Alex pointed out:
If the players are individual voters than any result with a margin greater than 1 vote is a Nash equilibrium. That isn't a very useful result, however. I reply: Yes, that seemed a bit unfair to me. Surely not the intent of the idea of the Nash equilibrium. In my example for wv & margins, the players were the 3 factions. It seems that the original Nash equilibrium definition is for when there are a few players each with different interest and strategies. I suggest that that intent is best served, for voting systems, with a different wording. No doubt someone has already written such a definition, but until I find out what he called it, I'll call it Many Voter Equilibrium: A strategies-configuration & outcome from which no set of voters whose utilities and strategy are the same can gain by changing their strategy. [end of definition] Again, I make no claim that that's original. I understand that game theorists have defined many kinds of equilibrium. Maybe that's one of them. Maybe some of the others will turn out to have relevance to voting systems too. >From now on, when I say equilibrium, I mean many-voter equilibrium. Alex continued: It could still turn out that in a 3-way race all Nash equilibria elect the CW. However, I haven't proven it. I reply: I've been wondering the same thing. I read that Riker proved that when voters have complete information about eachother's preferences, and act to optimize their immediate outcome, the sincere CW will win, no matter what (nonprobabilistic?) voting system is used. If so, does that mean that the sincere CW is the only candidate who can win at equilibrium? This subject is new to me, but it seems important for evaluating voting systems. How about this classification of voting systems' sincerity: (Again, by equilibrium, I mean many-voters equilibrium) If there's a sincere CW, and voters have complete information about eachothers' preferences then: A voting system that has situations where the only equilibria have order-reversal is a falsifying method. Any method that isn't falsifying is nonfalsifying. A voting system in which there are always equilibria in which no one reverses a preference or votes a less-liked candidate equal to a candidate he's voted for is an expressive method. A voting system in which, if falsification of preferences is ruled out as a strategy, there are always eqilibria in which everyone sincerely ranks all of the candidates is a conditionally completely expressive method. [end of definitions] Since, as I said, this subject is new to me, all I can say now is that Plurality, IRV, RP(m), & BeatpathWinner(m) are falsifying methods; and Approval, PC(wv), BeatpathWinner(wv), CSSD(wv), & RP(wv) are nonfalsifying methods. The wv Condorcet versions are expressive and conditionally completely expressive. Of course, with anything new (to me) there's always the possibility of an erroneous statement, or something overlooked when writing a definition. But right now, the definitions seem good, and the statements seem correct. Mike Ossipoff Alex _________________________________________________________________ Chat with friends online, try MSN Messenger: http://messenger.msn.com
