Arrow’s Theorem doesn’t say anything about strategic voting. The only reasonable non-strategic voting system I know of is random ballot (pick a random voter; they decide who wins). I’m currently trying to figure out a voting system that is based on finding the Nash equilibrium (which may be mixed) of approval voting, and this system might also be strategy-free.
When I said linear combination of utility functions, I meant that you fix the scaling factors initially and don’t change them. You could make all of them 1, for example. Your voting system (described in the last paragraph) is a combination of range voting and IRV. If everyone range votes so that their favorite gets 1 and everyone else gets −1, then it’s identical to IRV, and shares the same problems such as non-monotonicity. I suspect that you will also get non-monotonicity when votes aren’t “favorite gets 1 and everyone else gets −1”.
EDIT: I should clarify: it’s not 1 for your favorite and −1 for everyone else. It’s 1 for your favorite and close to −1 for everyone else, such that when your favorite is eliminated, it’s 1 for your next favorite and close to −1 for everyone else after rescaling.
Arrow’s Theorem doesn’t say anything about strategic voting. The only reasonable non-strategic voting system I know of is random ballot (pick a random voter; they decide who wins). I’m currently trying to figure out a voting system that is based on finding the Nash equilibrium (which may be mixed) of approval voting, and this system might also be strategy-free.
When I said linear combination of utility functions, I meant that you fix the scaling factors initially and don’t change them. You could make all of them 1, for example. Your voting system (described in the last paragraph) is a combination of range voting and IRV. If everyone range votes so that their favorite gets 1 and everyone else gets −1, then it’s identical to IRV, and shares the same problems such as non-monotonicity. I suspect that you will also get non-monotonicity when votes aren’t “favorite gets 1 and everyone else gets −1”.
EDIT: I should clarify: it’s not 1 for your favorite and −1 for everyone else. It’s 1 for your favorite and close to −1 for everyone else, such that when your favorite is eliminated, it’s 1 for your next favorite and close to −1 for everyone else after rescaling.