[Election-Methods] Proposed "Bucklin Ratio" methods

[Dan Bishop]

DEFINITION: The Jth Bucklin ratio for candidate C = (# of ballots on 
which C is ranked in the top J) / J

Example:

42%: Memphis > Nashville > Chattanooga > Knoxville
26%: Nashville > Chattanooga > Knoxville > Memphis
15%: Chattanooga > Knoxville > Nashville > Memphis
17%: Knoxville > Chattanooga > Nashville > Memphis

The first Bucklin ratio for each candidate is:

Memphis: 42% / 1 = 42%
Nashville: 26% / 1 = 26%
Chattanooga: 15% / 1 = 15%
Knoxville: 15% / 1 = 17%

The second Bucklin ratio for each candidate is:

Memphis: 42% / 2 = 21%
Nashville: (42% + 26%) / 2 = 34%
Chattanooga: (26% + 15% + 17%) / 2 = 29%
Knoxville: (15% + 17%) / 2 = 16%

The third Bucklin ratio for each candidate is:

Memphis: 42% / 3 = 14%
Nashville: (42% + 26% + 15% + 17%) / 3 = 33.333...%
Chattanooga: (42% + 26% + 15% + 17%) / 3 = 33.333...%
Knoxville: (26% + 15% + 17%) / 3 = 19.333...%

The fourth Bucklin ratio for each candidate is 100%/4 = 25%.

DEFINITION: The "greatest Bucklin ratio" (GBR) of candidate C = max((Jth 
Bucklin ratio for C) for J in {1, 2, ..., N}), where N is the number of 
candidates.

In the same example,

GBR(Memphis) = max(42%, 21%, 14%, 25%) = 42%
GBR(Nashville) = max(26%, 34%, 33.333...%, 25%) = 33.333...%
GBR(Chattanooga) = max(15%, 29%, 33.333...%, 25%) = 33.333...%
GBR(Knoxville) = max(17%, 16%, 19.333...%, 25%) = 25%

**** METHOD PROPOSAL #1: "GREATEST BUCKLIN RATIO" (GBR method) ****

Elect the candidate with the highest GBR.  (Or, in a multi-seat race 
with S seats, elect the S candidates with the highest GBR.)

This method meets the Majority Criterion (like Buckling) and 
Monotonicity (unlike Bucklin), but as you can see from the example, it 
can elect the last choice of a majority!  Perhaps some kind of instant 
run-off would help.

*** METHOD PROPOSAL #2: "BUCKLIN RATIO RUNOFF" (BRR method) ****

Eliminate the candidate with the lowest GBR until only S candidate(s) 
remain.

In the example, we start by eliminating Knoxville.  The ballots are now 
treated as:

42%: Memphis > Nashville > Chattanooga
26%: Nashville > Chattanooga > Memphis
32%: Chattanooga > Nashville > Memphis

And the new Bucklin ratios are:

Memphis: max(42%, 42% / 2, 100% / 3) = 42%
Nashville: max(26%, (42% + 26% + 32%) / 2, 100% / 3) = 50%
Chattanooga: max(32%, (26% + 32%) / 2, 100% / 3) = 33.333...%

We eliminate Chattanooga, leaving a two-candidate race between Memphis 
and Nashville, which Nashville wins 58-42, so Nashville is the winner.

Nashville happens to be the CW.  I don't know that BRR meets the 
Condorcet Criterion, but it clearly meets the Condorcet Loser Criterion 
(because the CL by definition cannot win the final pairwise race).