FPTP family theory, REDLOG shadowing

[David Catchpole]

> I wrote a computer subroutine to cast (P1) B-wins shadows.  It might have a
>  bug, since I found B-must-win region/shadow that intersected with a
>  B-must-lose shadow/region. Here is the code to cast shadows of B-wins
>  regions using a simplex (P1) light source:

It may be you've found a paradox brought on by (p1). Can you give us a
proper description of where the shadows intersect?

> 
> ------------------------------
> kknoa := (0<ab)and(0<ac)and(0<ad)and(0<b)and(0<c)and(0<d);
> 
> procedure bwsh(bw2); begin  % cast shadow of B-wins
>   scalar f,kon;
>   clear r,u,v,w,x,a,b,c,d,ab,ac,ad,tab,tac,tad,tb,tc,td;
>   bw2 := sub({a=ab+ac+ad},bw2);  bwshf := {};
>   kon:=(0<=ab+r-w-x)and(0<=b-r-u-v)and(0<=r)and(0<=u)and(0<=v)and(0<=w)and(0<=x);
>   f := rldnf(kon and sub({tab=ab,tac=ac,tad=ad,tb=b,tc=c,td=d},
>     sub({ab=tab+r-w-x,ac=tac+w,ad=tad+x,b=tb-r-u-v,c=tc+u,d=td+v}, bw2)));
>   bwshres := rlsimpl rlqe ex({r,u,v,w,x}, f);
>   write "Finished, the answer is in bwshres. Checking...";
>   if 'true = rlqe rlex (kknoa and bw2 and not bwshres) then begin
>     write "BWSH FAILED CHECK", rlqe rlex (kknoa and bw2 and not bwshres);
>     bwshf := gs3 (kknoa and bw2 and not bwshres);
>   end else
>     return rlsimpl sub({ab=a-ac-ad}, kknoa and bwshres);
> end;
> ------------------------------
> 
> The papers are only these 6: AB, AC, AD, B, C, D. (1 winner) 4 candidates.
> 
> Notes: sub({x=y},z) replaces all x variables with y.
>   rlex creates ("There Exists") for all free variables.
>   rlqe rlex  returns true if the region is not empty.
>   rlsimpl is REDLOG's fast simple simplifier.
>   rldnf expands the equation.
> 
> Casting shadows using simplex light sources is obviously very easy in
>  REDLOG.
> 
> (REDUCE is used in mathematics departments apparently.
> REDLOG is LISP code package for REDUCE. REDUCE runs in LISP.)
> 
> 
> -------------------------------------------------------------------------
> 
> A rule of Mr Catchpole is examined and rejected for requiring that a
>  method (where the papers are constrained to be just 3) is FPTP...
> 
> 
> At 15:39 08.12.99 , David Catchpole wrote:
> >Can I call 'em the Metameucil systems, can I? Can I?
> >
> ...
> >
> >The addition (removal) of a candidate does not, for any other candidate, 
> >increase (decrease) the probability of that other candidate winning.
> >
> 
> That is clear enough. It is not something people will agree is desirable
>  since it does not match what happens in elections. In reality with
>  actual preferential voting where preferences lists are not full, and
>  people don't mind to what happens to all the candidates for which no
>  preference was expressed. They are only interested in the candidates that
>  they vote for.
> Perhaps election methods could allow them to cast votes having a power of
>  10 votes, and allow them to cast upto 10 positive votes or upto 10 papers
>  with negative votes to help some candidates lose.
> 
> Add a constraint. Let the papers be just these, and find the 1 winner solution:
> 
>  AB a
>  B  b
>  C  c
> 
> A wins = (b<a) and (c<a)
> B wins = (A loses) and y(a,b,c)
> C wins = (A loses) and not y(a,b,c)
> 
> Any method with those three papers can be represented as coloured regions
>  inside a triangle, say with AB at the top, B at the left, and C on the
>  right.
> 
> The region for B on the lower left is coloured red, and C's region is coloured
>  green.
> 
> Where A wins is the same as in FPTP (say) (otherwise the method makes
>  outcomes for a candidate to be affected by subsequent preferences.
>  It can be asserted that the method must be not worse than STV).
> 
> All that is needed to define a method then, is the curve or piecewise
>  line segment between the centre point, and the midpoint in the middle
>  of the bottom edge.
> 
> Consider a point on the bottom edge of the triangle, say this
> 
>  a=0, b=0.501, c=0.499
> 
> What happens when papers for AB are added?. If proportionality is the aim
>  then the winner would flip from B to C. That certainly happens in
>  1 winner STV and IFPP, so proportionality is some achievable. STV goes
>  bad near the centre (B's region is too large) and Mr Catchpole's rule
>  is found wrong very close to the bottom edge, so ignoring IFPP and using
>  STV is OK for the argment that follows.
> 
> 
> This was written (repeated) [might have seen this rule before]:
> 
> Rejected 'Metameucil' or 'Regularity'(?) rule:
> -------------------------------------------------------------------------
> >The addition (removal) of a candidate does not, for any other candidate, 
> >increase (decrease) the probability of that other candidate winning.
> -------------------------------------------------------------------------
> 
> Probability probably means a blurring or fuzzifying of the B-C boundary.
> 
> If so, the boundary between B (coloured red) and C (green, on the lower
>  right), would be a blurry strip coloured red-pink-yellow-yellow_green.
> 
>  If papers for AB are added then the point that is the midpoint of the
>  bottom B--C edge of the triangle, is dragged upwards
> 
>  Hence, the divide between the regions of B and C is a line. There does
>  not seem to be any involvement of probability at all.
> 
> A possibility is that the author of the rule had never tested a method
>  against it ever.
> 
> FPTP makes it past the rule. Perhaps David could tell us if FPTP is the
>  only method that is passed by the rule, for all numbers of candidates
>  (or maybe the method no longer needs to be considered(?)).
> 
> 
> >Now, say you know how the system works for a two-candidate contest- for
> >simplicity, assuming no dependence on the number of voters- that is, you
> >know the function relating the ratio of probabilities of those two
> >candidates winning to the ratio of voters ranking one over the other to...
> >oh, blah, you get the idea.
> 
> 
> >Now, let's make it a three candidate election. It's easy enough to see...
> ...
> 
> The rule seems to be just the sort of rule that would propogate a
>  requirement that the method be 'similar to FPTP', throughout all election
>  problems involving a larger numbers of papers or candidates.
> 
> FPTP =? "The family of "regular" probabilistic (stochastic) electoral systems"
> 
> -------------------------------------
> 
> Note to Mr Davison: in replying to my message, you quoted the message I had
>  replied to:
> 
> At 00:12 02.12.99 , Donald E Davison wrote:
> >Greetings EM list,
> >
> >     Craig Carey asked about someone from the Freedom Party to comment on MMP.
> >     The following letter is about MMP and by Bill Frampton, Vice President
> >of the Freedom Party.
> 
> Also: New Zealand got a better electoral system (MMP) and wanted one. While
>  the government did change in some elections in the past, and at other times
>  it did not, FPTP was insulating politicians from public political opinion.
>  The public had an opinion that it didn't matter which party it was. 
>  Proportionality has improved that. The choice of MMP allowed a near minimal
>  change (with the electorates did drop in number from 90-something to 60).
>  That's one viewpoint.
> 
> (Errata: I said that the NZ National party was losing for no good reason.
> There must have been a good reason given how much they lost.)
> What was a little interesting was how the "Green" party so closely lost in
>  Coromandel (near Auckland city on the peninsular on the east): the National
>  party tried to win there (and did) and the leader of the Green party might
>  have won that seat (and hence seats in parliament too [which they never
>  got since under the 5% threshhold]), but the Labour party never withdraw
>  their candidate from the Coromandel seat. They withdrew one in another seat
>  so they must have intended to risk havign the Greens lose. The labout party
>  got a really quite small number of votes in Coromandel and prompted the
>  Green party to lose. The Green party didn't exist as a separate party last
>  election (1996). Labour can deal with the right using its coalition majority.
> 
> 
> 
> 
> Mr G. A. Craig Carey, research at ijs.co.nz
> Auckland, New Zealand.
> Snooz Metasearch: <http://www.ijs.co.nz/info/snooz.htm>
> 
> 
> 

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Nothing is foolproof given a talented fool.