Re: [EM] Strategy free method proposal
On 12/17/05, Simmons, Forest [EMAIL PROTECTED] wrote: Rob, in my experience typically when there is a Condorcet Cycle there is no Approval Strategy A style equilibrium of the kind you posit in your second message on this topic.It had been under the understanding that there will always be at least one equilibrium, but that iterative limit-seeking programs don't always find it. Maybe I misinterpreted something someone else said. I haven't yet written a program to test it. However, the method you propose in your earlier message might indeed approach an approval strategy A configuration as a limit without this limit configuration itself being an equilibrium configuration. If it does converge on a limit that is not an actual nash equilibrium, I assume it would be creating Range ballots in the end, not having converged all the ballots to be Approval ballots. That is not ideal, but not so bad either. The problem I have with Range ballots is they are not stratigically optimal in the last round of voting (from the point of view of each voter), and having the agent vote in a way that is not strategically optimal for the voter it represents makes this a bit less pure. This is analogous to the fact that a function with a jump discontinuity will approach a different value then the value of the function. * Got it. Your example, of course, converges toward a jump discontinuity (what I was calling a cusp). Now certainly there will be conditions like this if there are true ties, which should be less and less common as the number of voters gets larger. If these cases happen that rarely, I don't see it as a problem. However, if they are relatively common (like Condorcet cycles), then this technique is nowhere near as interesting. I have no way of knowing which is the case, without writing something to test it. Anyway, I think your first approach is worth exploring further.I'm sure there is something valuable there to be learned. Thanks-rob election-methods mailing list - see http://electorama.com/em for list info
[EM] [ER] FBC-complying Margins-like method (?)
Kevin, Warren, other FBC freaks, I've recently had an idea for a FBC-complying Margins method. Voters rank the candidates, equal-ranking and truncation allowed. (1) Make pairwise comparisons. Treating pairwise defeats by margins that are smaller than or equal to the number of ballots on which both candidates are ranked equal-top as pairwise equalities, eliminate candidates that are not in the smallest non-empty set of candidates that are pairwise undefeated by any outside-the-set candidates. (2)If more than one candidate remains, drop eliminated candidates from the ballots and then delete ballots that make no ranking distinction between remaining candidates, and repeat step 1. (3) Repeat steps 1 and 2 as many times as possible. (If at any stage only one candidate remains then that candidate is the winner.) (4) If after step 3 more than one candidate remains, then with ballots that rank both the candidates in a pairwise comparison equal-top used to modify the defeat margins by counting as whole single votes for the pairwise loser (so that some margins can be negative, but not so that any pairwise defeats can be reversed); change the pairwise defeat by a smallest margin to an equality; and as in step 1 again eliminate candidates that are not in the smallest non-empty set of candidates that are pairwise undefeated by any outside-the-set candidates. (5) If more than one candidate remains, then again drop eliminated candidates from the ballots and then delete ballots that make no ranking distinction between remaining candidates. (6) Keep repeating steps 4 and 5 until only one candidate remains. In common with MDD,ER-Bucklin(whole) I think it meets Majority for Solid Coalitions and Condorcet(Gross). But unlike that method, it meets Independence from Irrelevant Ballots. Chris Benham election-methods mailing list - see http://electorama.com/em for list info
Re: [EM] [ER] FBC-complying Margins-like method (?)
Kevin, Yes I am sure you are right, thanks. Probably then I'll stick with MDD,ER-Bucklin(whole) as my favourite FBC method. Chris Benham Kevin Venzke wrote: Chris, --- Chris Benham [EMAIL PROTECTED] a écrit : Kevin, Warren, other FBC freaks, I've recently had an idea for a FBC-complying Margins method. Voters rank the candidates, equal-ranking and truncation allowed. (1) Make pairwise comparisons. Treating pairwise defeats by margins that are smaller than or equal to the number of ballots on which both candidates are ranked equal-top as pairwise equalities, eliminate candidates that are not in the smallest non-empty set of candidates that are pairwise undefeated by any outside-the-set candidates. (2)If more than one candidate remains, drop eliminated candidates from the ballots and then delete ballots that make no ranking distinction between remaining candidates, and repeat step 1. There are two reasons why I don't believe this can work. 1. You're using a beatpath concept. Although you're replacing certain wins with pairwise ties, it could be that a pairwise tie between X and Y is what causes them to be excluded from the top tier. Replacing wins with ties only helps to satisfy FBC when it's clear that a tie between X and Y is at least as good for them as one of them beating the other. 2. You're eliminating candidates and recalculating. I think all you can afford to do is disqualify candidates without recalculating anything. Elimination makes it difficult to foresee what a specific vote is capable of doing across multiple rounds. It's much the same issue as Raynaud or Nanson failing monotonicity. Kevin Venzke election-methods mailing list - see http://electorama.com/em for list info
