FPTP family theory, REDLOG shadowing

[Craig Carey]

At 00:34 11.12.99 , Markus Schulze wrote:
>Dear Craig,
>
>you wrote (9 Dec 1999):
>> I am never going to promote Borda, actually.
>
>I bet you will promote some positional [*] election
>method. Saari's geometrical model will force you
>to do that.

What about that modified Condorcet method you mentioned?. That is
 going to have an amplke quantity of bad theoretical properties.


Borda doesn't is failed by the 'SPC' rule, defined here:


------------------------------------------------------------------------
At 08:04 24.April.99 , Blake Cretney wrote:
>Donald E Davison wrote:
...
>I have the following criterion defined in my Election Method Resource
>
>Name:  Secret Preferences Criterion: SPC 
>Definition:  If alternative X wins, and some of the ballots are
>modified in their rankings below X, X must still win.

The word "win" can be changed into "wins and loses respectively"
 (since (u impl v) = ((not v) impl (not u))).
The (P1) rule includes SPC and adds a requirement saying that if the
 preference itself is discarded, then candidate can't change from a
 winner into a loser [and then (P1) fails STV].

...
>So, my point is this.  Any method that meets SPC can be described
>using a procedure that never looks at any of the choices that are
>lower on ballots than the eventual winner.
>
>The procedures that you have been suggesting of late all look beyond
>an alternative on people's ballots without actually eliminating that
>alternative as a potential winner.  Unless this is unnecessary, we can
>conclude that they all fail SPC.

(The implication in the reverse direction there can't be concluded)
>
>My advice is either reject SPC, or stop suggesting methods that work
>like this.
>
>---
>Blake Cretney
>See the EM Resource:  http://www.fortunecity.com/meltingpot/harrow/124
>
------------------------------------------------------------------------

If a voting method doesn't satisfy SPC then it might as well be rejected.

[The only methods I favour are IFPP, FTP, and STV. The last is a bad method
 to be elected under, but it has no particular competition that satisfies
 SPC. IFPP is undefined for 4 candidates.

A positional method, by the definition given, is defined so that it has
 the same number of faces on boundaries to win-lose regions, as FPTP has.

The boundaries are shifted and distorted. That is bad: a surface flat can't
 be bent out into a region when proportionality would be better satisfied
 if that happened.

>
>If you think that you didn't yet walk into
>Saari's trap, then: (1) Could you -please- give me
>a concrete example of an election method that is
>not a positional election method and that can be
>described geometrically for any number of
>candidates? (2) Could you -please- give a
>geometrical description of Alternative Voting for
>102 or 103 candidates (or whatever your favorite
>number of candidates was) and [3] explain how a violation
>of the monotonicity criterion or the participation
>criterion [**] looks like geometrically?

If any preferential voting method can't be described geometrically in
 the simplex of all possible ballot paper count ratios, then the method
 is not defined. The existence of equations is not reduced if they are
 very long.

Therefore an answer to the first question is: 'every defined method
 that is not a "positional method"'.

Regarding question (2), the Alternative Vote most certainly has a
 algebraic polytope formula for 103 candidates. It would be symmetric
 with the number of similar regions equalling the factorial of 103.

Regarding the 3rd: Monotonicity roughly says that when a point moves along
 a line parallel to an edge and from one simplex corner to another corner
 having on the preference for the candidate unde consideration nearer to
 the first preference, then the candidate never changes from a winner into
 a loser.
 Monotonicity seems to be defined in public to apply to changes in a
 single paper only. In that case, a point may be able to move along (say)
 three directions parallel to edges, but it may not move along any other
 direction that is a weighted average of those three lines. That constraint
 can be removed simply and without problem.

Regarding participation...

>[**] The participation criterion says that an additional
>voter who strictly prefers candidate A to candidate B
>must not change the winner from candidate A to candidate B.
>In other words: A voter must not be punished for going to
>the polls and voting sincerely.

Adding or discarding a paper shifts a point in the simplex, along a line
 that passes through the paper's vertex.

IFPP with 3 candidates (1 and 2 winners) satisfies this 'participation
 rule'. However it is a very weak rule when there are only 3 candidates.
 (Imposing the 'particpation' rule would not lead to unwanted terms: e.g.
 if the vertex is d=1, then the rule would tend to create terms lacking
 the variable d.)




>You wrote (9 Dec 1999):
>> ... a cube?
...

>
>[*] A "positional" election method (e.g. FPTP, Borda)
>is an election method where a voter ranks the candidates
>and where a candidate gets A1 points for every first
>preference and A2 points for every second preference etc.
>and where A1 >= A2 >= ... and where that candidate is
>elected that has got the largest number of points.
>
[text shifted up]


--------------------------------------------------

Here's my URL giving computer code allowing polytopes to be exactly
 simplified:

   http://www.ijs.co.nz/polytopes.htm


Here's part of a REDLOG session showing testing of code that could be
 used to find the Boolean/geometric/algebraic formula for multiwinner
 FPTP:

1: procedure bigger(k,x,s); begin     % returns true iff exactly k of s
1:   scalar y,y2,h;                   %  are more positive than x
1:   if {} = s then return if 0 = k then true else false;
1:   if k = 0 then return rlsimpl foreach z in s mkand (z <= x);
1:   y:=first s;  y2:=rest(s);
1:   return rldnf (((y<=x) and bigger(k,x,y2))
1:               or ((x<y) and bigger(k-1,x,y2)));
1: end;

bigger

2:
2: bigger(0,a,{b,c,d,e});

a - b >= 0 and a - c >= 0 and a - d >= 0 and a - e >= 0

3:
3: bigger(4,a,{b,c,d,e});

a - b < 0 and a - c < 0 and a - d < 0 and a - e < 0


Craig Carey, Auckland