Date: Wed, 2 Feb 2005 14:45:57 -0800 (PST) From: Rob LeGrand <[EMAIL PROTECTED]> Subject: [EM] Re: simulating an Approval campaign/election
Forest wrote:Actually, DSV with Strategy A can sometimes converge to a stable equilibrium even when there is no Condorcet Winner:
4900 C 2400 B 2700 A>B
Rob replied:
You're right. I forgot to disclose my assumption of strict preferences. A better way to say what I meant: When for every candidate X there is another candidate Y that is strictly preferred by a majority of voters to candidate X, there is no equilibrium under Approval when every voter uses strategy A. Also, I forgot to mention that the results of my example election under the different DSV modes assume that all voters use strategy A.
[In my opinion this is unfortunate, since it would be better to have B and C win with about equal probability in this example.]
Why, do you think? Candidate A can't win under Approval no matter what (unless some voters are completely irrational), so the A-first voters have strong motivation to approve B. Plus, a majority strictly prefers B to C.
That's the way I thought about it until Kevin pointed out that making B the sure winner in this case would give high incentive for truncation among B supporters if their sincere preference happened to be B>A>C, thus demoting A from Condorcet Winner and promoting B to (wv) cycle winner.
Some folks solve this dilemma by going with the margins winner C for the truncated case.
I believe that a coin toss between B and C would be better in this case.
I wonder how good it would be in general to decide between the margins winner and the wv winner with a coin toss.
Here's another solution: do a symmetric completion of the ballots before applying your DSV. There would still be a cycle, and with strict orderings your DSV would cycle among the candidates, so that all had a chance at winning.
What would be their winning probabilities?
For a refinement of that idea use a refined symmetric completion:
4900 C becomes
2450 C>>A>B 2450 C>>B>A
Then interpret the first subfaction to mean that C, A, and B, respectively, have relative utilities of 3, 1, and 0.
How would this change the probabilities?
In some other election where a faction voted
1000 A=B>C
the refined symmetric completion would be
500 A>B>>C 500 B>C>>A
where the first of these subfactions would impute relative utilities of 3, 2, and 0 to candidates A, B, and C, respectively.
For another approach, using "lotteries," see my recent posting under that title.
Forest ---- Election-methods mailing list - see http://electorama.com/em for list info
