Hi Warren, I'm eagerly awaiting your reply on this message.
Rob On Sat, 2005-08-13 at 14:12 -0700, Rob Lanphier wrote: > Hi Warren, > > I'm not following your theorem. Can you give an example of what you are > referring to, showing a set of sincere preferences, followed by a set of > tactical ballots which illustrate your point? > > E.g.: > Sincere preferences: > Group 1 - 18 votes: A>B>C > Group 2 - 18 votes: A>C>B > Group 3 - 16 votes: B>C>A > Group 4 - 16 votes: B>A>C > Group 5 - 16 votes: C>B>A > Group 6 - 16 votes: C>A>B > > Tactical ballots: > Group 1 - 18 votes: A>B>C #sincere > Group 2 - 18 votes: A>C>B #sincere > Group 3 - 16 votes: C>B>A #B>C>A voters that bury B to help C win > Group 4 - 16 votes: B>A>C #sincere > Group 5 - 16 votes: C>B>A #sincere > Group 6 - 16 votes: C>A>B #sincere > > This particular example isn't a good one, since Group 3's strategy > doesn't affect the outcome of the election (A wins no matter what). I > hoping to see an example that's more illustrative of the point you are > making. > > Thanks > Rob > > On Sat, 2005-08-13 at 09:39 -0400, Warren Smith wrote: > > On the probability that insincerely ranking the two frontrunners max and > > min, is > > optimal voter-strategy in a Condorcet election. > > ----------Warren D. Smith Aug > > 2005---------------------------------------------- > > > > MATHEMATICAL MODEL: 3-candidate V-voter Condorcet elections > > with random voters (all 3!=6 permutations=votes equally likely). > > > > QUESTION: Is there a subset of identically-voting voters, who, by changing > > their vote > > to rank the two "perceived frontrunners" max and min ("betraying" their > > true favorite "third party" candidate) can make their least-worst > > frontrunner win > > (whereas, their true favorite cannot be made to win no matter what they do)? > > > > THEOREM: The answer to the above question is "yes" with probability > > at least 25% = 1/4 in the V=large limit. > > > > PROOF SKETCH: > > 1. The probability of a condorcet cycle is exactly 25% if we > > ignore situations that include exact ties. > > (Since: Assume A>B wlog. Then B>C with prob=50%, then with prob=50% C>A.) > > > > 2. The probability in the V=large limit tends to 1 that all the pairwise > > victory margins are of order approximately sqrt(V), and that all the 6 > > kinds of voters > > occur with counts approximately V/6 each, i.e. much larger than sqrt(V). > > > > 3. So assume there is a condorcet cycle, wlog it is A>B>C>A, and wlog the > > smallest > > margin of victory is C>A so that A is the winner (according to all the usual > > Condorcet methods, since they all are the same in the 3-candidate case). > > > > 4. Choose a subset, of cardinality of order sqrt(V), of the voters of type > > "B>C>A". > > (More precisely, we must choose the cardinality*2 to lie above the > > previously-mentioned victory margin.) > > If they betray their favorite B by insincerely switching to "C>B>A", > > then C becomes the Condorcet winner, > > which from their point of view is a better outcome. > > Q.E.D. > > > > STRENGTHENING. > > Note our "B>C>A"-type voter subset can argue that obviously, nothing they > > can do > > will elect B, since when they rank B top honestly that fails to do it. > > Therefore, > > their only chance for an improvement is to go for electing C. And the only > > way they can try is to raise C in the rankings. As we've seen, this > > reasoning > > yields success for them, 25% of the time. However, given their > > preconception > > that B has essentially no chance of victory, it actually makes sense > > for them to rank C top 100%, not 25%, of the time, even though we know > > this will only be successful for them with probability 25%. Because given > > their belief B has no chance, this cannot hurt them, and they know there is > > a > > 25% chance it will help them. So we conclude from this that in fact, the > > "betray B" > > strategy is plausibly better than honesty, 100% of the time. > > > > Summary. > > Adam Tarr in previous posts had questioned my claim that this this > > plurality-like voter > > strategy could ever be optimal in Condorcet elections. He said > > "I can't easily imagine a scenario where it is useful in Condorcet." > > He demanded that I "make some simulations that demonstrate > > this, or at least show some examples." He apparently had not noticed the > > fact I had already exhibited an example on the CRV web site, > > http://math.temple.edu/~wds/crv/RangeVoting.html > > "Why range is better than Condorcet" discussion, perhaps because said > > example was in the > > subpage http://math.temple.edu/~wds/crv/IncentToExagg.html. > > But now the present discussion shows that Tarr was maximally wrong: this > > strategy > > is ALWAYS the right move. > > > > Given that this is the case, we now can take it to be 100% certain that > > Condorcet voting methods will lead to 2-party domination, just like the > > flawed > > plurality system those methods were supposed to "fix", and just like > > experiemntlly > > is true with IRV. So anybody who is interested in third parties ever having > > a chance, would be advised NOT to foolishly advocate either IRV or > > Condorcet, > > but insetad would be advised to advocate RANGE VOTING (which experimentally > > favors third parties far more than either plurality or approval, > > incidentally, > > see the CRV web site). > > -wds > > ---- > > Election-methods mailing list - see http://electorama.com/em for list info > > ---- > Election-methods mailing list - see http://electorama.com/em for list info ---- Election-methods mailing list - see http://electorama.com/em for list info