[EM] Is no Clones rule but EMers that it was used?

[Markus Schulze]

Dear Craig,

"Independence from Clones" is a criterion for single-winner elections.
"Independence from Clones" has been introduced by Tideman. Please read:

[1] T. Nicolaus Tideman, "Independence of Clones as a Criterion for
    Voting Rules," SOCIAL CHOICE AND WELFARE, vol. 4, pp. 185-206, 1987,
[2] Thomas M. Zavist, T. Nicolaus Tideman, "Complete Independence of
    Clones in the Ranked Pairs Rule," SOCIAL CHOICE AND WELFARE, vol. 6,
    pp. 167-173, 1989.

My definition of "Independence from Clones" is identical to Tideman's
definition except (1) for the fact that Tideman discusses only strict
rankings and (2) for the fact that Tideman discusses only decisive
situations.

Tideman's definition of "clones" looks as follows:

> Definition ("clones"):
>
>   A[1],...,A[m] are a set of m clones if & only if the following
>   two statements are valid:
>
>   (1) For every pair (A[i],A[j]) of two candidates of this set,
>
>       for every voter V, and
>
>       for every candidate C outside this set
>
>       the following two statements are valid:
>
>       (a) V strictly prefers A[i] to C,
>           if & only if V strictly prefers A[j] to C.
>       (b) V strictly prefers C to A[i],
>           if & only if V strictly prefers C to A[j].
>
>   (2) For every candidate A[k] of this set and
>       for every candidate D outside this set and
>       for every voter W:
>       W either strictly prefers A[k] to D
>       or strictly prefers D to A[k].

My definition of "clones" looks as follows:

> Definition ("clones"):
>
>   A[1],...,A[m] are a set of m clones if & only if the following
>   two statements are valid:
>
>   (1) For every pair (A[i],A[j]) of two candidates of this set,
>
>       for every voter V, and
>
>       for every candidate C outside this set
>
>       the following two statements are valid:
>
>       (a) V strictly prefers A[i] to C,
>           if & only if V strictly prefers A[j] to C.
>       (b) V strictly prefers C to A[i],
>           if & only if V strictly prefers C to A[j].
>
>   (2) For every candidate A[k] of this set and
>       for every candidate D outside this set
>       there is at least one voter W, who either
>       strictly prefers A[k] to D or strictly prefers D to A[k].

Statement 2 is only needed to exclude situations where all voters
are indifferent about all candidates. In this situation each neutral
election method is necessarily identical to "Random Candidate". And
obviously "Random Candidate" can be manipulated by running a large
number of identical candidates.

"Independence from Clones" says that if a given set of clones
(A[1],...,A[m]) is substituted by a single macro-candidate A such that

   for every voter V and

   for every candidate C outside this set

   if voter V strictly preferred candidate C to candidate A[1]
   then voter V strictly prefers candidate C to candidate A and

   if voter V strictly preferred candidate A[1] to candidate C
   then voter V strictly prefers candidate A to candidate C

then for every candidate D outside this set the probability that
candidate D is elected must not change.

"Independence from Clones" is met e.g. by IRV, Tideman's Ranked Pairs
method, and my beat path method.

Markus Schulze

******************************************************************

> From: Craig Carey <research at ijs.co.nz>
> Date: Sun, 12 Jan 2003 06:46:40 +1300
> Subject: Is no Clones rule but EMers that it was used?
>
> This message asks a question to Mr Schulze (a subscriber).
>
> This is not a mailing list for the weak minded persons that make
> untrue comments and who steadfastly refuse to fix wrong definitions
> on rules.
>
> Question/Request: to Markus Schulze: I request the definition of
> the (apparently implied to exist) "Independence ... Clones" rule,
> you wrote on this month.
>
> The quoted text here is copied from the Election Methods List.
>
> http://groups.yahoo.com/group/election-methods-list/message/10669
> 
>  > From:  Markus Schulze <markus.schulze at a...>
>  > Date:  Wed Jan 8, 2003  10:25 am
>  > Subject:  Re: [EM] Ties (was Condorcet Voting)
>  >
>  > Dear Alex,
>  >
>  > you wrote (7 Jan 2003):
>  > > Markus Schulze wrote (7 Jan 2003):
>  > > > However, consider the following example:
>  > > >
>  > > > 40 voters vote A > B > C > D > E.
>  > > > 40 voters vote B > C > D > A > E.
>  > > > 40 voters vote C > A > D > E > B.
>  > > >
>  > > > Although this is not a symmetric situation, the used election
>  > > > method must violate Neutrality or Anonymity or Decisiveness or
>  > > > Local Independence from Irrelevant Alternatives in this example.
>  > >
>  > > I'm not sure I see why. And what is the local version of
>  > > Independence from Irrelevant Alternatives?
>  >
>  > The "Smith set" is the smallest set of candidates such that for
>  > each candidate A in this set and for each candidate B outside
>  > this set the number of voters who strictly prefer candidate A
>  > to candidate B is strictly larger than the number of voters who
>  > strictly prefer candidate B to candidate A. "Local Independence
>  > from Irrelevant Alternatives" says that adding a candidate who
>  > doesn't get in the Smith set must not change the result of the
>  > elections.
>  >
>  > In the example above, candidate D and candidate E are not in the
>  > Smith set. Therefore, they must not change the result of the
>  > elections. However, when we ignore candidate D and candidate E
>  > we get:
>  >
>  > 40 voters vote A > B > C.
>  > 40 voters vote B > C > A.
>  > 40 voters vote C > A > B.
>  >
>  > Now, every anonymous neutral election method is necessarily
>  > indecisive. Therefore, every anonymous neutral election method
>  > that meets Local Independence from Irrelevant Alternatives
>  > is necessarily indecisive in the original example.
>  >
>  > ******
>  >
>  > Here is another scenario:
>  >
>  > 40 voters vote A > B > C > D > E.
>  > 40 voters vote B > D > E > C > A.
>  > 40 voters vote E > C > D > A > B.
>  >
>  > This is not a symmetric situation. (E.g.: The Copeland method
>  > chooses candidate B decisively. The Borda method chooses
>  > candidate B decisively.)
>  >
>  > The candidates C, D, and E are a set of clones. Therefore,
>  > Independence from Clones says that when the candidates C, D, and
>  > E are substituted by a single macro-candidate F then for each
>  > candidate X outside this set of clones the probability that
>  > candidate X is elected must not change. Therefore, we get:
>  >
>  > 40 voters vote A > B > F.
>  > 40 voters vote B > F > A.
>  > 40 voters vote F > A > B.
>  >
>  > Now, every anonymous neutral election method is necessarily
>  > indecisive. Therefore, every anonymous neutral election method
>  > that meets Independence from Clones is necessarily indecisive
>  > in the original example.
>  >
>  > Markus Schulze
>  >
>
> It looks like the Election Methods List was given a false claim that
> Mr Schulze had used an Independence from Clones rule.
>
> I intend to expell Mr Markus Schulze if the topic here does not
> produce some interest that leads to perhaps a response that it is
> true. I suppose I should ask if Mr Schulze is learning something or
> other from this mailing list, or whether there seems to be a chance
> of that happening.
>
> I am content to overlook the words "anonymous" and "neutral". I really
> don't know what their meaning might be. It is becoming obvious that
> the Election Methods List is a practice zone for people that use
> utterly worthless acronyms or definitions that maybe have not been
> clarified for 3 years or more, if ever.
>
> Mr Schulze can write in and say whether he made a slip or got the
> wording wrong, or whether there was a belief that EM should be told
> that an 'Independence of Clones' rule would be described as having
> been used when that was untrue.
>
> Here is what happened to the rule that was a little difficult to get.
>
> $-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$
>
> At 02\12\20 02:48 +1300 Friday, Craig Carey wrote:
>  >More of this nonsensical notation that seems to be unthinkable unless
>  >the author permanently lost in darkness. It is actually of interest since
>  >it correlates a likelihood of many blunders in statement making.
>  >
>  >However I shall try to clean up this definition:
>  >
>  >A first thing to do is to delete the word voters. On the face of it, it
>  >means that ballot papers are being grouped in an unspecified way (or as
>  >more likely the case, the author can't can't access the meaning any
>  >better than readers). Often that is no impediment to correcting it and
>  >I have a go at doing that here.
>  >
>  >
>  >At 02\12\19 12:10 +0200 Thursday, Markus Schulze wrote:
>  > >I wrote (28 Aug 1998):
>  > >> Definition ("clones"):
>  > >>
>  > >>   A[1],...,A[m] are a set of m clones if & only if the following
>  > >>   two statements are valid:
>  > >>
>  > >>   (1) For every pair (A[i],A[j]) of two candidates of this set,
>  > >>
>  > >>       for every voter V, and
>  > >>
>  > >>       for every candidate C outside this set
>  > >>
>  > >>       the following two statements are valid:
>  > >>
>  > >>       (a) V strictly prefers A[i] to C,
>  > >>           if & only if V strictly prefers A[j] to C.
>  > >>       (b) V strictly prefers C to A[i],
>  > >>           if & only if V strictly prefers C to A[j].
>  > >>
>  > >>   (2) For every candidate A[k] of this set and
>  > >>       for every candidate D outside this set
>  > >>       there is at least one voter W, who either
>  > >>       strictly prefers A[k] to D or strictly prefers D to A[k].
>  > >
>  >
>  >
>  >A is a list (the same as an array or vector). It shall be auto-converted
>  >to a set as needed to get the correct input for the operators. The
>  >"x precedes y in paper p" means both x and y are named by the paper and
>  >the prerfence, x, is closer to the 1st preference.
>  >
>  >Statment (1) is this:
>  >
>  >(For all A, A is a list, all members of A are candidates, #A>=2)
>  >(For all ai, aj)[ (ai in A, aj in A, not (ai=aj)) implies [
>  >. . (For all p, p a ballot paper)
>  >. . (For all t, t a real)  % now the paper ("voter") V equals t.p
>  >. . (For all C) [ (not (C in A)) implies [  % try having "this set" mean "A"
>  >
>  >. . . . [ (both ai and C appear in p and ai precedes C) equals
>  >. . . . . (both aj and C appear in p and aj precedes C)
>  >. . . . ] and [
>  >. . . . . (both C and ai appear in p and C precedes ai) equals
>  >. . . . . (both C and aj appear in p and C precedes aj) ] ]] ]]
>  >
>  >That looks a real rule since it is processing a method. Unfortunately
>  >for the credibility of the rule, it never evaluates the method.
>  >Two other little problems are:
>  >
>  >(1) There is no use of a "(There Exists)". Thus the tiniest defect in the
>  >  rule allows it to be obvious that the whole rule has to be rejected.
>  >
>  >(2) The variable "t" appears only one. The rule seems to consider only
>  >  kinds of ballot papers rather than kinds of ballot papers that had a
>  >  weight/count property.
>  >
>  >That rule did not last long.
>  >
>  >The instant collapse of theories that have entangled in them a notion
>  >that it is best to use the word "voter" as if meaning "human" is not
>  >actually advancing the cause of those that would assert that humans are
>  >relevant.
>  >
>  >The definition was last about 52 months ago.
>  >
>  >----------------------------------------------------------------------
>  >
>  >Let's get to the second. It does not seem to be linked up with the
>  >first in any way, so we have 2 definitions. Hence easily neither is
>  >the information sought ("independence from clones"). Or are they to be
>  >AND-ed ?.
>  >
>  >This definition also appears to not actually evaluate the winner findin
>  >function that it is supposed to be checking.
>  >
>  >
>  > >>   (2) For every candidate A[k] of this set and
>  > >>       for every candidate D outside this set
>  > >>       there is at least one voter W, who either
>  > >>       strictly prefers A[k] to D or strictly prefers D to A[k].
>  >
>  >
>  >
>  > >An election method violates "Independence from Clones" when there are
>  > >situations where you can increase or decrease the probability that a
>  > >given candidate is elected by introducing additional clones to a set
>  > >of clones to which this candidate doesn't belong.
>  > >
>  >
>  >I remind Mr Schulze that polytopes do not have surfaces that get soft
>  >or that start to vibrate when someone suggests that probability
>  >exists. Until Mr Schulze fixes up the rule so that it does not completely
>  >ignore the function that it is supposed to be checking,
>
> $-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$-$=$
>
> The definition of something that was supposed to test a method but which
> seemed to not even actually evaluated it, was last given in 1998.
>
> Naturally the rule of economist Tideman is definitely not being considered
> here. That must be regarded as irrelevant unless, perhaps, it is proven
> to be the same Clones rule. They can't be demonstrated to be the same while
> Mr Schulze's is in an unusably ill-defined state. I am sure we can require
> that even if the rules are completely identical (and I guess that they
> definitely are not).
>
> The original text is obviously wrong since trying to make a general
> conclusion from a particular and it uses the word "therefore" to suggest
> reasoning led to that conclusion that reasoning would not lead to:
>
>  > Now, every anonymous neutral election method is necessarily
>  > indecisive. Therefore, every anonymous neutral election method
>  > that meets Independence from Clones is necessarily indecisive
>  > in the original example.
>
> Isn't a 1998 definition very much too out of date?.
>
> If Mr Schulze is on top of the topic of the algebra of some Clones rule
> then this is going to be an easy message to make a very satisfactory
> reply to. Note that I use the words "perfect precision" and they are
> quite hostile to any blinding attempt to pastel-shade up politicians
> or voters as being important.
>
> The conclusion seem pointless and not even worth knowing: that a
> method is 'indecisive'. I am not aware that that is defined (obviously
> it is some definition of a set/Boolean valued function that receives
> 'votes' (the usual weighted preference lists).
>
> It looks like the word "decisive" will be part of a recheck of the
> reasoning that exposes whether or not Mr Schulze did indeed use a
> definition of some Clones rule.
>
> There can't be both errors around here and secrecy on why they are
> sustained in the sense that they get spotted reappearing at the
> Election Methods List within a few months.
>
> Craig Carey
> Auckland



> Date: Mon, 13 Jan 2003 09:48:54 +1300
> From: Craig Carey <research at ijs.co.nz>
> Subject: Ulitmatum explained: 7 days to start answer the request for Clones info
>
> I privately alert you to the possibility that I will expell you from
> my Politicians-and-Polytopes mailing list.
>
> The reasoning for that is in the last message sent to the mailing
> list named the Politicians and Polytopes mailing.
>
> I will set a time limit of 1 week for you to respond to my requests
> for reasoning and so on, that is present in my last message to the
> Politicians-and-Polytopes mailing list (hosted with groups.yahoo.com).
>
> I have no doubts about the correctness of expelling you from the
> mailing list. I am personally affected by seeing you merely being
> subscribed to my mailing list while making seeming sticking with a
> principle of not correcting errors out at another mailing list.
>
> If you have got some personal concern then you may openly document
> it to me. E.g. these events are irritating and that is why you have
> to go with the Election Methods List. They only pretend to listen,
> and that is hand in hand with gliding past an error no matter how
> small. The quality of that list is not an issue here.
>
> Evading and segregating out has no place at the Politicians and
> Polytopes mailing list.
>
> Note that it is me who is writing thus the whole topic can rotate
> on a detail of reasoning. With me writing first, I was getting
> close to saying that you had lied, which is not really significant
> by the looks of it. E.g. my reacting would have seemed unlikely.
>
>            Deadline for delivery of results
>
> I shall ban your address from the Politicians and Polytopes mailing
> list maybe a day after 20 Jan 2003 if you have not replied to the
> questions and got the whole matter corrected, and I do want to
> see the wrong definition of that wrong Clones rule, unless you
> have some something other than the usual EM List that justify
> the withholding of the equations.
>
> If I ask a question (I did that) and then you do not reply, then
> I don't think you will be freely writing again for years to my
> Politicians and Polytopes mailing list. It seems to be a really
> 'sensible' way to run a mailing list, although over time the
> details will fade.
>
> My list defends justice and fairness and I have not read any
> comments by you that say single individuals should only be
> justly and precisely regarded when they have no power except at
> internal ties of the winner-selecting function(s).
>
> If you have something that is a rule that can't test a sequence
> of polytopes, then you are also outside of the first rule of the
> welcome message of my Single-Transferable-Vote mailing list. That
> is another mailing list. I am not considering your freedom to
> write there. But if have an unclear answer, I may moderate your
> messages to the Single-Transferable-Vote mailing list.
>
> I will have to wait for your comments. I don't wish to say that
> I expect you to remain (it is mere estimation of the future).
>
> It does seem to me that you fancy yourself to have an interest
> in history but surely historians will get petty over small
> details just like what is happening here. It could be that in this
> matter, it is I that defends history. The EM-ers justr lose
> questions.
>
> I suppose you have some destructive idea of having to batter at, and
> smash against, all thinking that is false. I am not seeing much of
> any such thing. However I consider only the archives of my
> Politicians and Polytopes mailing list for this topic.
>
> I guess you dislike my messages. I can suspect that an end is
> coming, for I have really somewhat withdrawn from your Ossipoffist
> philosophy mailing list (of claiming to have a definition when
> there wasn't one and it wasn't used too, and in fact never will
> be).
>
> Craig Carey

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