[EM] A method satisfying Minimal Defense and much Later-no-harm

[Kevin Venzke]

Hello,

I have been trying to design a method that satisfies Minimal Defense
and as much Later-no-harm as possible. I believe the method I'll
describe in this message is as close as I can get.

A method with similar properties is MinMax (Pairwise Opposition),
which strictly satisfies Later-no-harm, but fails Minimal Defense
when there are more than three candidates. My new method also
satisfies both criteria when there are only three candidates.

First let me define some terms:
Candidate A "dominates" candidate B if more than half of the voters
rank A above B.
Candidate A has a "majority-strength beatpath" to candidate B if
A dominates a candidate who dominates B, or who dominates someone
else who dominates B, etc.

"Minimal Defense" says that if more than half of the voters rank
A above B, and don't rank B above anybody, then B must be elected
with 0% probability. This implies Mike Ossipoff's SDSC.

When only pairwise contests are considered, we have to strengthen 
Minimal Defense to say that if some A dominates B, and B dominates 
nobody, then B must be elected with 0% probability.

"Later-no-harm" says that adding another preference to a ballot
(i.e., modifying a ballot so that a candidate previously ranked
above no one is now ranked above the other candidates who had been
ranked above no one, but still strictly lower than the same
candidates as before) must not decrease the probability of winning
of any candidate ranked above this new preference.

When only pairwise contests are considered, it's impossible to
determine the preference order of individual ballots, so we have
to strengthen Later-no-harm to say that increasing the number of
votes for A over B must not reduce the probability of election 
of any candidate except for B.

Now I'll define the method. First, determine a ranking of the
candidates using a method that satisfies Later-no-harm. You
could use Random Candidate, Random Ballot, FPP, IRV, Woodall's
DSC, or the MMPO method mentioned above. DSC probably has the
best properties (alone, it satisfies Clone Independence, Mono-
raise, Mono-add-top, and Participation) but is hard to count.
MMPO only needs the pairwise matrix, but it can be indecisive.
The simplest way to see the properties of the method would be
Random Candidate or Random Ballot, though.

Elect the highest-ranked candidate who has a majority-strength
beatpath to any and all candidates which dominate him. The set of
all such candidates is the interesting part of the method.

(By the way, you shouldn't eliminate the candidates not in the
set and then try to use a LNHarm ranking method, since that will
probably needlessly fail Mono-raise. You don't want the candidate
ranking to be influenced by which candidates are in the choice
set.)

When the winner always comes from this set, then Minimal Defense
is guaranteed, since a "Minimal Defense loser" will be dominated
by somebody, but won't possess any majority-strength beatpaths to
anyone.

When there are only 3 candidates, Later-no-harm is guaranteed:
Suppose your preference order is A>B>C and you alter your vote
from just A to A>B>C. The only way this can alter the set is if
you cause B to dominate C. Only two changes could result: C moves 
out of the choice set; or A and B both move into the choice set.

With more than 3 candidates, Later-no-harm can't be guaranteed
anymore, since new preferences could give other candidates (liked
less than A or B) majority-strength beatpaths which move them
back into the choice set. Here is an example, where ">>" means
"dominates":

Situation 1:
C>>D, D>>B. A isn't involved in any dominations.
The members of my set are {a,c}.
Situation 2:
You change your vote from A to A>B>C>D.
The domination B>>C is created.
Now the members of my set are {a,b,c,d}.
So adding B>>C has potentially hurt A.

Here's an example showing why I don't think Minimal Defense and
Later-no-harm can be satisfied simultaneously with 4+ candidates:

Situation 1:
A>>B, B>>C, C>>A, A>>D
The members of my set are {a,b,c}. D is a Minimal Defense loser.
Situation 2:
Add in D>>B.
Now the set members are {a,b,c,d}. But potentially this has reduced
the odds of winning of candidates other than B, so Later-no-harm
is failed.

I want to emphasize that a LNHarm failure can only occur when there
are majority-strength cycles. Cycles created by truncated ballots
can't create these. There have to actually be cyclic preferences
in the electorate.

My hope is that this degree of LNHarm failure is small enough to
permit the voters to ignore the possibility that additional
preferences could harm earlier preferences. 

Minimal Defense essentially guarantees that a majority of the
voters won't have to rank their compromise choice insincerely
high in order to defeat someone they prefer him to. In other words,
if the race is essentially between two strong candidates, then the
entrance of weaker candidates, and the willingness of voters to
rank these weaker candidates high, shouldn't confuse the method
into electing the wrong one of the two strong candidates.

I feel these two criteria together make for an impressive pairing
of guarantees. They should reduce voters' incentive to truncate
and uprank compromises, in particular, and as a result this should
reduce disincentive for candidates to enter the race.

I appreciate any thoughts or criticism.

Kevin Venzke



	

	
		
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