In the last days, I thought about some form of strategy-proofness like the following:
Criterion: Suppose that, with all voters voting sincerely, the method elects A, but some voter prefers B to A and can get B elected by voting insincerely. Then those voters not preferring B to A must have a way of voting which ensures that A or some option C gets elected which the first voter ranks *below* A, so that either the sincere result can be guaranteed or the incentive to vote insincerely is removed. However, I then came up with the following, very simple 3-by-3-example which seems to render those thoughts ridiculous... Problematic Example: Sincere preferences Voter 1: A>B>C Voter 2: B>C=A Voter 3: C>A>B If voter 2 would have B>C>A instead, all three options would necessarily get a winning probability of 1/3 if only the method satisfies neutrality and anonymity. Hence in the sincere case where voter 2 has C=A instead of C>A, A must get a winning probability larger than 1/3, while C must get a winning probability below 1/3, if only the method satisfies some weak version of monotonicity. Almost all methods would elect A with probability 1 here, I guess. This means that voter 2 can improve the chances of its favourite B by voting insincerely C>A instead of sincerely C=A, a strategy which has been called "burying" here. Now the only voter who does not profit from this strategy is voter 1 since also C's (voter 3's favourite) chances have been improved. The only thing voter 1 can do about this is to vote B=C or even C>B instead of B>C, thus producing a situation where C gets elected. However, this does not remove voter 2's incentive to vote insincerely because C is no worse to him than the original winner A... What I think is noteworthy here is that this example seems to affect *all* acceptable methods whatsoever (more precisely, those which satisfy neutrality, anonymity, and some small amount of monotonicity and decisiveness)! I sincerely hope I missed some essential point in that example... Can anyone tell what this would look like with Approval (I mean, what is a sincere Approval vote in the first place?)? Jobst ---- Election-methods mailing list - see http://electorama.com/em for list info
