Condorcet Truncation Example

[Markus Schulze]

Hugh Tobin wrote:
> Markus Schulze wrote:
> >
> > Dear Hugh Tobin,
> >
> > I cannot agree with your statement, that (if a Condorcet Criterion
> > method is used) only omnissent voters can vote strategically.
> >
> 
> I did not make the statement attributed to me.  I did, I believe, show
> that rational strategic voters would not use truncation as their
> strategy, even if the tiebreak system were not "truncation resistant."
> I believe you have conceded that point.
> 
> Regards,
> 
> Hugh Tobin

Sorry!

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

Rob Lanphier wrote:
> Though I find the above scenario troubling, it doesn't change
> my support of Smith//Condorcet[EM] for three reasons:
> 
> *  It doesn't encourage order reversal
> *  One can't seem to upward truncate and cause someone who wasn't in the
> Smith set to enter the Smith set and win the election (this is gut
> instinct; I could be wrong, but I'd like to see a counter-example).
> *  Any other proposal I've seen has worse problems

If I cannot dissuade you from supporting Smith//Condorcet[EM], perhaps
I can propose some changes to Smith//Condorcet[EM].

In my opinion, if you don't want to harm Condorcet, you should try to
get as far as possible with pairwise comparison and you should use a tie
breaker only as a very very very last possibility to get rid of a tie.
There are two reasons for not using a tie breaker too early:

(a)If you use a tie breaker too early, then the voters will always
   think: "What will happen, if candidate A comes into the Smith set
   and has his highest defeat against a candidate B, which I prefer?"
   or "What will happen, if candidate C, who I don't like, comes
   into the Smith set? How will I have to vote to make him
   have a high defeat?" or ...
   In short: If the voter is urged to decide in the same run, who will
   be in the Smith set and who becomes the winner due to the used tie
   breaker, the voter will always believe, that he has to vote
   strategically.

(b)Suppose there is no Condorcet winner, than you have to use a tie
   breaker. As the tie breaker methods proposed in this election
   methods list are rather complicated (and as to every voting
   situation every party could propose a tie breaker, which could be
   justified and which would make their own candidate win), there is
   always the danger, that the supporters of a defeated candidate
   will feel being cheated.

Proposal 1:

  You said, you support Smith//Condorcet[EM], because the tie breaker is
  truncation resistent.

  But I believe, most problems caused by downward truncation can already
  be eliminated by using a truncation resistent definition of
  the Smith set.
  To differ between the usual definition of the Smith set and the
  truncation resistent definition of the Smith set, I will call the
  latter one "Smith//Truncation set".

  A possible definition of the Smith//Truncation set looks as follows:

    The "majority of the voters" is calculated as follows: You look at
    the pairwise comparison with fewest truncated votes. Suppose this
    is the pairwise comparison of X versus Y. Then you add the number
    of votes for X and the number of votes for Y (in the pairwise
    comparison of X versus Y) and divide it with 2.

    If a majority of the voters likes candidate A more than candidate B,
    then A wins against B. If a majority of the voters likes candidate
    B more than candidate A, then B wins against A. If neither a
    majority likes A more than B nor B more than A, than A=B.

    Now you have a set of wins, defeats and equalities. If you now
    calculate the Schwartz set of these wins, defeats and equalities,
    you get the Smith//Truncation set.

  Example:

  Markus Schulze wrote:
  > 
  > Here is an example to show why voters would truncate, if
  > a Condorcet Criterion Method is used:
  > 
  > Case 1:
  > 
  > 47 voters vote ABC.
  > 10 voters vote BAC.
  >  8 voters vote BCA.
  > 35 voters vote CBA.
  > 
  > A:B=47:53.
  > A:C=57:43.
  > B:C=65:35.
  > 
  > B wins against A and against C in the pairwise comparison. Thus B
  > is the Condorcet winner.
  > 
  > Case 2:
  > 
  > Now the 47 voters, who prefer A most, do truncate.
  > 
  > 47 voters vote A.
  > 10 voters vote BAC.
  >  8 voters vote BCA.
  > 35 voters vote CBA.
  > 
  > A:B=47:53.
  > A:C=57:43.
  > B:C=18:35.
  > 
  > Now there is a tie between A, B, and C. Whether A is elected,
  > depends on the used tie breaker. But if lowest defeat margin is
  > used, then A is elected.

  In case 2, B wins against A because more than half of the voters like
  B more than A. A wins against C because more than half of the voters
  like A more than C. But neither more than half of the voters like
  B more than C nor more than half of the voters like C more than B;
  thus B=C.
  Thus B is the only candidate of this Schwartz set and of
  the Smith//Truncation set. Thus B wins.

  In the above mentioned example, there is a unique winner without
  having to use a tie breaker. I only used a truncation resistent
  definition of the Smith set.

Proposal 2:

  In my opinion: If there is no Condorcet winner, than the best method
  is simply to have a second run. And in this second run, only the
  candidates in the Smith set (resp. of the Smith//Truncation set,
  if you support my first proposal) should be allowed to run for
  election.

  A second run has many advantages:

  (a)A candidate can withdraw (e.g., if he recognizes, that he only
     got into the Smith set, because some voters used order reversal
     to defeat another candidate and the candidate has to recognize,
     that he has no chances to win the second run).

  (b)The voters wouldn't be that encouraged to vote
     strategically, because they don't have to fear that another
     candidate enters the Smith set and defeats their most
     favoured candidate in the first run. The voters will just use the
     first run to decide, which candidate enters the Smith set.
     And only in the second run, they will try to vote strategically.

  (c)Surprising election are not that probable: If a Condorcet
     criterion method is used, it can always happen that a party
     gets many seats with only a few first preferences. If this
     party is only the third or fourth largest party due to
     the first preferences and it gets the most seats, than
     many voters will feel being cheated. But if there is a 
     second run, than this party will have many candidates in
     the second run and it will get much more first preferences in
     the second run. And the win of this party wouldn't be that
     surprising.

  You can combine proposal 1 and proposal 2 in many different ways.
  For example: You can have a second run with all the candidates
  of the Smith set of the first run and a third run with
  all the candidates of the Smith//Truncation set of the second run.
  If there is no unique winner in the third run, you can still
  use Smith//Condorcet[EM].

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

New Democracy wrote:
> C4PR: Many voters in Australia and Ireland have simplified their vote by picking
> a party's candidates by alphabetical order. This defeats the best aspects
> of the Single Transferable Vote. Furthermore a party candidate who receives
> too many first place votes could in fact harm other party members chances
> of becoming elected since the redistribution of votes may penalize the
> other candidates. It is C4PR's conclusion that the Single Transferable Vote
> system does not work well with partisan elections which is a vital aspect
> of Canadian politics.
> 
> This system has much to offer Canadians but should not be used in our
> parliamentary elections. It would be better used in municipal elections
> where partisan politics play a minor role. C4PR feels that in electing a
> non-partisan Senate, Privy Council or Supreme Court would such a system be
> appropriate.
> <snip>
> The Party List PR system would be the most ideal choice for Canada. The
> arguments made by its opponents can be easily defeated once we better
> understand the nature of this system.
> End of these C4PR's evaluations:
> 
> Donald: I am disappointed because I feel that STV should be used and Party
> List should not be used.

In Germany, only few literature is available about Hare's PR-STV method.
But it is always said, that the voters tend to vote alphabetically
[i.e., they give only a few preferences sincerely and than distribute
the remaining preferences in the same order as the candidates are
ordered on the ballot]. For example: You can read "Democracy or
Anarchy?" by Ferdinand Aloys Hermens.

The tendency to vote alphabetically can have catastrophal consequences
for a Condorcet criterion method:

(a)If Hare's PR-STV is used, than a worse preference is used only
   if all the better preferences were already used. Thus the worse the
   preference, the smaller the probability that this preference gets
   relevant.
   But if a Condorcet criterion method is used, than each preference
   becomes relevant with the same probability. Thus the effect
   of alphabetical voting becomes very important.

(b)New Democracy wrote (in http://www.mich.com/~donald/more.html):
   > The city of Cambridge, Mass has had a proportional representation
   > method of electing their city council since 1943. The law requires
   > that the names of all candidates be in cyclic alphabetical order,
   > with each candidate's name appearing at the top of an equal number
   > of ballots at each polling place.

   If the candidates would be placed on the ballot like this and the
   voters would distribute their worse preferences the same order
   as the candidates are ordered on the ballot, then they would 
   create very much more ties between the different candidates.

(c)Rob Lanphier wrote:
   > 
   > On Tue, 6 May 1997, Hugh Tobin wrote:
   > > Markus Schulze wrote:
   > > > But the way you have defined "truncation resistant" and
   > > > Smith//Condorcet [EM], the following problems will occur:
   > ....
   > > > (2)Distributing the worst preferences random among the least
   > > >    favoured candidates becomes a usefull strategy. That means:
   > > >    If the voter doesn't care, who of his least favoured
   > > >    candidates wins [if one of them should win], then it is
   > > >    nevertheless the best strategy to give them different
   > > >    preferences to maximize the chances of a favoured
   > > >    candidate to win.
   > >
   > > You are right.  I have objected to the [EM] tiebreaker on this
   > > ground. Other tiebreakers, such a margins-of-defeat, do not
   > > create an incentive to vote randomly, though (at least in
   > > theory) a voter who anticipates a circular tie and thinks
   > > he knows who will win the pairwise race between
   > > his less favored candidates may cast an insincere vote to make
   > > the margin larger.
   > 
   > I think that the tendency to vote randomly for the bottom
   > candidates is a small price to pay for the added truncation
   > resistance. Furthermore, if it pays to vote randomly,
   > it pays even more to study the differences among
   > the candidates who one would otherwise scatter random votes
   > upon and determine how best to rank them sincerely.
   > 
   > This does bring up an interesting point, though, which is that
   > Smith//Condorcet[EM] seems to encourage voters to be less
   > differentiating of their favorite candidates, and then get
   > more and more particular of candidates as one moves down
   > the list.  At the bottom of the list, one is
   > encouraged to find the most minute differences and rank otherwise
   > identical candidates differently.
   > ---
   > Rob Lanphier
   > robla at eskimo.com
   > http://www.eskimo.com/~robla
  
   If Smith//Condorcet[EM] is used, voting alphabetically seems to be a
   very good possibility to achieve the same results as with what I 
   called "distributing the worst preferences random among the least
   favoured candidates". If the voters vote alphabetically, then each
   of the disliked candidates get a high defeat against the candidate 
   who is listed above him in alphabetical order.

The catastrophal consequences of alphabetical voting are one of the
reasons, why I don't like Smith//Condorcet[EM]. As Smith//Condorcet[EM]
urges the voters to give to every of the least favoured candidates a
different preference, Smith//Condorcet[EM] will increase the tendency
to vote alphabetically.
  
************************************************************************************

Steve Eppley wrote:
> Here's a variation of ranked ballots (which purists will rightly
> consider to be something other than ranked ballots).  The method
> might be named "BeatsAll//WeakMeansIndifferent//Condorcet".
> 
> The voters would be able to express two strengths of relative
> preference (plus indifference), not just a preference order.
> 
>    Example ballot:
>    A > B >> C > D=E >> F
> 
> The symbol '>' can be read as "is SOMEWHAT preferred to".
> The symbol '>>' can be read as "is STRONGLY preferred to".
> 
> Choices left unranked would be treated as if the voter had ranked
> them last and had used the stronger preference '>>' to separate them
> from those explicitly ranked.
> 
> In the pairwise tallying, '>' is initially treated the same as '>>',
> a full strength preference.  If there's a choice which beats all
> other choices pairwise, it wins.  (In other words, exaggeration of
> preference intensities won't defeat a choice which beats all others
> pairwise.)
> 
> If there's no choice which beats all others pairwise, the first
> (circular) tie-breaker treats the weak preference '>' the same as
> indifference '=' in a re-tally.
> 
>    Example:
>    23:  A >> B >> C
>    23:  A >> C >> B
>    10:  B >> A >> C
>    10:  B >> C >> A
>    34:  C  > B >> A          <--- these use the "weaker" preference
> 
>                prefer A more      prefer B more     prefer C more
>    A vs B            46                 54W
>    A vs C            56W                                  44
>    B vs C                               43                57W
> 
>    No choice beat all others.
>    Then the ballots are treated as:
>    23:  A >> B >> C
>    23:  A >> C >> B
>    10:  B >> A >> C
>    10:  B >> C >> A
>    34:  C  = B >> A          <--- the '>' becomes a '='
> 
>    and re-tallied:
>                prefer A more      prefer B more     prefer C more
>    A vs B            46                 54W
>    A vs C            56W                                  44
>    B vs C                               43W               23
>    B beats all others and is elected.
> 
> Though I've expressed the pairwise matrix in the reverse sense as
> "preferences for" instead of "preferences against", I'd suggest using
> Condorcet(EM)'s "against" to resolve a continued circular tie.
> 
> By expressing a "weaker" preference for the favorite over the
> "lesser evil", the 34 voters have chosen to protect (elect) their
> compromise choice.
> 
> The method seems very similar to the "fancy" iterative methods we
> discussed months ago, where the algorithm iteratively translates
> preferences to indifference when necessary to make a voter's
> compromise choice less beaten than the voter's greater evil.
> 
> It appears at first glance that the weaker preference '>' can't be
> used in offensive voter strategy.  The voter who uses it instead of
> the full strength '>>' would be making some less preferred choice
> less beaten than otherwise, with no reduction in the "beatenness" of
> a more preferred choice.  So the '>' appears to be useful only for
> defensive strategy.  It might be used unnecessarily, but its use
> wouldn't entail a misrepresentation of preference order.
> 
> I haven't had time to examine this method much.  At first glance,
> it appears to perform reasonably well on some of the standards and
> criteria some of us have been advocating.  And the distinction
> between the "weaker" and "stronger" preferences may be a reasonably
> compelling way to break circular ties.
> 
> On the other hand, it would be harder for the voters to vote than
> simple rankings, and the tally algorithm appears more complex to
> understand than Condorcet since there's an extra step.  And I
> haven't really looked for possible strategy problems and mandate
> interpretation problems.
> 
> ---Steve     (Steve Eppley    seppley at alumni.caltech.edu)

Hmmm, this sounds interesting. I want to make a similar
proposal:

The voters don't only make main preferences 1,2,3,4,5,6,.....,
they may also make sub-preferences 1a,1b,1c,2,3a,3b,4,5a,5b,5c,5d,6,...

I already said in my last e-mail, that Smith//Condorcet[EM]
motivates the voters to truncate upwards [i.e., the voter gives
his best preference to more than one candidate]. And Rob Lanphier
was friendly enough to spend some time to create an example to
demonstrate this.

Now, if e.g. Smith//Condorcet[EM] is used, then the problem, that
a voter would have to truncate upwards if he fears, that one of 
his most favoured candidates has his highest defeat against
another of his most favoured candidates and thus a less
favoured candidate wins, could be prevented as follows:

   If there is at least one candidate in the Smith set,
   who got a different main preference of the voter than another
   candidate of the Smith set, then only the main preferences are
   counted to determine the defeats of the candidates.
   The result would be the same as if the voter would have
   truncated upwards.

   If all candidates in the Smith set have got the same main 
   preference of the voter, than the sub-preferences are 
   counted to determine the defeats of the candidates.

This proposal would make the Condorcet criterion method
become a little bit more similar to a rating method.
But perhaps, the optimal election method is somewhere
between Steve Eppley, Mike Ossipoff and Mike Saari.      :-)


Markus Schulze (schulze at speedy.physik.tu-berlin.de)