# [Election-Methods] Free withdrawal vs. agreed structure/rules (was: Smith + mono-add-top?)

Juho juho4880 at yahoo.co.uk
Tue Jan 1 12:17:06 PST 2008

```Steve wrote:
> MinMax + CandidateWithdrawal would be a very good method, thanks
to its simplicity, the incentive it would give candidates to try to
be the best compromise, and the full-bore competition it would
facilitate

This is an interesting approach to reduce the strategic problems in
Condorcet and Minmax. The vicious cycle in the given ABCD example is
also good in pointing out the potential problems/benefits in Diego's
method (I don't discuss that method much here, maybe another time,
the basic idea is very interesting at least).

You mentioned the risk of creating a vicious cycle (ABC) within one
faction, and the possibility to elect D when the vicious cycle
exists. This is typical to non-path-based methods (and non Smith set
compliant) like Minmax.

There are two explanations to the ABCD example. If A, B and C are
from one faction then one could assume that the loop among these
candidates is weaker than the preference of the ABC group over C. I
mean that even if the pairwise comparison matrix indicates that the
ABC loop preferences are stronger than the D related preferences
maybe e.g. the first four voters wanted to say something like
A>B>C>>>D etc. This need not be the case but this may be probable.

The other alternative explanation is that A, B and C are actually
from three different parties and there really is a strong cycle
between them. The actual votes in your ABCD example do not really
support this interpretation since A, B and C are clones in the sense
that they are always next to each others in the ballots. It is
however possible to get similar pairwise matrixes also with votes
where A, B and C are not clones.

This second explanation offers another approach also to the results.
Now the loop is (probably) an expression of a real strong cycle. It
is maybe not good to elect any of A, B and C but to elect the good
compromise candidate D instead. (Btw, I think winners should usually
come from the Smith set but not necessarily in all cases.)

Now back to your proposal. The candidate withdrawal option offers the
option to react to these two different situations in different ways
after the election. If A, B and C are from the same party we could
expect them to use the withdrawal option if that gives the victory
back to one's own party (that was actually also preferred by the
voters if we see A, B and C as one "group candidate").

The candidate withdrawal option is a strong mechanism that has both
benefits and problems (they have been discussed on this list).
Another interesting and strong mechanism that has similar
implications but that will be set up already before the election is
to allow the candidates/parties to declare the party relationships
among the candidates (also discussed on this list before). In this
case the method gives different results depending on the declared
relationships.

The structure could consist of different parties simply listing their
candidates, but to be complete the structure should maybe be seen as
a tree. The method could simply consider party (or subgroup) internal
defeats to be weaker than party external defeats. In the ABCD example
if A, B and C form a party one of them will win even if otherwise the
method would be Minmax. If the parties would be AB and CD then B
would win since its only defeat is party internal.

Note that if the tree is a complete binary tree then there will be no
groups of three and therefore there can be no loops of three either.
This eliminates quite efficiently the strategic voting risks in this
method. One could also form coalitions if one feels there is a risk
of successful strategic attacks. This feature allows the method to
adjust to different more or less strategic elections (and maybe
reduce the fears that strategic voting would be a real threat in
Condorcet methods).

This method to some extent takes power away from the voters by
declaring beforehand the relationships between the candidates. On the
other hand this is partially a good thing since this adds information
that the voters get about the commitments of the candidates.
Declaring the relationships before the election may be better than
doing party structure and strategy related withdrawals after the
election (voter control vs. party/candidate control).

One could also develop systems where candidates may form whatever
personal relationships instead of being bound to a tree like
structure, but I'll skip this option now.

This method is of course not Condorcet compliant in cases where the
agreed party borders weaken the party internal defeats. But it is
Condorcet compliant between the parties.

I tend to think that strategic threat scenarios are inevitable in all
Condorcet methods. If that is not considered to be a meaningful risk
(in the environment in question) then using plain Condorcet methods
could be good enough. If the risks are considered real and a threat
then there may be a need for strong enough countermeasures that
eliminate these problems. The up to binary tree level defensive
alliance system would be one efficient method to cut down many of the
threats. It may not be possible to agree the structure down to binary
three level between the parties/candidates but at least they had the
option.

What do you think of this? Is it a good approach to use this type of
means to reduce the strategic threats in Condorcet based systems? Can
they offer an adjustable level defence mechanism against strategic
environments/elections?

Juho

On Jan 1, 2008, at 17:24 , Steve Eppley wrote:

> I think the method Diego Santos is considering can elect outside the
> Smith set (a.k.a. top cycle), depending on the tie-breaker.  Here's an
> example with 21 voters and 4 candidates:
>
>     4    4    4    3    3    3
>    ---  ---  ---  ---  ---  ---
>     A    B    C    D    D    D
>     B    C    A    A    B    C
>     C    A    B    B    C    A
>     D    D    D    C    A    B
>
> {A,B,C} is a set of clones in a "vicious" cycle. (By vicious, I
> mean all
> margins in the cycle are large.  I think Mike Ossipoff may have been
> first to use the term, many years ago.)  What makes this scenario very
> rare (assuming many voters) is that the margins in the vicious
> cycle are
> equal:
>
>    A over B by (4+4+3+3) - (4+3) = 7
>    B over C by (4+4+3+3) - (4+3) = 7
>    C over A by (4+4+3+3) - (4+3) = 7
>
> The Smith set is {A,B,C}.  Can D win?  If I understand Diego's
> definition, D is not eliminated since the margin in D's pairwise
> defeats
> is smallest (12 - 9 = 3).  I think A and B and C are also not
> eliminated
> since there's a tie in their cycle's margins.  Thus the set of
> non-eliminated candidates is {A,B,C,D}.  Among {A,B,C,D} there is no
> Condorcet winner.  So, a tiebreaker must select from {A,B,C,D}.  If
> the
> tiebreaker can select outside the Smith set, D can be elected.
> Typical
> tiebreakers (Random, Random Voter's Ballot, Chairperson's Choice) can
> select outside the Smith set.
>
> D would win given plain MinMax even if the margins in the vicious
> cycle
> are unequal.  Thus, given plain MinMax the elite political actors
> might
> limit competition, to eliminate the chance of a vicious cycle among
> their faction.  A consequence of limiting competition is increased
> corruption, for instance by the use of primary elections which require
> large amounts of money to win nomination.  That's unfortunate, since
> MinMax might be relatively simple to sell: "Elect the candidate that
> minimizes the number of voters who prefer someone else."  (I believe
> Diego's method is too complicated to be adopted in public elections
> for
> the foreseeable future.)  However, MinMax + CandidateWithdrawal
> would be
> a very good method, thanks to its simplicity, the incentive it would
> give candidates to try to be the best compromise, and the full-bore
> competition it would facilitate.  Even Instant Runoff +
> CandidateWithdrawal would be a decent method, and considering the
> progress Instant Runoff has been making, it makes sense to propose
> patching it with CandidateWithdrawal.  Please take some time to do
> that.
>
> --Steve Eppley
> --------------------------
> Diego Santos wrote:
>> Happy new year to all!
>>
>> Perhaps my previous definition was not enough clear, for the possible
>> confusion between "potential winner" and "winner" on its final.
>> Then, I
>> reformulated it:
>>
>> "Some candidate X is eliminated if a) exists Y that beats X and b)
>> the
>> margin of Y against X is greater than the greatest margin of another
>> candidate against Y. The winner is the Condorcet winner among non-
>> eliminated
>> candidates".
>>
>> An example (from http://www.mcdougall.org.uk/VM/ISSUE6/P4.HTM):
>>
>> 5:a>d>c>b
>> 5:b>c>a>d
>> 8:c>a>b>d
>> 4:d>a>b>c
>> 8:d>b>c>a
>>
>> Notation:
>> Candidate X(minimax score of X): Candidate Y(margin of Y against
>> X, minmax
>> score of Y):
>>
>> a(12): c(12,4)           eliminated
>> b(4): a(4,12), d(4,6)
>> c(4): b(4,4), d(4,6)
>> d(6): a(6,12)
>>
>> d beats either b and c, then d is elected.
>>
>> Another example (from Markus' paper):
>>
>> 3:a>d>e>b>c>f
>> 3:b>f>e>c>d>a
>> 4:c>a>b>f>d>e
>> 1:d>b>c>e>f>a
>> 4:d>e>f>a>b>c
>> 2:e>c>b>d>f>a
>> 2:f>a>c>d>b>e
>>
>> a(5): c(1,5), d(1,3), e(1,9), f(5,7)
>> b(7): a(7,5), d(1,3)                           eliminated
>> c(5): b(3,7), e(5,9)
>> d(3): c(3,5)
>> e(9): b(1,7), d(9,3)                           eliminated
>> f(7): b(7,7), c(1,5), d(1,3), e(1,9)
>>
>> c beats a, d and f, then c is elected.
>> ________________________________
>> Diego  Santos
>>
> ----
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