Condorcet(x( ))

[Steve Eppley]

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