I believe that Arrow's Theorem has been widely misinterpreted, and with a slightly different interpretation it makes more sense.
Here is my argument:
For every collection of preferences, there must be some outcome that satisfies Arrow's Theorem. Like, for voting there must be some winner or there is some sort of tie. Either there is a winner or there is not. Arrow's Theorem can't demand that there has to be a winner and there cannot be a winner.
Since for any collection of preferences there must be some outcome that satisfies Arrow's Theorem, we can make an acceptable voting system by simply looking at each collection of preferences and going through the finite list of outcomes until we find one that satisfies Arrow's Theorem.
The reason people say that no voting system can always work, is that they imagine that voting systems cannot have ties. Obviously, no voting system that never results in a tie can be completely fair, nor will it satisfy Arrow's Theorem.
When there are more than two candidates and more than one voter, you can have a tie in new ways, it isn't necessary to have two candidates with exactly the same number of votes. You can also have a tie when -- one way or another -- the voters collectively prefer A to B, B to C, and C to A. Then of course there is no valid way to say which they prefer most. If the voting system prefers A then it's wrong because they prefer C to A, etc.
So a good voting system should not declare a winner in that case.
Here's a further conclusion. A voting system which does not collect enough information, cannot detect that A > B > C > A and so it will not know to declare a tie. So voting systems which satisfy Arrow's Theorem must somehow ask the right questions.
I believe that Arrow's Theorem has been widely misinterpreted, and with a slightly different interpretation it makes more sense.
Here is my argument:
For every collection of preferences, there must be some outcome that satisfies Arrow's Theorem. Like, for voting there must be some winner or there is some sort of tie. Either there is a winner or there is not. Arrow's Theorem can't demand that there has to be a winner and there cannot be a winner.
Since for any collection of preferences there must be some outcome that satisfies Arrow's Theorem, we can make an acceptable voting system by simply looking at each collection of preferences and going through the finite list of outcomes until we find one that satisfies Arrow's Theorem.
The reason people say that no voting system can always work, is that they imagine that voting systems cannot have ties. Obviously, no voting system that never results in a tie can be completely fair, nor will it satisfy Arrow's Theorem.
When there are more than two candidates and more than one voter, you can have a tie in new ways, it isn't necessary to have two candidates with exactly the same number of votes. You can also have a tie when -- one way or another -- the voters collectively prefer A to B, B to C, and C to A. Then of course there is no valid way to say which they prefer most. If the voting system prefers A then it's wrong because they prefer C to A, etc.
So a good voting system should not declare a winner in that case.
Here's a further conclusion. A voting system which does not collect enough information, cannot detect that A > B > C > A and so it will not know to declare a tie. So voting systems which satisfy Arrow's Theorem must somehow ask the right questions.