[EM] determinism (was New Condorcet/RP variant)
Forest W Simmons
fsimmons at pcc.edu
Thu Nov 11 15:11:18 PST 2004
Jobst Heitzig wrote:
>Dear Paul!
>
>you asked:
>> Is clone independence strictly more important than determinism?
>
>I would rather say, a sufficient *non-determinism* is much more
>important than clone independence since all deterministic majoritarian
>methods suffer from serious strategy problems whenever there is no
>sincere Condorcet Winner (in such a situation there is always a majority
>which has an incentive to change the winner by voting strategically). As
>I have pointed out a number of times, only methods which resolve cycles
>with a sufficient amount of randomness are more stable against such
>strategic threats!
The tricky part is finding the right probabilities for the candidates.
If you give all of the candidates equal probability, then it is to the
advantage of non-CW supporters to distort their votes in a way that
will artificially enlarge the Smith set (eg. from one, to more than one
member) so that their less popular candidate will be as likely to win
as any other.
Somehow more popular candidates need to have greater probability, but
exactly how to measure popularity, and how to best assign the
probabilities is an unsolved problem.
The simplest way to (at least nearly) pull this off seems to be Rob
LeGrand's ballot-by-ballot approval method applied to a set of ordinal
ballots that has been shuffled into a random or pseudo-random order.
[Rob sets the approval cutoff (on the current ballot) next to the
current approval champ on the side of the current approval runner up.]
In Rob's simulations this method has always picked the CW (in cases of
its existence) even though the method doesn't strictly satisfy the
Condorcet Criterion, since it is possible that highly "non-random"
sequences of ballots can be chosen at random, just as it is possible,
though unlikely, to generate the sequence
HHHHHHHHHTTTTTTTTT
of H's and T's randomly.
Forest
P.S. Here's an example:
40 ABC
30 BCA
30 CBA
If these are run through ballot-by-ballot in the order that they appear
in the above list (all 40 of the ABC ballots followed by all 30 of the
BCA ballots followed by the CBA ballots), then clone C will end up with
the highest total approval (60), and B (the CW) will barely beat out A,
the Condorcet Loser, by half a point (40.5 to 40).
[These totals could vary by plus or minus a point depending on how you
initialize the approval and where you put the cutoff when all three
candidates have the same current approval.]
If you randomize the ballot sequence, B will win with a probability of
more than 99.99999%.
This should be good enough for practical purposes, but it will take
extremely enlightened voters to understand the benefits of
randomization or pseudo-randomization.
Pseudo-randomization is best because of its reproducibility. The fact
that pseudo-random is really deterministic (like the result of
successive perfect shuffles) might make this more palatable to some
voters. But most of them will rightly consider the number of shuffle
steps as an arbitrary feature of the method.
If some information theorist could show us how to come up with an
optimal number of shuffles (of a certain type) relative to some
information theoretic measure of randomness (entropy, say) we could
claim that the number of shuffles was not arbitrary.
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