[EM] election-methods Digest, Vol 37, Issue 5

Thu Jul 5 12:50:56 PDT 2007



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>Today's Topics:
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>   1. Re: election-methods Digest, Vol 37, Issue 4
>      (raphfrk at netscape.net)
>
>
>----------------------------------------------------------------------
>
>Message: 1
>Date: Thu, 05 Jul 2007 09:29:17 -0400
>From: raphfrk at netscape.net
>Subject: Re: [EM] election-methods Digest, Vol 37, Issue 4
>To: election-methods at electorama.com
>Message-ID: <8C98D1F9ED2B8FC-E24-8E00 at FWM-D30.sysops.aol.com>
>Content-Type: text/plain; charset="us-ascii"
>
>
> Forest W Simmons <fsimmons at pcc.edu> wrote
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>> 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)
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>It depends on how the system works.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>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.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>Actually, there is also a method for batch processing.? I would
>prefer that as it treats all votes equally.? 
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
> DSV seems to use plurality as its 'base' voting method.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>Maybe the system should allow strategies to share information about
>their internal content, if they wish.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>I would suggest the process for each round would be that
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>- 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
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>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.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>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.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>The iteration in this system is multi-dimensional.? The
>values in each dimension are the total number of votes given to
>each candidate.
>???????????????????????????????????????????????????????????  
>e.g.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>X(n) = [a(n), b(n), c(n), ...]
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>a(n) = candidate a's vote total after round n
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>The simple iterator:
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>X(n+1) = f(X(n))
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>This can be improved by finding X such that the below is true.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>X - f(X) = 0
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>The iterator is:
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>X(n+1) = X(n) - f(X(n))/f'(X(n))
>
>It would apply to each dimension individually.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>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. 
????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
??????????  
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>> 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.]
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>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.
>????????????????????????????????????????????????????????????????????????
????????????????????????????????????????????????????????????????????????
???????????? 
>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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>End of election-methods Digest, Vol 37, Issue 5
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