Jobst Heitzig wrote: > > 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 > [...] > > 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?)?
You would need information about utilities to predict how Approval voters should behave, but if you assume that voters 1 and 3 have utilities of 1.0, 0.5, and 0.0 for the three choices, and voter 2 has 1.0, 0.0, 0.0, and ties are resolved randomly, then: The utility of bullet voting is: Voter 1: 1/3 + .5 * 1/3 = 1/2 Voter 2: 1/3 + 0 = 1/3 Voter 3: 1/3 + .5 * 1/3 = 1/2 If only voter 1 approves of A & B, B wins so his utility is still 1/2 (but voter 2's is 1.0 and voter 3's is 0.0). If only voter 2 approves of any two candidates, the 2nd candidate wins, so his utility is 0.0 (but the beneficiary's is 1.0). If only voter 3 approves of C & A, A wins so his utility is still 1/2 (but voter 1's is 1.0). If both voters 1 & 3 approve of 2 candidates,the result is a tie between A and B, so the utility to voter 1 is 0.75, and the utility to voter 3 is only 0.25. It looks like voter 1 should approve of two candidates, since he won't be worse off (and gets a more stable outcome, since least-favorite C is guaranteed to lose). In addition, this gives voter 3 strong incentive to approve two candidates as well (to increase utility from 0.0 to 0.25). I can't find any incentive for voter 2 to approve of a 2nd candidate. So the likely outcome should be: Voter 1: AB Voter 2: B Voter 3: CA Result: Tie between A and B. Of course if the utility of a 2nd choice is something other than 0.5, strategies could change. Anyone else see anything different? Bart ---- Election-methods mailing list - see http://electorama.com/em for list info
