### A right to have a vote and Multiwinner Monotonicity

It seems plausible to equate the idea of equal suffrage with the idea that the power of all ballot papers be inside of the range 0 to 1. A power of 1 is when it is only just possible to get a positive sum of FPTP papers of the same weight to obtain the same winner or the same satisfaction number (e.g. the satisfaction of (ABC) with (Winners={A,C}) is 0.101 (base 2). There is a multiwinner version of monotonicity that moves 2 ordered groups of preferences through each other without having a preference pass over the end of the paper. (See also [[Suffrage]], [[International Covenant on Civil and Political Rights]], Article 25(b): http://www.unhchr.ch/html/menu3/b/a_ccpr.htm).

Single Transferable Vote is failed by monotonicity

Also the 2 winner 4 candidate (Meek) STV ([[Single Transferable Vote]])) method fails monotonicity with 16.6% of the papers being involved in the defect, as this example shows:

ACB:16668,BA:16667,CB:33331,D:33334, Winners={A,D}

ACB:33333,BA:16667,CB:16666,D:33334, Winners={B,D}

### Monotonicity, the 1/3 quota (of IFPP method)

Abstract:

The page derives a 3 candidate 1 winner method. Firstly P2 or truncation resistance eliminates papers having exactly three preferences. Then truncation resistance and the rule on getting the number of winners right, allows the method to be factorised into two 5 paper examples, e.g. with one having these papers: (A),(AB),(AC),(B),(C). Then P2 can eliminate 2 more papers leaving (AB),(B),(C). Then another dimension can be removed by dividing the counts by the total weight. The 2 winner 3 candidate solution can be found by a similar argument (appeared at the PaP mailing list in late 2002) or alternatively by negating the votes and swapping winners with losers.

Introduction: A method that results in preferential voting methods that have desirable principles that are not rejected by a very critcal skeptical public audience the inevitable idea that the winners can be found by extending ties between candidates as if the ties were just like shock waves or shadows. The idealism is as simple to grasp as the algebraic consequences of it are difficult to make progress with.

Most of the argument is in the 3 pictures so the . The pictures themselves are not so good in that a dimension was reduced using a x/y division, and to derive a 4 candidate solution, it would seem better with the extra dimension than with the division. A problem is that an “its obvious” argument is used done at the same time. Anyway the results are OK. [The division projects 3-space onto a 1=a+b+c triangle.]

Here a method for selecting a winner in a 3 candidate election is derived. electing A method for electing winners where the ballot papers specify a rank for 1 or more candidates, i.e. the numbers 1, 2, 3,…, can be derived from principles, at least where there are 3 candidates.

### The Rule of Monotonicity in Preferential Voting

(where the papers are like the ones of the Single Transferable Vote and the Alternative Vote)

Monotonicity: the rule requiring that support rises shall not harm

In [[voting system]]s, the a voting system can be said to be ”’monotonic”’ if it satisfies the following rule:

If candidate X loses, and the ballots are changed only by shifting preferences for X further away from the 1st (includes: dropping off the end of the paper), and without changing the relative position of all other preferences, then candidate X must still lose. The weight of papers does not change when shifting of X preferences occurs; but an exception is for shifting a preference naming X out of an “(X)” paper that names only candidate X. In that case the weight can reduce (and more votes can be removed than the “(X)” paper started with). The definition copes with papers with negative weights (counts).

There may be no authoritative definition. The definition above has some ideas distinctly stated: set of winners, candidate, ballot paper. and preferences on a ballot paper.

### Criticism against the Alternative Vote

A possible criticism against the Alternative Vote could be along the lines of saying that it transfers too much support from deeper preferences onto some minor candidate who is not actually that minor at all, but who has enough 1st preference support to be in a close contest with the biggest 2-3 candidates. Referring to the diagram, that is an extrapolation of how AV is getting the winners wrong around the centre of the triangle. Having AV/IRV be the cure to ‘vote splitting’ problems, would mean that the concern is about the bottom of the triangle. Fixing the vote splitting of FPTP is a trojan horse for introducing new unwanted defects resulting in the wrong winners being selected by the so called IRV method of Time Magazine and others.

After a long day at work, catch up with the latest news, and thereafter play a round of slot games at Slotsexpert.

No principle seems to be more incompatible with the AV/IRV than the principle of equal suffrage. [There seems to be no point in using that term when part of it would suffice. The part that fails the AV method is said to be generalized monotonicity (support gains don’t harm even when it is a group of candidates that are harmed).] Equal suffrage provides a just basis for rich, right of centre, voters or candidates, to complain in a court when the winner is wrong. Equal suffrage is a very slack rule. The author found that under one view, it allowed one normal vector of a region where a candidate won, to slip around between -45 degrees and +45 degrees. So slack is the human right in its requirements on governments that it is easily possible that the “Unfairness to the 4th” method near the top of the webpage is passed by a test of equal suffrage. The Alternative Vote is some unusually uncommon find — a stupid voting method that chooses a time and place to be meaninglessly and abjectly (perhaps cruelly) unfair notwithstanding the wide laxity of the mathematical principles describing fairness. Notwithstanding the lobbyists advancing IRV can’t master them apparently, the principles are not that difficult. E.g. the public does not want fairness to left-wing candidates on sunny days only. Instead the theorist or the council nudges the idea over to the ideal of providng fairness on all days, but pro-IRV/AV refomers are out of contact despite even today producing propaganda for whomever.

### Compliance of some other methods

The Borda method is monotonic and the Alternative Vote method is not. The [[Approval voting]] method is not tested by monotonicity, though it could be tested and presumably passed by a monotonicity rule that handled a ballot paper having only 1 preference with that preference being a set of candidates (rather than a single candidate).

External link

Axioms of Preferential voting and symbolic algebra of [[polytopes]]: http://groups.yahoo.com/group/politicians-and-polytopes/

Comments welcome (see edit history for contact addresses).

### Centre of Research into Wrong Mayors in US cities

Worst yet found out about the USA (Maryland state) IRV (Instant Runoff Vote) method. A new finding is that with the Alternative Vote, if candidate A is winning with only about 1/2 of the votes in, then on adding the other half, with ALL but 2 papers naming candidate A using only a 1st preference, then candidate A switches from winning into losing even though nearly unanimously supported. The 11 candidate election showin that (i.e. amplified cascading): http://groups.yahoo.com/group/single-transferable-vote/message/245 [“Losing if a 49.9%…\ middots added; Cumulative Vote”, 19-Jul-2003]

Good methods don’t do that of course, and methods designed by man (excluding CVD staff here) should do better with the 1st preference hopefully. Some comments for the above URL are in the message before the previous.

I am unsure about this next. Montonicity is a rules that is maybe public or popular but it simply produces unnecessary complexity in arguments carefully using it. Better is to have monotonicity make its statement with trailing preferences scrambled in any way.

### A derivation of a 3 candidate 1 winner

preferential voting method follows. Axioms that allow the conclusion to be made are stated.

[1] An axiom shall be that the method returns the right number of winners.

[2] An axiom of the method is P2. P2 is defined so that the set of winners is not altered when the following papers are added:

P2(0): {t:(A), t:(B), t:(C), t:(D),…};

P2(1): {-(N-1)t:(A), t:(AB), t:(AC), t:(AD),…} [a different ‘t’];

P2(2): {-(N-2)t:(AB), t:(ABC), t:(ABD),…};

P2(3): {-(N-3)t:(ABC), t:(ABCD),…}; and so on.

[3] An axiom is that there is an ‘aim of proportionality’. This axiom removes degrees of freedom remaining after the other axioms have imposed their strict constraints. Proportionality (here) is the idea that for each candidate, there is a internal total that has added to it the weight (or quantity) of every paper naming that candidate. If nw winners needed to be selected, then this proportionality axiom would make the nw winners be the nw candidates that have the most positive totals. This proportionality aim would ignore the order of the preferences. However the P1 axiom (introduced below) is strict and applied first and it makes the order of the preferences be not ignored.

[4] The proportionality aim is defined to select the First Past the Post winner when 1 winner is to be found and the papers have no 2nd preferences. So the winner of the 1 winner election with the 3 papers {a:(A),b:(B),c:(C)} is candidate A if and only if (b<a)&(c<a). [The notation “x:(AB+{C+)” means that there are x (A=1st,B=2nd) papers and the intersection of the set of winners with {A,B,C} is {B,C}. When there is no “+” then the winners might not be being noted.]

[5] The method is truncation resistant. Truncation resistance is defined to require that for each candidate, e.g. B, alterations of preferences after any preferences for candidate B, do not affect whether candidate B wins or loses. (“After” means preferences on the “right” side distant from the 1st preference.)

[6] An axiom of the method is that it satisfies P1. A definition of P1 is this: for each candidate, e.g. B, alterations of preferences after any preferences for candidate B, and optionally including preference B, never change candidate B from a loser into a winner. For papers where candidate B has the 1st preference, some (positive) part of its weight may be discarded. The definition is very similar to the definition of truncation resistance except that the preference itself may be altered.

[7] Using [3] and [4], in the 1 winner election, with papers {a:(AB),b:(B),c:(C)}, candidate A wins if and only if (b<a)&(c<a). The region where candidate A wins, is shown in Figure 1.

[8] Property P1 is about the same as both truncation resistance and monotonicity. Monotonicity is (here) defined to allow preferences to be shifted to the right (i.e. away from the first) with that never resulting in the candidate changing from a loser into a winner. Truncation resistance allows the preferences on the right to be reordered and added to and removed from, while the preferences of the candidate under consideration are being shifted to the right.

[9] Figure 2 illustrates axiom P1 casting a B-must-lose shadow, of the A wins region. Axiom P1 requires that candidate B must not change from a loser into a winner when (B) papers are altered in any way at all. This alteration includes making the weight more negative but no votes may be gained.

Note: P1 can’t be replaced with monotonicity and truncation resistance for the (Figure 2) alteration (B)-(C{B+), since (BC) papers have been excluded. If it were not for that, this argument could have been made:

(a) Must have the same win-lose state for B: (B)–>(BC); by truncation resistance.

(b) B won’t change from a loser into a winner for the alteration (BC)–>(C), by monotoncity.

[10] Using the axiom that the number of winners is correct, candidate C wins in the triangle (a<c)&(1/3<a) since both A and B lose there. Temporarily assume that the weights are normalised so that 1=(a+b+c

### Dual of shadowing stays-losing polytope

is a normal vector constraining polytope

Preferential voting rules that define rights for single voters, e.g. monotonicity, can be merged together with other rules.

A rule of the form “if X loses here then X loses there” defines a raw shadow (or a light source). Any number of those rules can be combined. To do that, the dual of the product of the duals ([[Dual polyhedron]]) of each rule is calculated. So rules are able to lose their identity: they can be merged together and pulled apart in different ways.

A voting dual transform: Dual_Of_N(x)=(All y)[N(y) implies (x*y <= 0)].

In 3-D, the preferential voting dual function of a convex pyramidal “normal vector constraining” touching shadow polyhedron is scanned with lines along its surface. Perpendicular to those lines, are planes also containing the contact point. Those planes/flats define the shadow polyhedron on the other side. To use the shadow polytope: its tip is put at at point where X is known to lose. Then X is known to lose in the body of the shadow polytope.

In the case that there is a 2 winner election and the rights of an ordered pair of candidates is being upheld, then the base 2 dissatisfaction of the 2 candidates with the winners is at the smallest at the tip of the polytope.

Extra inferring is required as multivalued shadows intersect, before the winners of the non-proportional parts of the solution can be inferred. If all goes well, the proportional solution can be non-arbitrarily dropped in and an ideal preferential voting method results.

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### Political prisoner numbers disputed

At least 252 Tibetan political prisoners are detained in Chinese jails, more than double the number claimed by Beijing, a Tibetan rights group said yesterday. Official Chinese media said in May only 115 of the 2,300 inmates incarcerated in Tibet were political prisoners, convicted of “espionage, subversion or terrorism”.

However, the Tibetan Centre for Human Rights and Democracy compiled a study which found 252 known Tibetan political prisoners as of June.

It said 129 – including 26 women – were kept in Drapchi Prison, recognised as the harshest prison in Tibet.

The prisoners have been charged with endangering state security, which involves political activities such as pro-independence demonstrations and poster-pasting, or for possessing photos and videos of exiled Tibetan spiritual leader the Dalai Lama.

“The denial of basic human rights is apparent from the reasons for arrest and the inhuman treatment meted out to them while in detention and during imprisonment,” the centre said.

It said prisoners were subjected to severe beatings.

Since 1987, 27 prisoners had died in Drapchi Prison and 47 political prisoners’ sentences had been increased for non-conformity or disobedience, the centre said.

There had also been 21 protests within the prison.

The largest and most violently suppressed protest occurred in May 1998 and resulted in the deaths of eight political prisoners and longer sentences for many more, the centre said. “The only recourse that the prisoners can take to vent their pent-up feeling is through protest and demonstration,” said Lobsang Nyandak Zayul, the centre’s executive director. No one knows the exact number of Tibetan political prisoners jailed by China, as the information is not revealed by the Chinese Government.

London-based Amnesty International estimates there are hundreds of Buddhist monks and nuns in Tibet’s jails for political reasons.