[EM] Full Majority as Part of Strong Defeat

[Forest Simmons]

Jobst, the more I think about it, the more I like your idea (influenced by 
Kevin) of requiring full majorities for strong defeat.

I don't think that we lose any of the basic properties, and it solves 
Kevin's 49C, 24B, 27A>B problem without the additional randomness that I 
was beginning to accept as inevitable.

The method can be described briefly as follows:

List the candidates from top to bottom in order of decreasing total 
approval.

Eliminate every candidate that is beaten pairwise on more than half of the 
ballots by some candidate higher up on the approval list.

If more than one candidate remains, then resolve the fundamental ambiguity 
by democratically giving each candidate a fair chance: the remaining 
candidate that is ranked highest on a randomly drawn ballot wins.

In Kevin's example the sincere ballots are

49 C>>A=B
24 B>>A>C
27 A>B>>C

The approval order (from top to bottom) is B>C>A .

Candidate C is eliminated because B beats C majority pairwise, as well as 
in approval.

Random ballot between A and B gives respective probabilities of 27/51 and 
24/51.

If the B supporters truncate A, then the approval order is unchanged, and 
C is still the only candidate beaten from above by a full majority, since 
the C>A defeat is only on 49 percent of the ballots.

The probabilities remain the same, so the truncation gives no advantage to 
the B supporters.


Note that if the A supporters didn't like B that much the sincere ballots 
would be

49 C
24 B>>A>C
27 A>>B>C

In this case the approval order would be C>A>B, and no candidate would be 
eliminated, since the two full majority defeats go uphill (against 
approval).

If the B supporters could anticipate this, they might lower their approval 
cutoffs:

49 C
24 B>A>>C
27 A>>B>C

Then A would be both the approval winner and the ballot CW.

The approvals order would be A>C>B.

Only C would be eliminated by strong defeat, and the probabilities would 
be the same as in the first example.

It is to the advantage of the B faction to (somewhat insincerely) approve 
candidate A.

So the method doesn't completely do away with strategy.

Note that A and B are a majority clone set, and that as long as they give 
a reasonable amount of support to each other they both have a fair chance 
of winning.

But their combined victory over C is by such a slim margin, that if they 
do not cooperate, then C also gets a share of the probability.

In other words, a loose clone set doesn't have as much force as a tight 
clone set.

Forest