[EM] I have to rescind my method (a better one exists)

[Kevin Venzke]

Hello,

I was afraid this might happen. Today, I found that the choice set
I defined fails Mono-raise. Example:

Situation 1:
A>>B, B>>C, C>>A, D>>C.
The set is {a,b,d}.
Situation 2:
A is raised on some ballots so that A>>D.
Now the set is {a,b,c,d}.
So if e.g. Random Candidate generates the ranking, A's probability of
election has decreased.

As it happens, Mono-raise can be satisfied if the choice set definition
is changed to "every candidate who has a majority-strength beatpath to
every candidate with a majority-strength beatpath to him," rather than
just requiring a beatpath to each candidate dominating him. With this
changed definition, the sets are {b} and {a,b,c,d}.

I thought that perhaps my old definition did a better job of satisfying
Later-no-harm, so I wrote a simulation to compare the two sets. I could
accept Mono-raise failures in exchange for better Later-no-harm compliance.
Unfortunately, not only does this more strict set not fail Mono-raise,
it has half or fewer the Later-no-harm failures of the first set.

Also unfortunately, I can't claim to have invented this set. It may have
been used on EM before. I think it is equivalent to Woodall's CDTT, the
"Condorcet doubly-augmented top tier," which is defined as the union of
all minimal nonempty sets where no candidate in the set is dominated by
any candidate outside the set.

So I have to rescind my method, and instead advocate electing that CDTT
member who appears highest in a ranking generated by a method satisfying
Later-no-harm.

My preference is CDTT with Random Ballot. Unless voters truncate extensively
(which they are not supposed to have much incentive to do in this method),
this should pick a decent winner. (Remember that if more than half of the
voters vote X over Y, Y normally can't win.) Also, I think randomness is
a deterrent to burial strategy which might otherwise be a problem.

Kevin Venzke



	

	
		
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