oh wow, I really got drawn into a discussion here. now I'm not an expert on QV, but my recollection is this: cardinal voting allows people to express the strength of their preferences rather than just the order. so if there's an issue that a minority cares about a lot while the majority is just indifferent, then the minority can concentrate their votes on that issue and move it higher up the ballot (more so than in an ordinal voting system).
so in your example, if the wolves have a 2/3 preference for sheep for dinner and 1/3 for apples, and the sheep has a 100% preference for apples, then with linear voting the group decision would be 5/3 vs 4/3 in favour of apples. with quadratic voting, the vote weights would be sqrt(1)+2sqrt(1/3) for apples and 2sqrt(2/3) for sheep, and apples would still win. so in this example, the sheep can push the group decision because its preference is stronger than the wolves', something it couldn't express by just ranking the options (if the wolves had a 100% for sheep, that would still win)
so as we just saw, there are different ways to implement cardinal preferences, and QV is one of them that has been shown to be optimal under some standard (but still debatable) utility assumptions.
that being said, if we were going to seriously consider implementing QV, I'd still have a lot of questions - I mentioned collusion, but also about manipulation resistance: what if the wolves anticipate this outcome and strategically misrepresent the relative strength of their preferences to make sure that their absolutely preferred option wins? they could still do that even without collusion. it's another common problem in voting mechanisms, I don't know how strong that problem is in QV (maybe the manipulation would be exponentially hard to compute, or very easy, Idk rn)
so in your example, if the wolves have a 2/3 preference for sheep for dinner and 1/3 for apples, and the sheep has a 100% preference for apples, then with linear voting the group decision would be 5/3 vs 4/3 in favour of apples. with quadratic voting, the vote weights would be sqrt(1)+2sqrt(1/3) for apples and 2sqrt(2/3) for sheep, and apples would still win. so in this example, the sheep can push the group decision because its preference is stronger than the wolves', something it couldn't express by just ranking the options (if the wolves had a 100% for sheep, that would still win)
so as we just saw, there are different ways to implement cardinal preferences, and QV is one of them that has been shown to be optimal under some standard (but still debatable) utility assumptions.
that being said, if we were going to seriously consider implementing QV, I'd still have a lot of questions - I mentioned collusion, but also about manipulation resistance: what if the wolves anticipate this outcome and strategically misrepresent the relative strength of their preferences to make sure that their absolutely preferred option wins? they could still do that even without collusion. it's another common problem in voting mechanisms, I don't know how strong that problem is in QV (maybe the manipulation would be exponentially hard to compute, or very easy, Idk rn)