[EM] election-methods Digest, Vol 37, Issue 4

Thu Jul 5 06:29:17 PDT 2007

 Forest W Simmons <fsimmons at pcc.edu> wrote
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> Designated Strategy Voting (DSV) methods relieve the voter of repeated
> returns to the polls for each iteration of the feedback loop, and also
> solve the anonimity requirement, but as has been noted, methods that
> are supposed to iterate unto an equilibrium may not converge.
>
> But linear order convergence can be speeded up by making use of a
> convergence accelerator.
>
> If? a, b, and c are successive numbers in a sequence of iterates where
> the convergence is linear (as per the usual case) then? the quantity
>
> Q=(a*c - b^2)/(a+c-2*b)
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It depends on how the system works.
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My understanding of DSV is that each vote is considered in sequence.
The voting algorithm has access to the vote totals for all the
candidates to date.? This doesn't lend itself to the above process.
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Actually, there is also a method for batch processing.? I would
prefer that as it treats all votes equally.? 
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 DSV seems to use plurality as its 'base' voting method.
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Maybe the system should allow strategies to share information about
their internal content, if they wish.
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I would suggest the process for each round would be that
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- each strategy is reset (no memory of previous rounds)
- each strategy is given access to the result of the last round (State)
- each strategy does not know which strategy other strategies are using
- each strategy indicates which communication protocols it supports
- each strategy can open a communication channel with any other strategy
- voting occurs using a number of sub-rounds (perhaps 20)
- no communication is allowed for the sub-rounds
- the sub rounds are
-- each strategy votes
-- results so far are made known
- the end result is the total of all the votes for all the voting sub-rounds
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This gives a clear (though complex) mapping of any state to another
state.? The communications would also have to be in rounds so that
they don't change based on voter ordering.? Also, a random number
generator would not be made available to the strategies as that
would also make the system undefined ... though that may 
eliminate some of the benefits of DSV.

The communication channels are essential.? Otherwise, the base election
method would have a major effect on the result.
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If it is plurality, then centre squeeze will still happen.? If there
are 2 candidates with 45% and 47% each, then strategies for the middle
candidate are going to switch.? If they can communicate, maybe they can
get the 45% group to side with them as a condorcet/lesser of 2 evils.
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The iteration in this system is multi-dimensional.? The
values in each dimension are the total number of votes given to
each candidate.
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e.g.
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X(n) = [a(n), b(n), c(n), ...]
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a(n) = candidate a's vote total after round n
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The simple iterator:
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X(n+1) = f(X(n))
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This can be improved by finding X such that the below is true.
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X - f(X) = 0
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The iterator is:
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X(n+1) = X(n) - f(X(n))/f'(X(n))

It would apply to each dimension individually.
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I am not sure what the convergence rules are for that formula though.

Also, since it is multi-dimensional and possibly discontinuous, there may
not be a solution. ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????  
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> In other words, if you were to run the sequence backwards, so that you
> are iterating the inverse of f instead of f itself, then the sequence
> would converge, since if the graph of f has a slope with abs value
> greater than one, the graph of the iinverse of f will have a slope with
> abs value less than one. [Their slopes are reciprocals at the
> equilibrium point.]
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I am not sure an unstable equilibrum point is actually a good thing for a
voting method.? It seems to me that it would be making the result worse
on each step instead of better from the perspective of the voters.
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The strategy in the forward direction is designed to improve the current
situation, if you reverse it, then each voter's declared strategy is
trying to make the situation worse for the voters.

I would probably include a rule that once the point has been found, it
is tested to confirm convergence.? (When in the X(n+1) = f(X(n)) 'mode').

Raphfrk
--------------------
Interesting site
"what if anyone could modify the laws"

www.wikocracy.com

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