The following is an edited version of the triangle that showed up in my derivation of the ideal/IFPP 2 winner 3 candidate method. That was at the [EMAIL PROTECTED] list and also at the: single-transferable-vote mailing list, in a message dated "2002\10\21 00:45 +1300 Monday". ... : : � � � � � � � � � � � � � � � � a:(AB) : � � � � � � � � � � � � � � � � � /\ : � � � � � � � � � � � � � � � � �/� \ : � � � � � � � � � � � � � � � � / � �\ : � � � � � � � � � � � � � � � �/� � � \ : � � � � � � � � � � � � � � � / � � � �\ : � � � � � � � � � � � � � � �/� � � � � \ : � � � � � � � � � � � � � � / � � � � � �\ : � � � � � � � � � � � � � �/� � � � � � � \ : � � � � � � � � � � � � � / � � � � � � � �\ : � � � � � � � � � � � � �/� � � � � � � � � \ : � � � � � � � � � � � � / � � � � � � � � � �\ : � � � � � � � � � � � �/� � � � � � � � � � � \ : � � � � � � � � � � � / � � � � � � � � � � � �\ : � � � � � � � � � � �/� � � � � A wins� � � � � \ : � � � � � � � � � � / � � � � � � � � � � � � � �\� � (a+b = c) : � � � � � � � � � �/� � � � � � � � � � � � � � � \ �/ : � � � � � � � � � / � � � � � � � � � � � � � � � �S/ : � � � � � � � � �/� � � � � � � � � � � � � � � � �/\ : � � � � � � � � / � � � � � � � � � � � � � � � � / �\ : � � � � � � � �/� � � � � � � � � � � � � � � � �/� � \ : � � � � � � � / � � � � � � � � � � � � � � � � / � � �\ : � � � � � � �/� � � � � � � � � � � � � � � � �/� � � � \ : � � � � � � / � � � � � � � � � � M___________U � � � � �\ : � � � � � �/� � � � � � � � � � � �\� � � � � � � � � � � \ : � � � � � / � � � � � � � � � � � � \ � � � � � � � � � � �\ : � � � � �/� � � � � � � � � � � � � �\� � � � � � � � � � � \ : � � � � / � � � � � � � � � � � � � � \ � � � � � � � � � � �\ : � � � �/� � � � � � � � � � � � � � � �\� � � � � � � � � � � \ : � � � / � � � � B wins� � � � � � � � � T � � � �C wins � � � �\ : � � �/� � � � � � � � � � � � � � � � �/� � � � � � � � � � � � \ : � � / � � � � � � � � � � � � � � � � / � � � � � � � � � � � � �\ : � �/� � � � � � � � � � � � � � � � �/� � � � � � � � � � � � � � \ : � / � � � � � � � � � � � � � � � � / � � � � � � � � � � � � � � �\ : �/� � � � � � � � � � � � � � � � �/� � � � � � � � � � � � � � � � \ : +---------------------------------R----------------------------------+ : b:(BA) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �c:(C)
Option [1] : � � � � � � �/� � � � � � � � � � � � � � � � �/� � � � \ : � � � � � � / � � � � � � � � � � M � � W_____U � � � � �\ : � � � � � �/� � � � � � � � � � � � � �/� � � � � � � � � \ : � � � � � / � � � � � � � � � � � � � / � � � � � � � � � �\ : � � � � �/� � � � � � � � � � � � � �V� � � � � � � � � � � \ : � � � � / � � � � � � � � � � � � � � \ � � � � � � � � � � �\ : � � � �/� � � � � � � � � � � � � � � �\� � � � � � � � � � � \ : � � � / � � � � � � � � � � � � � � � � T � � � � � � � � � � �\ : � � �/� � � � � � � � � � � � � � � � �/� � � � � � � � � � � � \
Option [2] : � � � � � � �/� � � � � � � � � � � � � � � � �/� � � � \ : � � � � � � / � � � � � � � � � � M � � � � � U � � � � �\ : � � � � � �/� � � � � � � � � � � � � Y______/� � � � � � \ : � � � � � / � � � � � � � � � � � � � �\� � �Z� � � � � � �\ : � � � � �/� � � � � � � � � � � � � � � \ � � � � � � � � � \ : � � � � / � � � � � � � � � � � � � � � �\X � � � � � � � � �\ : � � � �/� � � � � � � � � � � � � � � � �/� � � � � � � � � � \ : � � � / � � � � � � � � � � � � � � � � T � � � � � � � � � � �\ : � � �/� � � � � � � � � � � � � � � � �/� � � � � � � � � � � � \
Option [3] : � � � � � � �/� � � � � � � � � � � � � � � � �/� � � � \ : � � � � � � / � � � � � � � � � � M___� � � � U � � � � �\ : � � � � � �/� � � � � � � � � � � �\� +---+__/� � � � � � \ : � � � � � / � � � � � � � � � � � � \_� � � �N� � � � � � �\ : � � � � �/� � � � � � � � � � � � � � \_� � � � � � � � � � \ : � � � � / � � � � � � � � � � � � � � � \_� � � � � � � � � �\ : � � � �/� � � � � � � � � � � � � � � � �/M � � � � � � � � � \ : � � � / � � � � � � � � � � � � � � � � T � � � � � � � � � � �\ : � � �/� � � � � � � � � � � � � � � � �/� � � � � � � � � � � � \
Option [4] : � � � � � � �/� � � � � � � � � � � � � � � � �/� � � � \ : � � � � � � / � � � � � � � � � � M� � � �____U � � � � �\ : � � � � � �/� � � � � � � � � � � � P____/� � � � � � � � \ : � � � � � / � � � � � � � � � � � � �\� � � � � � � � � � �\ : � � � � �/� � � � � � � � � � � � � � + � � � � � � � � � � \ : � � � � / � � � � � � � � � � � � � � | � � � � � � � � � � �\ : � � � �/� � � � � � � � � � � � � � � �\� � � � � � � � � � � \ : � � � / � � � � � � � � � � � � � � � � T � � � � � � � � � � �\ : � � �/� � � � � � � � � � � � � � � � �/� � � � � � � � � � � � \
What I called "option (a)" is not shown. What I called "option (b)" is now shown by the diagram of "option [4]".
Those show slices of a parameterized hybrid method.
Mr Schulze can surely see that the diagrams show how to transform 1 winner IFPP into a method compliant with the rule of Mr Dummett.
If there was a slackness checkbox in only the (AB) papers, then that would affect only the lower bulge along line segments R-T-M or just T-M.
For each flat like M-T, it is necessary to be able to identify the paper that owns that flat.
>Dear Craig,
>
>you wrote (11 March 2003):
>> Case S = 0 : MNW = Min{Ceil(1*0.65)-1; 2} = min{0; 2} = 0
>> Case S = 1 : MNW = Min{Ceil(2*0.65)-1; 2} = min{Ceil(0.3); 2} = 1
>> Case S >= 2: MNW = Min{Ceil(3*0.65)-1; 2} = min{Ceil(1.95); 2} = 2
>
>You meant:Could say: "It is now corrected, and the text is..."
>> Case S = 0 : MNW = Min{Ceil(1*0.65)-1; 2} = min{0; 2} = 0
>> Case S = 1 : MNW = Min{Ceil(2*0.65)-1; 2} = min{Ceil(0.3); 2} = 1
>> Case S = 2 : MNW = Min{Ceil(3*0.65)-1; 2} = min{Ceil(0.95); 2} = 13*0.65 = 1.95; Ceil(1.95)=2; 2-1 = 1; Min(1,2) = 1 So the error is shown.
>> Case S >= 3: MNW = Min{Ceil(4*0.65)-1; 2} = min{Ceil(1.6); 2} = 2
>I ask Mr Markus Schulze:
(1) which of the shown 4 ways of combining a Dummet method with IFPP, are failed by truncation resistance ?. (2) Do you reject that particular rule yourself?. A yes/no reply should be OK.
--
Option 3 seems rejectible, in that a narrow cone can arise.
I am not CC-ing this to the Politicians and Polytopes mailing list since this material is too trivial, poorly formed, and too easy for my PaP mailing list. It is some freebies for the EM List, but doubtless the lack of replies would show that it is too tough a topic for the Election Methods List. Presumably their "space" theorist, Mr Forest Simmons can't comment since I got the physical dimensions of the space correct (it is the weight of papers, normalised or not).
G. A. Craig Carey
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