Steve Eppley SEppley at alumni.caltech.edu
Tue Jan 1 07:24:06 PST 2008

```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
>

```