I've been trying to find methods that are maximally resistant to strategy, but I'm not sure how manipulability should be defined in the case of ties. My definition for non-tied elections is that a method, call it X, is manipulable in election eA if X elects a candidate (A), but there exists a way for the voters who prefer B to A to alter their ballots so that, in the post-altering election (eB), B wins. The manipulability of X for a certain number of voters and candidates is just the mean manipulability for X over every election. If A wins with certainty in eA, and B wins with certainty in eB, there's no problem; X's manipulability in election eA is 1. But if A wins in eA with positive probability (but not certainty), and/or B wins in eB with positive probability, but not certainty, then what should the manipulability measure be? Suppose A's win probability in eA is 0.6, and B's win probability in eA is 0.4; and in eB, B's win probability is 0.9, and A's win probability is 0.1. Furthermore, say it's possible for B>A strategists to manipulate eA and turn it into eB, but no other manipulation from eA is possible. Then I see three possible ways to quantify the manipulability, plus another pragmatic one: i.The manipulability is 1, because there still exists a way to manipulate eA in favor of some candidate so as to increase the win probability of that candidate at the expense of some other candidate with positive win probability in eA; ii. The manipulability is 0.6, because the manipulation only ever has an effect if A wins in eA, which happens with probability 0.6; if B wins, there's no need to do any manipulation, and none has any effect; iii. The manipulability is 0.9 - 0.6 = 0.3 because that's the benefit to B by executing the strategy. (iv. Whatever makes my optimization easier to do. Two of these measures require integer programming to optimize for, and one can be done by linear programming alone.) The first definition could be called "count the number of manipulable elections", the second "count the probability that the outcome can be manipulated", and the third "count the expected benefit to manipulators". What do you think would be the appropriate definition?

I think I rank these as 4>1>2>3. 4. However you do it, I think the results will be meaningful, but probably not clear-cut or intuitive enough for anyone to take them as the total truth. So it's most important just to get to at least one set of results. 1. If you view the outcomes as the probability distributions (and I suppose the B voters would have to perceive that they are) then even though B  voters may not benefit from the strategy, they still have the incentive to try to benefit. (But with ties I suppose B need not win outright in the post-strategy scenario. So the question is whether B's odds improve, or (different) some group of strategizing voters feel the mix of win odds in the new tied state is overall better than the old mix.)  2/3. These acknowledge that if there is less possibility to benefit then there should also be less incentive. I feel like this is a messy concept though. Intuitively I wonder, if there is any incentive at all to use the strategy, what do we suppose is the opposite incentive to stop them from using it? A voter's desire to be sincere seems like it falls outside of the scope of the simulation. And uncertainty about the other voters' ballots seems totally out of scope. Kevin Le mardi 24 novembre 2020 à 16:58:28 UTC−6, Kristofer Munsterhjelm <km_elmet@t-online.de> a écrit : >Suppose A's win probability in eA is 0.6, and B's win probability in eA >is 0.4; and in eB, B's win probability is 0.9, and A's win probability >is 0.1. Furthermore, say it's possible for B>A strategists to manipulate >eA and turn it into eB, but no other manipulation from eA is possible. >Then I see three possible ways to quantify the manipulability, plus >another pragmatic one: > >i.The manipulability is 1, because there still exists a way to >manipulate eA in favor of some candidate so as to increase the win >probability of that candidate at the expense of some other candidate >with positive win probability in eA; > >ii. The manipulability is 0.6, because the manipulation only ever has an >effect if A wins in eA, which happens with probability 0.6; if B wins, >there's no need to do any manipulation, and none has any effect; > >iii. The manipulability is 0.9 - 0.6 = 0.3 because that's the benefit to >B by executing the strategy. > >(iv. Whatever makes my optimization easier to do. Two of these measures >require integer programming to optimize for, and one can be done by >linear programming alone.) > >The first definition could be called "count the number of manipulable >elections", the second "count the probability that the outcome can be >manipulated", and the third "count the expected benefit to manipulators". > >What do you think would be the appropriate definition?