Hi,
The message defining Dissimilar(g) (posted May 5 2000) included
a comment that the ambiguity in step 2.1 (about which similar
set is picked) probably means that the definition of similarity
needs patching for Dissimilar to satisfy complete independence
from similar alternatives (ISA).
The message also included an example showing the operation of
Dissimilar. It should be noted that that example exposes the
ambiguity and the failure to satisfy ISA, since {y1,y2} is not
the only similar set in step 2.1 which could be picked if the
definition of similarity isn't patched. For easier reference I
repeat the voters' rankings:
100 voters rank the alternatives as follows:
1: y1 > x > y2
24: x > y1 > y2
25: x > y2 > y1
25: y1 > y2 > x
24: y2 > y1 > x
1: y2 > x > y1
Since all three pairings are ties, {x,y1} and {x,y2} could just
as well be chosen as similar sets in step 2.1, and the lottery
would indeed be affected by this choice. (And a poor choice
could be worse than Random Dictator, if we are considering
robustness.)
For the patch to provide robustness, {y1,y2} should be a similar
set but {x,y1} and {x,y2} should not, since y1 and y2 are nearly
clones. (Except for 2 voters, y1 and y2 would be clones.) If
the definition is patched so the only similar set is the
trivially similar set {x,y1,y2}, then the lottery will be
(1/3,1/3,1/3). That's not an exhibition of robustness, since
the lottery in the nearly identical 25/25/25/25 scenario is
(1/2,1/4,1/4).
Presumably a good patch would result in the choosing of the
"most clone-like" similar set in step 2.1. If the naked eye can
detect that y1 and y2 are nearly clones, perhaps an algorithm
can too. If anyone wants to take a look at this, I offer the
reminder that it's not Dissimilar which we want to patch, it's
the definition of similar set (or perhaps the definition of
independence from similar alternatives) which was posted on May
4 2000 (in the message "Re: [EM] Question about complete clone
independence"). Otherwise the claim that Dissimilar is
completely independent from similar alternatives is lost.
---Steve (Steve Eppley [EMAIL PROTECTED])
