Dear Craig,
you wrote (13 Dec 1999):
> I'll simplify that "multiwinner participation axiom" definition I put
> in a previous message.
>
> I had written this:
> -------------------------------------------------------------------------
> 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))]]
> -------------------------------------------------------------------------
>
> If 'SPC' holds then Q can be removed.
> SPC says that a candidate's win-lose state is unaffected by subsequent
> preferences.
> For example, p=(ABC), Q=((ABCD),(ABCE)), t=(2,5)
> V+t*Q= V+(1*t)p=
> 2 ABCD 7 ABC
> 5 ABCE S
> S
> In these 2 examples, if A (or B, or C) wins one then it wins the other.
> Similarly for B and C but not D
> and E.
> Hence p.W(V+t*Q) = p.W(V+(1*t)p), in this case:
> {A,B,C}.W(V+t*Q) = {A,B,C}.W(V+7p)
>
> Quite probably those persons that would want to impose a participation
> axiom would also impose SPC too. If SPC is not imposed then the rule
> has been made weaker.
>
> Also the rule can be altered and made more restrictive, by altering
> #(p.W(V))=#(p.W(V+tp)) into #(p.W(V))<=#(p.W(V+tp)). That would
> would require that the satisfaction value increases when the number
> of winners increases.
>
> -------------------------------------------------------------------------
> Definition: (Q2), "multiwinner participation axiom" rule. (13-Dec-99):
>
> (For All V)(For All p)(For All t) [
> (0<=t)(#(p.W(V))<=#(p.W(V+tp))) .=>
> (Satisf(W(V),p) <= Satisf(W(V+tp),p))]]
> -------------------------------------------------------------------------
>
> The Satisf function can be removed.
>
> (Satisf(W,p) <= Satisf(X,p)) = (For All j=1..length(p)).[ G(W,X,p,j) ]
>
> G(W,X,p,j) = ((W.trunc(p,j-1) = X.trunc(p,j-1)) => (W.{p[j]} <= X.{p[j])})
>
> ("<=" is subset, so: (W.{x} <= X.{x}) = ((x in W) => (x in X))
>
> The rule as defined may not be beyond being improved.
>
> Here are some example alterations. (Papers are not remarked.)
>
> Write a n:A+B+CDEF to mean that the paper (ABCDEF) has weight n and that
> {A,B} = ({A,B,C,D,E,F}.Winners).
>
> Let Z(p) = [#(p.W(V)), #(p.W(V+tp)); Satisf(W(V),p), Satisf(W(V+tp),p)]
> The rule fails the particular alteration if this is false:
> (z1<=z2)=>(z3<=z4).
>
> (0:AB)--(9:AB+): Z(AB)=(0,1; 0,1/4), Allowed
> (0:AB+)--(9:AB): Z(AB)=(1,0; _,_), Allowed
> Allow the adding of AB to causes B to start losing
>
> (0:ABC+DE+FG)--(9:ABC+DE+FG+): Z(ABCDEFG)=(2,3;5/32,5/32+1/128), Allowed
> (0:ABC+DE+FG+)--(9:ABC+DE+FG): Z(ABCDEFG)=(3,2; _, _), Allowed
> (0:A+BCD)--(9:AB+C+D+):
> Z(AB) =(1,1; 1/2, 1/4), Not allowed
> Z(ABCD)=(1,3; 1/2, 1/4+1/8+1/16), Not allowed
>
> I do consider this rule to be worth imposing.
Example (1 seat; 21 voters; Alternative Voting):
7 voters vote A > B > C > D.
6 voters vote B > A > C > D.
5 voters vote C > B > A > D.
3 voters vote D > C > B > A.
Candidate A is elected. But if the three DCBA voters had
truncated their votes and had voted only for candidate D,
then candidate B would have been elected.
Could you -please- explain why Alternative Voting meets your
multiwinner participation axiom so that Alternative Voting is
not "too defective to be used in practice" (20 Oct 1999)?
Markus Schulze
