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