats j also pairwise beats i, and there 
is at least one candidate that pairwise beats i but does not pairwise beat j.

Hare:  is sequential deletion by least first-place rankings, i.e., by the 
minimum of b(i,1) -- see Part 1 for the definition of b.  Hare is also called 
Majority Preference Voting, Single Transferable Voting (with one winner), and 
some other IMHO less desirable names.  In my notation, Hare is equivalent to 
/Plurality/1a/.

Kemeny:  Consider all n! permutations of the n candidates.  Give each 
permutation 1 point for every voter and every distinct pair of candidates such 
that both the voter and the permutation agree on the relative ranking of that 
pair of candidates.  Give each permutation 1/2 a point for every voter and every 
distinct pair of candidates such that the voter ranks the two candidates in that 
pair has been tied with each other.  Kemeny selects the candidate(s) that is 
(are) ranked first on the (any) permutation that receives the maximum number of 
these points.

Max-Tourneys:  Consider all of the meaningfully different ways of seeding a 
(balanced, non-adaptive) single elimination tournament among the n candidates.  
(I have proven that the number of such seedlings is much less than n!.)  
Max-Tourneys selects the candidate(s) that win(s) the most such tournaments 
(counting pairwise ties in the logical manner).

Nanson:  is sequential deletion by Borda score.  Using my notation, I define 
Nanson as being equivalent to /Borda/1a/ (i.e., delete at most one candidate at 
a time), but others define Nanson as being equivalent to /Borda/2a/.

Plurality:  selects the candidate(s) with largest number of first-place 
rankings, i.e., by the maximum of b(i,1).

Plurality-w/runoff:  selects the candidate that is the majority winner, if such 
a candidate exists; otherwise, it selects the pairwise non-loser(s) between the 
two candidates that receive more first-place rankings (i.e., higher b(i,1)) than 
any other candidate, if exactly two such candidates exist; otherwise, it selects 
any candidate such that no more than one other candidate receives more 
first-place rankings than it does and that is a pairwise non-loser in some 
matchup between it and some other such "first or second place" candidate.

Schwartz:  selects the candidate(s) that is (are) not a Schwartz losers.  A 
candidate, say i, is a Schwartz loser if there are one or more other candidates, 
say j1,...,jm, such that j1 pairwise beats ... who pairwise beats jn who 
pairwise beats i, but there are not one or more other candidates, say k1,...,kn, 
where k1 is j1, such that i pairwise beats kn who pairwise beats ... who 
pairwise beats k1 who is j1.

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

Smith:  A candidate is a Smith winner if it belongs to the smallest non-empty 
set of candidates such that each candidate in that set pairwise beats each 
candidate not in that set.

Young:  selects the candidate(s) that minimize the number of voters whose 
ballots need to be ignored in order to make that candidate not lose any pairwise 
matchup.

Beats-all:  selects the candidate that pairwise beats each other candidate, if 
such a candidate exists; otherwise, it selects all of the candidates under 
consideration.

Plurality-ext:  selects the candidate(s) with largest number of first-place 
rankings, i.e., by the maximum of b(i,1), with ties broken by the largest number 
of second-place rankings (i.e., by b(i,2)), and so on.

To be continued.

Bruce