>> --"Nash equilibria" are an attempt to salvage game theory in N-player >> games with N>2. >> But it works badly for voting purposes. >> My usual example is, suppose everybody realizes Adolf Hitler is the >> worst candidate but still (idiotically) everybody votes for Hitler, >> who wins. >> OK, this election is a Nash equilibrium representing, in the sense of >> Nash equilibria, >> "best voting strategy" for all. > > Huh? > > If everyone else follows the "vote for Hitler" strategy, your optimal > strategy isn't vote for Hitler. (Well assuming that the secret ballot > hasn't been compromised). > > You might as well vote for someone else.
--A game player situation is a "Nash equilibrium" if no player can get more reward by changing their strategy. In the all-vote-for-Hitler situation, no player can alter the election result by changing their vote. Therefore this is a Nash equilibrium. Is this "best strategy"? Well, only in the sense that Nash-advocates proclaim that in a Nash equilibrium situation, each player's strategy is "optimal"! The point is, as you yourself have just observed, this whole conception often is silly and useless when applied to voting. > In most real elections, you don't know the exact way the others are > going to vote. If all the other voters were using "vote for Hitler > with a 99% probability and a random candidate otherwise", then your > optimal vote is to vote for your favourite. --true. This actually sounds like a good way to perturb the Nash equilibrium notion/definition to make it become more sensible than the official definition. So the new improved Raphfrk+Nash notion would be, assume each player will play whatever strategy they select, or with probability epsilon they play a random strategy. Now we only have equilibrium if no player can improve their expected reward, and this includes improvements by very tiny amounts proportional to epsilon^4 or whatever. This would get rid of stupid equilibria like "all vote for worst." Here's another nasty Nash equilibrium which still applies for the Raphfrk-Nash version: In a plurality election consider say, "all vote for Gore or Bush about 50-50" but a higher reward would come if Nader won. In this scenario I guess you cannot improve expected reward, and in fact will worsen it, by switching your vote to Nader. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step) and math.temple.edu/~wds/homepage/works.html ---- Election-Methods mailing list - see http://electorama.com/em for list info
