[Election-Methods] Smith + mono-add-top?

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 

   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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