Forwarded
>In this message I provide a definition of One Man One Vote that I >wrote down earlier this year. > > > >From:� Richard Moore <[EMAIL PROTECTED]> > >Date:� Tue Jul 30, 2002� 2:20 pm > >Subject: �Re: What are we all about?, etc. >http://groups.yahoo.com/group/election-methods-list/message/9924 > > >It's simple to do this for IRV: If there are N candidates, and M > >candidates are eliminated before there is a pairwise winner against > >the remaining candidates, then it is possible that that winner > >is a pairwise loser in M contests. This looks very bad for IRV if M = > >N - 2 and N is large. IRV needs to test the winner against M > >additional candidates. > > > >The argument is based on an error which is that pairwise losing is somehow >able to provide information on the right winner. Whatever the argument is, >is rejected for using the idea of considering pairwise winners. > >My Ppos and Pneg equations take account of votes. Pairwise comparing loses >information by producing Boolean values that then lead to tiny little >"paradoxes". Some of the Condorcet fixes back out of the mistake. If your >argument is valid then repaired Condorcet methods would be rejected too >for paying too close an attention to "voters". [sic]. All we need then is >to assume that the possibly false idea saying that the Condorcet method is >like repaired Condorcet methods, and the idea that pairwise comparing is >like Condorcet, and then I have an argument saying that the assumption >that pairwise comparing is due respect, leads to a conclusion that it is >not. > >The comment "looks very bad" is simply rejected. No method is found to be >wrong by using a wrong test. It Mr Moore wants to cut the wrong parts of >a pairwise comparing test out of it then that could be attempted and then >the details posted up. > >------------------------------------------------------------ > >In the following equations > >W is the method >L is a truncated preference list >"." is set intersection or Boolean "AND", and in this equation >"#" returns the number of elements in a set or list. >"\" means the complement of the set. The universe set is the set of all >candidates. (Remember that candidates can win even when no paper names the >candidate). >a set "."-ed with a list, returns a set. > >H is a base election point. > >H+S is that H election that has added to it, the paper S that is having >its power tested. (Actually S is a combination of papers). > >H+T is the H election that has added to it, a weighted combination of FPTP >papers, where T is in that base of a simplex. What the "13Mar2002" formula >does is check that T can get an outcome as good as what S got. It is not >that obvious in this "P4" because the use of an inductive argument >involving the deletion of digits on both sides of a "<=" comparing 2 >binary numbers smaller than 1, has driven some of the meaning into other >formulae with different #L values. > >-------------------------------------------------------------------- >(P4 13Mar2002) = >not (Exists L, t, H, S) [ >� [ �(S perpendicular to #L FPTP axes for L) >� � � (All k)(0 <= t.S[k])(abs((Sum k)S[k]) <= abs(t)) >� � � (W(H+S) is defined) >� � � [ (0<t)({}=L.\W(H+S)) or (t<0)({}=L.W(H+S)) ] >� ] and >� � � (All T) [ >� � � � [ �(T is a sum of #L weighted FPTP axes for L) >� � � � � � (All k)(0 <= t.T[k])(t = (Sum k)T[k]) >� � � � ] implies [ >� � � � � � (W(H+T) is defined) >� � � � � � [ (0<t)({} <> L.\W(H+T)) or (t<0)({} <> L.W(H+T)) ]]]] >-------------------------------------------------------------------- > >(1) The P4 rule is adjusted so that it excuses a method for imposing a >need to vote for the wrong candidate. The aim is to find out if a paper >can have more power than 1 FPTP paper, without checking to see if the >paper has to name the candidate being advanced. [One Man One Vote >overlooks insincerity.] >To do that >� �"S perpendicular to #L FPTP axes for L" >is rewritten as >� �"S does not contain FPTP papers" >and >� �"T is a sum of #L weighted FPTP axes for L" >is rewritten as >� �"T is a sum of FPTP papers" >That changes the rule: let the new rule be called P5. > >(2) Also define a variable, p, called "power". "t" is the count of the >FPTP papers added, and "(Sum k)S[k]" is the count of the sample tested >paper. To find out what the power is, the weight of the T papers can be >divided by p to give a value to compare against the tested paper. > >(3) Assume that the method W has one winner. >Without effect, let the length of list L, equal 1. Let L = {g}: >� � � ({}=L.\W(H+S))� = �(g in W(H+S)) >� � � ({}=L.W(H+S)) � = -(g in W(H+S)) >� � � ({}<>L.\W(H+T)) = �(g in W(H+T)) >� � � ({}<>L.W(H+T))� = -(g in W(H+T)) > >-------------------------------------------------------------------- >P = not (Exists L, t, H, S) [ >� [ �(S does not contain FPTP papers) >� � � (All k)(0 <= t.S[k])(p.abs((Sum k)S[k]) <= abs(t)) >� � � (W(H+S) is defined) >� � � [ (0<t)(g in W(H+S)) or (t<0).-(g in W(H+S)) ] >� ] and >� � � (All T) [ >� � � � [ �(T is a sum of FPTP papers) >� � � � � � (All k)(0 <= t.T[k])(t = (Sum k)T[k]) >� � � � ] implies [ >� � � � � � (W(H+T) is defined) >� � � � � � [ (0<t)(g in W(H+T)) or (t<0).-(g in W(H+T)) ]]]] >-------------------------------------------------------------------- > >That can be split into 2 parts: P = Ppos & Pneg > >-------------------------------------------------------------------- >Ppos(p) = not (Exists g, t>0, H, S) [ >� [ �(S does not contain FPTP papers) >� � � (All k)(0 <= S[k])(p.(Sum k)S[k] <= t) >� � � (W(H+S) is defined) >� � � (g in W(H+S)) >� ] and >� � � (All T) [ >� � � � [ �(T is a sum of FPTP papers) >� � � � � � (All k)(0 <= t.T[k])(t = (Sum k)T[k]) >� � � � ] implies [ >� � � � � � (W(H+T) is defined) >� � � � � � (g in W(H+T)) ]]] >-------------------------------------------------------------------- >Pneg(p) = not (Exists L, t<0, H, S) [ >� [ �(S does not contain FPTP papers) >� � � (All k)(S[k] <= 0)(t <= p.(Sum k)S[k]) >� � � (W(H+S) is defined) >� � � not (g in W(H+S)) >� ] and >� � � (All T) [ >� � � � [ �(T is a sum of FPTP papers) >� � � � � � (All k)(T[k] <= 0)(t = (Sum k)T[k]) >� � � � ] implies [ >� � � � � � (W(H+T) is defined) >� � � � � � not (g in W(H+T)) ]]] >-------------------------------------------------------------------- > >Since Approval is expected to not fail any worse when checked with >negative numbers, it can be assumed that W is fully defined: > >-------------------------------------------------------------------- >Ppos(p) = not (Exists g, t>0, H, S) [ >� [ �(S does not contain FPTP papers) >� � � (All k)(0 <= S[k])(p.(Sum k)S[k] <= t) >� � � (g in W(H+S)) >� ] and >� � � (All T) [ >� � � � [ �(T is a sum of FPTP papers) >� � � � � � (All k)(0 <= t.T[k])(t = (Sum k)T[k]) >� � � � ] implies [ >� � � � � � (g in W(H+T)) ]]] >-------------------------------------------------------------------- >Pneg(p) = not (Exists L, t<0, H, S) [ >� [ �(S does not contain FPTP papers) >� � � (All k)(S[k] <= 0)(t <= p.(Sum k)S[k]) >� � � not (g in W(H+S)) >� ] and >� � � (All T) [ >� � � � [ �(T is a sum of FPTP papers) >� � � � � � (All k)(T[k] <= 0)(t = (Sum k)T[k]) >� � � � ] implies [ >� � � � � � not (g in W(H+T)) ]]] >-------------------------------------------------------------------- > >(Largest Power) = (Largest Power of a Paper in method W) = >� � �(The least positive p such that (All r>p)[ Ppos(r) . Pneg(r) ] > [(or "All r>=p")] > >('One Man One Vote' test pass mark) = ((Largest Power) <= 1). > >There is a definition of One Man One Vote that is a weaker test than >it, but maybe exactly right when the number of preferences on each >ballot paper equals two or less. > >If there is a problem with it, then do identify the exact parts of the >equation that are permitting any such concern to arise. > >The equations and Approval can be very easily checked with a QE solver, >(a Quantifier Eliminator system), e.g. REDLOG of Germany. > >Thus... One Man One Vote rejects Approval, and also One Man One Vote >is clarified up. Persons that like Approval but who ignore the >equations, likely could not be mathematicians inside of the topic of >preferential voting (or maybe some similar topic). > > >--------------------------------------------------------------------- > >The same message of Mr Moore > > >From:� Richard Moore <[EMAIL PROTECTED]> > >Date:� Tue Jul 30, 2002� 2:20 pm > >Subject: �Re: What are we all about?, etc. > >... > > > >However, Alex was merely developing an idea (in the collaborative > >environment of the list), not submitting it for publication in a > >mathematical journal, so most of Craig's pedantic comments in that > >post are uncalled for at this stage. > >I can't see how Alex's ideas threaten blacks in USA. But Appoval >could since it presumably disadvantages uneducated persons. > >Alex's ideas were unfixable and vague. >Something to be swept under a mat perhaps. > > > > >4. Craig's anti-approval diatribe (July 27) fails to point out any > >real problems. Examples presented on the EM list and elsewhere lack a > >The next step could be to say that he doesn't see a fault in my >equations that define power. > > >sufficient number of candidates for his taste. Yet the existence of > >small examples does not imply failure with large numbers of > >candidates. Will Craig point out how increasing the number of > >What a dud comment: once a method is failed by a Boolean-valued test then >the testing can stop since any further "AND"-ing with "False" won't alter >that fail mark. > > >candidates would make approval deteriorate? > > > >Is there anything else you want, Mr Moore, in addition to having equations >testing the reject Approval method, to have a nature that varies with >"time" ?. > >Of course increasing numbers of candidates make Approval deteriorate. > >I will check back every 10 years and see if the Approvalists have >cast comprehensible doubts on my power equation that is continued in >this message. You can test the method yourself, allowing 1 winner, >any number of candidates, and letting no paper have 3 or more >preferences. Some of the Approvalists would live a long time and may >want to investigate the method. Probably not many. > >-- > > >I would like Craig to present an argument, at least as coherent as the > >one above, for why Approval fails if the number of candidates is >large, >and to state what test it is failing. > > > >Approval is therefore a method that no government would want to use. >However the equation has not been used to test the method. There is >no doubt it will fail. It does not reject Rob Richie's IRV method >(maybe Alaska 's after a while) since it is not paying attention to >the appalling "power<0" problems. > >That "One Man One Vote" is not designed by seeing what passes IFPP >and what fails others methods. > >Readers can see how the Approvalist's undo the damage to their movement >that this new mathematical formula is providing. > > > >======================================================================= > >References > >[1] Use of the method on Borda: >http://groups.yahoo.com/group/politicians-and-polytopes/message/173 > >[2] An implementation of an older version of Ppos: > >-------------------------------------------------------------------------- > >167: procedure corrupt(mn); begin > >167: zp1 := rlqe all({ta,tb,tc}, > >167: � �( (0<=ta) and (0<=tb) and (0<=tc) and (t = ta+tb+tc) ) > >167: � �impl ( ( � � �isok(� � � Ha0+ta, Hab, Hac, > >167: � � � � � � � � � � � � � � Hb0+tb, Hbc, Hba, > >167: � � � � � � � � � � � � � � Hc0+tc, Hca, Hcb) ) and > >167: � � � (stri nott meth(mn,a, Ha0+ta, Hab, Hac, > >167: � � � � � � � � � � � � � � Hb0+tb, Hbc, Hba, > >167: � � � � � � � � � � � � � � Hc0+tc, Hca, Hcb) ) ) ); > >167: zp2 := zp1 and (� (0 < t) � and �(0<=Sab) and (0<=Sac) and > >167: � � � � � � � � � � � � � � � � �(0<=Sbc) and (0<=Sba) and > >167: � � � � � � � � � � � � � � � � �(0<=Sca) and (0<=Scb) and > >167: � � � ( � � � � �isok(� � � Ha0, Hab+Sab, Hac+Sac, > >167: � � � � � � � � � � � � � � Hb0, Hbc+Sbc, Hba+Sba, > >167: � � � � � � � � � � � � � � Hc0, Hca+Sca, Hcb+Scb) ) ); > >167: zp3 := gs2 stri zp2; > >167: zp4 := zp3 and (p * (Sab+Sac + Sbc+Sba + Sca+Scb) <= t) and > >167: � � � (stri � � �meth(mn,a, Ha0, Hab+Sab, Hac+Sac, > >167: � � � � � � � � � � � � � � Hb0, Hbc+Sbc, Hba+Sba, > >167: � � � � � � � � � � � � � � Hc0, Hca+Sca, Hcb+Scb) ); > >167: p5p := rlqe all({t, > >167: � � � � � � � � Sab, Sac, � � �Sbc, Sba,� � � Sca, Scb, > >167: � � � � � �Ha0, Hab, Hac, Hb0, Hbc, Hba, Hc0, Hca, Hcb }, nott stri zp4); > >167: > >167: if 'list = part(p5p,0) then begin � write "REDLOG: p5p = "; write p5p; > >167: end else begin > >167: � �p5p := gs2 p5p; > >167: � �write "CORRUPT: p5p = ", p5p; > >167: end; >[the other test I call the dual follows immediately after] > >From: http://groups.yahoo.com/group/politicians-and-polytopes/message/172 >Sample output showing Borda has a max power value of 2.0, which implies >that that method fails One Man One Vote since 2>1. > > >% CLASSICAL BORDA: > >x4 := 4/3; > >x2 := 2/3; > > > >> � � � � � � � � �2 > >>CORRUPT: p5p = 3*p �- 4*p >= 0 and p > 0 > >> > >> � � � � � � � � �2 > >>CORRUPT: p5m = 3*p �- 4*p >= 0 and p > 0 > >"nott" means 'not' >"stri" makes inequalities strict >"lax" makes inequalities lax >"rlqe all(Vars, Expr)" is "(All vars...)(Expr(vars)) >"rlqe ex(Vars, Expr)" is "(Exists vars...)(Expr(vars)) >"gs2" simplifies. >T*,H*,S*,t are symbols for Reals that do not contain a single Real >If "p5p" is false then the method > >-------------------------------------------------------------------------- If the formula is modified then the "p" power number can be negative and the power of the so called IRV (STV) method could be precisely checked. Here is a question: ----------------------------------------------------------------- Is the most negative power of any 3 candidate STV ballot paper in any given election, ever less than -1/2 ?. ----------------------------------------------------------------- >Craig Carey >Wed 31 July 2002 The headers (resent after being posted while I was not subscribed): >To: [EMAIL PROTECTED], > [EMAIL PROTECTED] >From: Craig Carey <[EMAIL PROTECTED]> >Date: Wed, 31 Jul 2002 18:52:41 +1200 >Subject: One Man One Vote in equation form; Approval is rejected >Reply-To: [EMAIL PROTECTED] ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
