[EM] Multiwinner participation rule. Geometric descriptions
Craig Carey
research at ijs.co.nz
Fri Dec 10 21:21:40 PST 1999
1. A multiwinner participation rule
2. A reply to the why use geometry question
[1]
Mr Shulze's definition of the participation axiom was worded in a way that
indicated that there would be only one winner. A repair attempt for that
problem is described here:
----------------------------------------------------------------------------
Definition: (Q2), "multiwinner participation (axiom)" rule. (11-Dec-99):
(For All V)(For All p)(For All Q)[
[(For All q)[(q in Q) => (p = trunc(q,length(p)))]] =>
(For All t) [(0<=t)(#t=#Q)(#(p.W(V))=#(p.W(V+t*Q))) .=>
(Satisf(W(V),p) <= Satisf(W(V+t*Q),p))]]
----------------------------------------------------------------------------
Notation:
p is a preference list (ballot paper) with a weight which equals 1.
The list part might be for example, "(AB)" (which could be "A>B>C").
Q is a set of papers that all start with p, but which can have anything
after p.
t is a vector of real numbers.
t*Q is the papers in Q with each having weights equal to the numbers in t.
V is a collection of papers (an election). It can be a vector of real
numbers;
W(V+t*Q) is the set of winners of V' when V' is V to which the papers in Q
are added with each having weights given by the vector t.
W(V) is the set of winners of V
#S is the number of elements in S, For All S
p.W(V) is the set intersection of the set of winners of V with all the
candidates that have a preference in the preference list p.
Satisf(W,p) measures the satisfaction of paper p with the set of
winners, W.
Satisf(W,p) is defined to be equal to the sum over i=1,2,3..,length(p)
of (2**(-i))*(if p[i] in W then 1 else 0).
The i-th preference in paper p is for candidate p[i].
length(p) is the number of preferences in paper p.
trunc(p,k) returns a weighted preference list which is p but with the
minimum of trailing preferences removed so as to cause this to hold:
length(trunc(p,k)) <= k.
Notes:
1. Both this (Q2) and Mr Schulze's allow 2 regions where the outcome alters
in the prohibited way, to be not adjacent.
I suppose this rule is identical to Mr Schulze's, except that the rule
of Mr Schulze didn't seem to me to be clear about whether the paper had
to have a preference for both candidates. Is that what the word "strict"
means in "[a paper/voter] strictly prefers candidate A to candidate B".
2. Q is introduced to cause 'SPC' style indifference to truncation. That is
what (P1) does but (P1) wouldn't allow any alteration of preferences
before the preference for the first winner that may change to cause the
participation axiom to be violated. (P1) allows only movement along
rays contained in a (possibly lower dimension) simplex with a vertex on
the point that has 'edges' parallel to the universe simplex's edges.
I guess it is plausible to tighten the rule's constraining power in the
way done by Q.
3. The term (#(p.W(V))=#(p.W(V+t*Q))) stops 'satisfaction' values being
compared when they are based on differing numbers of winners of
interest to the paper.
The term prevents the rule being too restrictive in circumstances like
this:
There are 3 papers, {(AB), (B), (C)}, and 1 winner is required. The
rule shall not find wrong the IFPP divide (between B wins and C
wins) that is the line segment (1/3=b)(1/3<c<1/2).
That line segment is parallel to the (AB)-(C) edge and a ray from the
(AB) vertex would, when crossing that divide, have the satisfaction
number for (AB) change from 0 (A and B lose) to 1/4 (A loses and B
wins). Similarly without that change the rule would seek to modify
STV into FPTP (STV has instead the divide at (a=b)(1/3<c<1/2)).
AB, a=1
/\
/ \
/ \
/______\
B,b=1 C,c=1
(This tends to suggest that there may be grounds for dispute on whether
paper (ABCDEFG) would prefer prefer winner set {C} over winner set
{D,E,F,G}.)
This participation axiom seems to be a little weak to me: in the diagram,
the A-wins region has to leave A's corner before there is violation. Maybe
all 3 candidate methods are passed by the rule. It might be rather weak in
effect in 4 candidate methods too. (Are there examples around motivating
this rule?.)
---------------------
[2]
At 08:37 11.12.99 , Markus Schulze wrote:
>Dear Craig,
>
>you wrote (11 Dec 1999):
>> 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.
>
>In so far as you couldn't define IFPP for all situations, you favour only
>two methods: FPTP and Alternative Voting.
>
>Didn't you say that Alternative Voting was "a method too defective to be
>used in practice" (20 Oct 1999)? [Alternative Voting can punish voters
>for voting.]
>
>What has caused this change of your opinion?
What was written is this:
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
At 03:03 20.10.99 , Craig Carey wrote:
>At 22:57 19.10.99 , Markus Schulze wrote: ...
>>Markus Schulze wrote (19 Oct 1999):
...
>>Do you question that some election methods sometimes punish voters
>>for going to the polls and voting sincerely? Or do you question that
>>a voter rather wants to have no influence on the election result than
>>to worsen the election result?
...
>There are two questions. To the first I reply: no, election methods
> can punish voters for voting. A proof could involve an analysis of
a
> voting method devised to allow the proof to proceed easily, e.g.
> a method too defective to be used in practice.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
That word "punish" is undefined (e.g. more than one type of strategic
voting problem or etc.).
>Markus Schulze wrote (11 Dec 1999):
>> Markus Schulze wrote:
>> > 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 favo[u]rite
>> > 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.
>
>Whether you can draw something or whether a given drawing has any
>geometrical meaningfulness are two different questions.
The rules specify simplices that touch win-lose boundaries and that
constrain their normal vectors. Rules in preferential voting need not do
that but so far about all of them discussed have. A possibility is that
someone has a favourite method: a geometric idea is to apply the rule and
clip back the offending faces (or extend out other faces).
>If you think that the "simplex" has any non-trivial geometrical
>meaning, then -please- explain this meaning for the above mentioned
>two situations.
>
>If you don't think that the "simplex" has any non-trivial geometrical
>meaning, then -please- explain why you always use geometrical terms.
Suppose I wrote: "the plane that contains the said point, and also the
(ABC) and (AC) vertices". That can be translated into a parametric form,
or into an matrix product inequality Operations Research form. The last
style of describing, using matrices can be quite wordy.
My writing as if some matter were just something about hyper-geometric
polytopes, allows (1) full precision and (2) many less words and (3) it
tends to easily understood. Any comments?.
...
>But does a violation of monotonicity or participation have any non-trivial
>geometrical meaning that could justify the introduction of your geometrical
>interpretation of elections?
("non-trivial" isn't defined well enough for me.)
As far as I know, the world's great experts have botched the definition of
monotonicity and is the public expected to wait for a book to be
published?.
My aim is to avoid numbers (and single election examples) because the aim
is to have rules hold, and rules impose constraints over an infinity of
winner sets. An obvious problem is: what happens when the problem is
found to be quite implicit in that it can't be solved until it is already
solved?. A method of 'guessing a method and testing it' is an option but
that method can keep failing for a decades. If numbers are replaced with
algebra (geometry) the implictness can vanish maybe.
Here is a geometric description for Mr Schulze to be specific about how it
could be reworded:
'if the "participation axiom" is enforced, then an attempted
'method tweak-up' would tilt each particular offending win-lose
boundary face minimally until it contained the paper's vertex.'
What non-geometric way easy to understand way of describing that is an
improvement?.
11 December 1999
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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