I think you’re misinterpreting Arrow there. Even a voting method that requires cardinal information entails that voters be able to say which one they prefer bet

I think you’re misinterpreting Arrow there. Even a voting method that requires cardinal information entails that voters be able to say which one they prefer between two candidates. As for how a ballot capturing cardinal methods be reduced to ordinal ones: it depends on the ballot. Can you give an example of a cardinal ballot that does not induce an order or otherwise implies something about a voter’s preferences (assuming they were voting sincerely)?