[EM] Proportional preferential voting

[David Catchpole]

Oh! It solves the thing on a case by case basis (like finding all examples
of inversion and using the rule to exclude answers)!

On Fri, 17 Sep 1999, Craig Carey wrote:

> I don't appear to have all the equations needed to get a
>  complete solution in any 4 candidate problem of my theory.
> 
> At 01:55 99/09/17 , you wrote:
> >Dear Craig,
> >
> >could you -please- explain your method using the
> >following example with 100 voters and 4 candidates
> >running for 3 seats?
> >
> >30 voters vote A > B > C > D.
> >26 voters vote B > D > A > C.
> >24 voters vote C > B > A > D.
> >20 voters vote D > B > A > C.
> >
> >Markus Schulze
> 
> OK.. here's a rough outline of this method. I am sure it will
>  seem defective:
> 
> "... solve
>  the subproblems, get the rules listed, and when candidate i
>  definitely won or lost at some point, cast shadows in every
>  direction allowed by the rules throughout the simplex. At some
>  points, the number of winners would rise to equal the number
>  that had to win the election, and therefore, at each such
>  point in the simplex, the losers would all be known. Then the
>  shadows for all those losers could be cast.
>  The iteration can stabilise without a solution being found."
> 
> I could give a sort of answer to the problem from Mr Schulze,
>  if the following three 1 winner 4 candidate sub-problems 
>  were solved (or partly solved)
> 
> 
> VAD
> -20 D...
> -24 CBA.
> -26 BD..
> -30 A...
>          
> VBC
> -20 DB..
> -24 C...
> -26 B...
> -30 AB..
>          
> VBD
> -20 D...
> -24 CB..
> -26 B...
> -30 [missing by accident]
> 
> 
> 
> Picking a single winner with votes that are negative.
> 
> Notation: V is election system
>  
> Try this notation : aV is true iff A wins V
> VA is V with votes after the preference for candidate A discarded.
> 
> Then aV = aVA = aVA.-bVAB.-cVAC.-dVAD. That holds for STV & FPP.
>  It may hold for Condorcet too? (but not some of the modifications?).
> 
> The 3 above because of the solutions I found for VBCD, VACD,...
> 
> ("-" is Boolean 'not'). 
> ("CBA." = "C > B > A > D")
> 
> ------------------------------------
> Simplifying concave polytopes would become a problem
>  probably. REDLOG seems to be the package to use.
> 
>      http://www.fmi.uni-passau.de/~redlog/htmldoc/rl_toc.html
> 
> One of the authors of REDLOG wrote:
> 
> :Try "rlsimpl((a<b) and (a<c) and (2*a<b+c));",
> :but the REDLOG simplifier cannot simplify the formula.
> :
> 
> [Note: Some inequality term extracting and expression parsing code,
>  can add a convex polytope simplification feature.]
> 
> :However, with the quantifier elimination of REDLOG you can prove, that
> :a<b and a<c implies 2*a<b+c:
> :
> :	12: rlqe rlall(a<B and a<c impl 2*a<b+c);
> :	---- (all a b c) [DFS: depth 3, watching 3]
> :	[0e] [DEL:0/1]
> :	true
> :
> :This allows you to drop the atomic formula 2*a<b+c from the above
> :conjunction. 
> :
> 
> G A Craig Carey, 08:46 Fri 17 Sept 1999 NZT
> 
> 
> 
> 
> 
> 
> 
> Mr Craig Carey
> 
> E-mail: research at ijs.co.nz
> 
> Auckland, Nth Island, New Zealand
> Pages: Snooz Metasearch: http://www.ijs.co.nz/info/snooz.htm,
>  & Public Proxies, MEDLINE Search, Multithreaded Add-URL
> _____________________________________________________________
> 
>