[EM] Probabilistic criteria. Participation & no-show.

Fri May 5 01:17:45 PDT 2000

Dear Mike,

you wrote (4 May 2000):
> Maybe for academic purposes, putting the criteria in terms
> of probabilities instead of actual winners would be good,
> because it would make the criteria applicable to probabilistic
> methods. Because, ideally, it's good to be completely general.
> But for actual practical purposes, it's more useful to
> talk about concrete outcomes rather than probabilities. For one
> thing, I know of no probabilistic method that is in actual use,
> or which has any significant amount of advocacy for public
> elections. I've only heard of one person advocating such a
> method, and he didn't have a proposal, only the suggestion
> that maybe a good method of that type could someday be found.

I guess that you are talking about Albert Langer. But remember
e.g. that Lucien Saumur promotes Smith//RandomCandidate.

******

You wrote (4 May 2000):
> p.s. Is is participation an adverse results criterion?
> Isn't it one of those criteria that says that participation in
> the election by a group of same-voting sincere-voting voters
> shouldn't cause their 1st choice to lose (or maybe it's that
> it shouldn't cause their last choice to win), if that wouldn't
> have happened without their participation?
>
> If that criterion specifies sincere voting, then it's worthless,
> because the methods that pass it are methods that discourage
> sincere voting.
>
> That's the purpose of SARC. Sincere voting isn't a realistic
> assumption for most methods, especially the ones that pass
> such criteria as "participation" or "no-show".
>
> Or, if I'm mistaken about "participation" & "no-show" assuming
> sincere voting, would you tell me what they say about how they
> assume people vote--or just quote those criteria here?

The participation criterion says that the participation in the
election by a same-voting group of voters should never worsen
(due to the opinion of this group) the result of the elections.

Of course, the participation criterion presumes sincere voting
because (1) if this additional group of voters votes insincerely
then -as we know only the reported opinion and not the sincere
opinion of this group- we cannot check whether it worsens the
result of the elections due to its sincere opinion and (2) if
this additional group of voters voted insincerely and worsened
the result of the elections due to its sincere opinion then this
wouldn't be considered as a problem because this fact would deter
this group from voting insincerely.

Markus Schulze
schulze at sol.physik.tu-berlin.de
schulze at math.tu-berlin.de
markusschulze at planet-interkom.de

FFrom election-methods-list-request at eskimo.com  Fri May  5 09:11:40 2000
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Subject: Re: [EM] Question about complete clone independence
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Markus S wrote:
-snip-
> This problem has been discussed in August 1998. The definition of
> clones was changed as follows (28 Aug 1998):
>     Definition ("clones"):
>     A[1],...,A[m] are a set of m clones if & only if the following
>     two statements are valid:
-snip-
>     (2) For every candidate A[k] of this set and
>         for every candidate D outside this set
>         there is at least one voter who either
>         strictly prefers A[k] to D or strictly prefers D to A[k].
> 
> The aim of statement (2) is to exclude explicitly those situations
> where every voter is indifferent.

Thanks for clearing that up.  My attempt to broaden independence 
from clones to independence from similar alternatives can't be 
patched as easily, unfortunately.

Roughly speaking, my current definition of similar alternatives 
states:

   A set Y which is a subset of the alternatives is a similar set
   if and only if both of the following conditions hold:

   1. For all y,y' in Y and all x not in Y,
      y beats x if and only if y' beats x, and
      x beats y if and only if x beats y'.

   2. For all y,y' in Y and every majority cycle MC such that
      y is in MC and no other alternative in Y is in MC,
      min(substitute(MC,y,y')) = substitute(min(MC),y,y').

Several expressions used above demand additional definitions:

   Given a pair of alternatives x and y, support(x,y) denotes
   the number of voters who voted x strictly preferred to y.

   Given a pair of alternatives x and y, x beats y if and only if
   support(x,y) > support(y,x).

   x1,x2,...,xn is a majority cyclic permutation if and only if
   xn beats x1 and for all j in {1,2,...,n-1} xj beats xj+1.

   MC is a majority cycle if and only if MC is a nonempty subset
   of alternatives such that there is at least one majority
   cyclic permutation of the alternatives in MC.

   Given a majority cycle MC, majorities(MC) denotes the set of
   ordered pairs of alternatives {(x,y) such that there is at
   least one majority cyclic permutation of the alternatives
   in MC such that x immediately precedes y in the permutation}.

   Given a majority cycle MC, min(MC) denotes the set of
   ordered pairs of alternatives {(x,y) such that (x,y) is
   in majorities(MC) and support(x,y) <= support(z,z')
   for all (z,z') in majorities(MC)}.

   Given a pair of alternatives x and y and a set S which
   is either a set of alternatives or a set of ordered pairs
   of alternatives, substitute(S,x,y) denotes the set obtained
   by replacing x with y everywhere x appears in S.

The conditions in the definition of similar set are based on 
aggregate properties which are robust under small perturbations 
of the votes.  Condition 1 embodies the basic intuition for 
broadening of clone sets, but it's too great a relaxation for 
any voting procedure to satisfy an independence based on it 
alone.  Imposing condition 2 restricts similarity enough that 
voting procedures can be "practically" independent of similar 
alternatives, being out-of-compliance only in indecisive 
scenarios which would be rare in large public elections.  I'm 
wondering if there's a reasonable patch for the definition so 
that complete independence can be obtained.

Similarity could also be significantly relaxed without 
compromising independence:  In condition 1, beats relations 
outside the top cycle could be neglected.  In condition 2, 
majority cycles outside the top cycle could be neglected.

There are other ways similarity could be defined, for instance 
by using margins instead of support in the definition of 
min(MC).  But my primary purpose is to show that independence 
from clones is robust under small perturbations of the votes 
which cause the clones premise to not strictly hold.  The point 
is to show that it would be harder to manipulate outcomes given 
procedures which are independent from clone (or similar) 
alternatives.  Institutional rules often make it very easy for 
small minorities to nominate more or fewer alternatives, and 
it's pretty easy to find potential candidates which are similar 
to candidates.

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)

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Subject: Participation & SARC (was Re: [EM] Probabilistic criteria. Participation & no-show.)
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Markus S replied to Mike O:
-snip-
> The participation criterion says that the participation in the
> election by a same-voting group of voters should never worsen
> (due to the opinion of this group) the result of the elections.

The wording of the participation criterion by Herve Moulin 
(Axioms of Cooperative Decision Making, Cambridge University 
Press, 1988, p.239) refers to a single voter, not a group:

   Participation: Say that candidate a is elected from the set A
   by the electorate N.  Next consider a voter i outside N. 
   Then the electorate N U {i} should elect a or some candidate
   whom agent i strictly prefers to a.

Since Moulin doesn't indicate how i will vote, sincerely or 
strategically, it is possible that i will vote in such a way 
that i is worse off.  So we must infer that Moulin meant either 
that i would vote sincerely or that we would measure compliance 
by i's voted preferences.

> Of course, the participation criterion presumes sincere
> voting because (1) if this additional group of voters votes
> insincerely then -as we know only the reported opinion and not
> the sincere opinion of this group- we cannot check whether it
> worsens the result of the elections due to its sincere opinion
> and (2) if this additional group of voters voted insincerely
> and worsened the result of the elections due to its sincere
> opinion then this wouldn't be considered as a problem because
> this fact would deter this group from voting insincerely. 

A criterion like SARC seems better than participation when the 
voters are sophisticated (and unconstrained by accountability, 
e.g., secret ballot), since it is unreasonable to compare only 
sincere voting and the abstention strategy when other strategies 
may be better than both of those.

For participation to be considered an important criterion in 
large public elections, wouldn't there need to be empirical 
evidence that significant numbers of voters will routinely not 
vote due to a procedure's failure to rigorously comply with 
participation?  (Voter turnout might actually increase overall, 
given a procedure which fails participation but does a good job 
of solving other problems like spoiling.)  Also, is there a 
plausible argument that society would be worse off using a 
procedure which doesn't rigorously comply with participation?

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)