Warren asked me to not write him anymore, but I want everyone to know that he has been unable to substantiate his claim that as much information can be obtained from approval ballots as from ranked or range ballots.
What's more interesting to a simple voter like me is the suggestion that my ballot could be encoded as "6" using the mapping of x>y>z -> n and then possibly COUNTED using a machine that has n <- a>b>c. I wouldn't like a ballot representation that gives different results depending upon which machine counted it. I.e., I don't care how possible it is to compact the information by tricks like lookaside tables, I want my ballot auditable and to contain all the information I put in. -----Original Message----- From: [email protected] [mailto:[email protected]] On Behalf Of Paul Kislanko Sent: Saturday, June 06, 2009 4:40 PM To: 'Kristofer Munsterhjelm' Cc: [email protected] Subject: Re: [EM] Some myths about voting methods All fairly interesting, but whatever that algorithm is is "off topic." Show me an algorithm that can take ONE ballot as input and return one of the permutations of {A B C} using 3 or fewer bits. Several people have noted that one can list all 6 permutations in order and assign each a number as in 1 = A>B>C 2 = B>C>A 3 = C>A>B 4 = B>A>C 5 = A>C>B 6 = C>B>A so since 6 = '110' it requires only 3 bits to record the voters' choices. BUT, and this is important, there are 6! possible relations of {1,2,..6} -> {x>y>z} so in order for one to retrieve the ranked order from a ballot the ballot must include the specification of WHICH of the 720 possible arrangements was used to form the ballot. At the very least the table requires 3 bits for the left-hand side of the table and six bits for the right hand side so using the "table lookup" trick requires 18 bits to be added to the 3 bits in the {1,2,...3} code or 24 bits per ballot. -----Original Message----- From: [email protected] [mailto:[email protected]] On Behalf Of Kristofer Munsterhjelm Sent: Saturday, June 06, 2009 2:58 PM To: Paul Kislanko Cc: [email protected] Subject: Re: [EM] Some myths about voting methods Paul Kislanko wrote: > No, no no no NO. Now you're introducing counting algorithms. Which have to > have pre-processed the ballots to produce the summary in a compacted format. > > You MUST consider how to use ONE ballot to represent A>B>C in a > three-candidate race. You cannot do it with less than 6 bits. > > ONE ballot. Three candidates. Express ONE VOTER's choices with the minimum > number of bits. Alright. I'm going to disregard equal rank so that it may be simple, and I'll take the three candidate example. The first choice of a voter is one of three candidates. The second choice of the voter is one of two candidates. The third choice is given, because it's whichever candidate remains. The algorithm is like the one you use to convert a decimal number to a hexidecimal number. We can make the observation that a decimal number is merely a group of digits that each range from 0..9, inclusive. For the above, we have one "digit" which ranges from 0..2, inclusive, and another which ranges from 0..1, inclusive. So, for three candidates, you'd get: Start with the set {A, B, C}, and the ballot number b. Get the remainder when dividing b by 3 - call the remainder r_0. Subtract this from b and divide by 3 to get q. Get the remainder when dividing b by 2 - call the remainder r_1. If r_0 is 0, then A is the first choice. If r_0 is 1, then B is the first choice. If r_0 is 2, then C is the first choice. Remove that candidate from the set. Now you have a set of two candidates. Call those candidates c1 and c2. If r_1 is 0, then c1 is the second choice, otherwise c2 is the second choice. Remove the second choice from the set. The choice that remains is the third choice of the voter. - Now we just have to show that this codes all the possible ballots: b r_0 r_1 first chosen second chosen third chosen 0 0 0 A B C 1 1 0 B A C 2 2 0 C A B 3 0 1 A C B 4 1 1 B C A 5 2 1 C B A and there we go. ---- Election-Methods mailing list - see http://electorama.com/em for list info ---- Election-Methods mailing list - see http://electorama.com/em for list info ---- Election-Methods mailing list - see http://electorama.com/em for list info
