Condorcet(x( ))
Steve Eppley
seppley at alumni.caltech.edu
Sun May 19 01:43:47 PDT 1996
Bruce Anderson wrote:
[snip]
>Taken all together, this gives 27 different possibilities.
>Using the numbering just above and the ordering:
>(ties-for-first;ties-in-the-middle;ties-for-last),
>these 27 possibilities are:
>(1;1;1), (1;1;1/2), (1;1;0),
>(1;1/2;1), (1;1/2;1/2), (1;1/2;0),
>(1;0;1), (1;0;1/2), (1;0;0),
>(1/2;1;1), (1/2;1;1/2), (1/2;1;0),
>(1/2;1/2;1), (1/2;1/2;1/2), (1/2;1/2;0),
>(1/2;0;1), (1/2;0;1/2), (1/2;0;0),
>(0;1;1), (0;1;1/2), (0;1;0),
>(0;1/2;1), (0;1/2;1/2), (0;1/2;0),
>(0;0;1), (0;0;1/2), (0;0;0).
[snip]
>At first, I thought that most of these 27 possibilities were
>obviously implausible. Then I realized that one of the ones I
>thought was implausible, (1/2;1/2;1), is used all the time in
>sports, and probably elsewhere too. So now I just think it's
>better not to jump to conclusions.
I think that the (xF;xM;xL) for which any of the following are true
are obviously implausible:
xF > xM
xF > xL
xM > xL
This eliminates 18 of the 27, if I eyeballed it right, but doesn't
eliminate (1/2;1/2;1).
But what about my other question--making x() a smoothly growing
function?
number_of_candidates_ranked_better
Such as x1() = K * ----------------------------------
total_number_of_candidates
rank_position - 1
or perhaps x2() = K * -------------------------------------
number_of_distinct_rank_positions - 1
If an election is between candidates A,B,C,D,E,F,G,H,I
and the voter's ballot is {A=B > C=D=E > F=G}
then the following terms would plug into the two formulae for x():
total number of candidates = 9
number of distinct rank positions = 4 (includes the unranked position)
A=B C=D=E F=G H=I
number of candidates ranked better 0 2 4 6
rank position 1 2 3 4
x1() 0 2K/9 5K/9 7K/9
x2() 0 K/3 2K/3 K
So, I wonder, would x1(), x2(), or something similar do what the
voters want better than x=0? Would x<>0 open the door to unwanted
properties like the LOE dilemma or voter strategies? Would the
results be "improved" enough to justify the added complexity?
--Steve
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