[EM] Re: chain climbing methods

[Forest Simmons]

My email server was down for a while, but I'm glad to see this message 
from Jobst.

I like the TACC option the best, but I would like to suggest the following 
variation (which I will call TACC+ if you don't mind):

After finding the (deterministic) TACC winner, create a lottery based on 
random ballot among the set of all candidates that have at least as much 
approval as the TACC winner.

This method is definitely monotone:  If the TACC winner or anyone above 
him in approval is the only candidate to move up in the rankings on the 
ballots, then the TACC winner remains the TACC winner (assuming that 
approval counts don't change), so the set on which the lottery is based 
remains the same, and only the TACC winner has a chance of increasing his 
winning probability in the lottery.

If the TACC winner A has an approval count increases, advancing him up the 
approval ladder, a candidate lower on this ladder (but not below A's 
original position) now has a chance of taking his place as the TACC 
winner, but that's OK, since A has greater approval than the new TACC 
winner, so is still included in the new lottery.

This new lottery is based on a subset of the original lottery members 
(those that are above the new TACC winner were all above A's original 
position), and the number of ballots on which A is first is the only one 
that might have increased, so A's winning probability in this new lottery 
is at least as good as in the old one.

Can you see any holes in that sketch of an argument?


Now here's another lottery method I favor because, although it tends to 
spread the probability around more promiscuously, it has nice properties:

First find the highest approval score alpha such that no candidate with 
approval less than alpha beats (pairwise) any candidate with an approval 
score of alpha or higher.

Then do random ballot relative to the candidates that have approval scores 
greater than or equal to alpha.


That's it.  Let's call it Viable Candidate Random Ballot (VCRB).

This method is monotone and clone proof.

It also has this nice property:

If the candidates with approval less than alpha are eliminated, and the 
method is run again, then it will produce the exact same winning 
probabilities as before (as long as approval scores are not adjusted to 
reflect the new realities).

What should we call this?  Independence from zero probability candidates? 
This would work if we assume that each candidate is ranked first on at 
least one ballot.

Or we could call it independence from non-viable candidates.

But the best thing about this method (VCRB) is that it seems to 
consistently punish insincere order reversals.

It handles Kevin's 49C, 24B(sincere>>A), 27A>B>>C, example quite nicely by 
giving the same lottery .49C+.24B+.27A for both the sincere and the 
insincere cases.  In neither case does any candidate have any approval 
below alpha=27.

But I'm afraid that neither TACC nor my modification TACC+ would dissuade 
the B voters from their truncation.

Still, TACC+, with its greater frugality in doling out probability might 
be better than VCRB for most applications.

Incidently, it was fun proving VCRB's monotonicity.  When I have more time 
I will write it up, but you might enjoy doing it yourself.

Forest