[EM] Son of Condorcet vs Approval

[Tony Simmons]

>> 111 ABC
>> 110 DE
>> 101 F
>> 100 GHIJ
>> 011 KLMNOP
>> 010 QRS
>> 001 TUVW
>> 000 XYZ

>> In the second representation, the score would be the place
>> of the most significant digit in which the two binary
>> labels differ.

Oops, this means that there's a whopping difference between
100 and 011, merely because of an artifact of the
representation.  That is, if you dropped ABC from the
election, 011 moves up to 100 and 100 moves up to 101, so the
score becomes 1 instead of 3.

How about just making the score the difference between the
two values?  Or if you want to give the score a logarithmic
flavor, as your second proposal does, how about using the
base 2 log of the difference?  This gives roughly the same
distribution of scores, but without the artifact.