Besides the obvious problem with the notion of a fraction of a bit, you're still confusing the number of possible ballots with the amount of information conveyed by a single ballot.
If there are 3 candidates, in approval a ballot only needs 3 bits. Ranked ballots need to carry the order the voter selected, and that requires 2 bits per alternative. I.e for ballots with ABC, you need 11, 01, 10 to indicate B>C>A. You cannot do that with fewer than 6 bits, even though it only takes 3 bits to count the 3! = 6 possible ballots. -----Original Message----- From: [email protected] [mailto:[email protected]] On Behalf Of Jonathan Lundell Sent: Saturday, June 06, 2009 1:44 PM To: Paul Kislanko Cc: [email protected] Subject: Re: [EM] Some myths about voting methods On Jun 6, 2009, at 11:10 AM, Paul Kislanko wrote: > The number of possible votes is not the same as the amount of > information in > a single ballot. With 3 candidates, there are indeed 8 possible > ballots, but > any one ballot can be encoded in 3 bits, since any particular choice > requires only that many to represent it. > > Ranked ballots require 2 bits per alternative (01 = 1st, 10 = 2nd, > 11 = 3rd) > so the minimum ballot representation is six bits, twice as much > information > as is contained in an approval ballot. If we disallow truncation and equal-ranking, we have 3! or 6 ballots, or ~2.6 bits. If we allow truncation but not equal-ranking (except as implied by truncation), we have: A > B [> C] A > C [> B] A ...and the symmetrical cases ranking B or C first, for 9 possible ballots. Add one for the empty (no preference) ballot and we have 10 ballots, requiring ~3.3 bits to represent. If we allow equality but not truncation (which would be redundant) we have: A = B = C A > B > C A > C > B B > A > C B > C > A C > A > B C > B > A A > B = C B > A = C C > A = B A = B > C A = C > B B = C > A Do I have them all? 13 possible ballots, ~3.7 bits. There are 7 possible approval ballots (if we consider approve-all and approve-none equivalent): ~2.8 bits. ---- Election-Methods mailing list - see http://electorama.com/em for list info ---- Election-Methods mailing list - see http://electorama.com/em for list info
