[EM] Droop fails the Markus Schulze Rule

[Craig Carey]

At 06:50 20.10.99 , Bart Ingles wrote:
>
>Craig Carey wrote:
...
>> It is simple to replace it: For example, if the set of winners in
>>  an election are {B, E, G}, and if a preferential voting paper
>>  under consideration is (A B C D E F), then the election outcome
>>  satisfies the paper by this amount given by this number:
>> 
>> (A wins)+(B wins)/2+(C wins)/4+(D wins)/8+(E wins)/16+(F wins)/32
>> = 0 + 1/2 + 0/4 + 0/8 + 1/16 + 0/32
..
   = 0/2 + 1/4 + 0/8 + 0/16 + 1/32 + 0/64 
   = 9/32.
Call this the "satisfaction" value that measures the satisfaction
 a paper ('voter') has with an election outcome.

>I hope this was intended as only one of many possible ways of measuring
>satisfaction with an outcome.  Why powers of two, and not a linear
>function?

Note that the numbers are not to be compared between papers.

 Since voters don't balance their satisfaction with the election
 outcome against the satisfaction of other voters (i.e. other papers),
 then similarly a paper's satisfaction number need not be compared
 with another paper's satisfaction number.

Probably I was unconvincing when I said that (Q1) [18-19 October, 'no
 paper may improve its base-2 satisfaction by strategically voting'] was
 too strong to impose on a method. Some people would say, if there is
 no doubt at all a voter would vote strategically, then a method ought do
 that for the voter. Then the question arises: does that damage
 proportionaily or does it lead to a contradiction in which two rules
 can't be imposed (e.g. (P1) and (Q1)). This example I gave doesn't on
 the face of it, rule out both being applied:

    5  A .
    4  B C
    2  C A             : STV, IFPP, FPTP: {A} = Winner

 In likelihood, the voter(s) that voted (B C) would probably actually
 strategically vote, and make C win.
 [My IFPP doesn't satisfy (Q1), and I am wondering if getting rid of
 strategic voting by removing preceding losers would result in FPTP.]

The numbers make each preference 'infinitely' preferable to the next,
 since/while the numbers are only compared.

Example: Some method will find six winners:

Case 1: paper=(ABCDEFGH), winner={C,D,E,F,G,H}, Satisfaction=63/256
Case 2: paper=(ABCDEFGH), winner={B,I,J,K,L,M}, Satisfaction=1/4

So the 2nd alternative is the alternative that the satisfaction value of
 the paper would prefer.

G. A. Craig Carey
Auckland