In Arrow's Impossibility Theorem, what's the difference between a rank vote and a cardinal vote? Arrow's impossibility theorem states that in any rank-based voting system involving three or more candidates, at least one of the following criteria will by necessity be violated:


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*If every individual prefers Choice A to Choice B, then the group prefers Choice A to Choice B

*If every voter's preferences between Choice A and Choice B remain unchanged, then the group's preferences between Choice A and Choice B remain unchanged

*There is no one "dictator," or individual who can alone sway the group's preferences


However, according to Dr. Kenneth Arrow his theorem does not apply to cardinal voting systems as inherently distinct from rank-based or ordinal voting systems.
What does this mean? What's the difference between a cardinal voting system and a rank-based voting system?
 A: Well, "in Arrow's theorem", there are no cardinal votes, since Arrow's theorem only applies to ordinal votes.  The main branches of voting systems are:


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*Ranked (ordinal) systems, where voters rank the candidates in order of preference: 1st choice, 2nd choice, 3rd choice, etc.

*Rating (cardinal) systems, where voters rate each candidate independently, so 100% approval, 0% approval, 73% approval, etc.


Rating systems provide more information, since they allow voters to give the same score to multiple candidates, if they feel the same way about each, and to express varying degrees of preference.  For instance, if the only three options on the ballot are


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*A: Eat strawberry ice cream

*B: Eat chocolate ice cream

*C: Eat rotting garbage


The preference ranking may be A > B > C, but B > C is a much stronger preference than A > B.  Ratings would be more like A: 100%, B: 99%, C: 0%.
If the society's preferences are strictly 1-dimensional, a ranking system carries sufficient information to determine the Condorcet ("beats-all") winner.  If there are two or more dimensions, it does not.
As I've been told, Gibbard's theorem applies to all conceivable voting systems, proving that they are all subject to strategic voting to some degree.
