As someone from one of those non-US democracies, the reason why we can mandate IDs for votes is because they're basically trivial to get, whereas from what I've heard of the American hodgepodge of state-level IDs and "this one counts while this one doesn't" shenanigans, that doesn't seem to apply to the US. Now, I know that over there, the several states are responsible for running the elections within their borders -- that being in the US Constitution -- but some states seem to try to (and succeed in) making it unnecessarily difficult, and that may or may not be on purpose. Civil Rights groups certainly seem to think that it is, and that it affects minority populations disproportionately, because of course it does.

For a point of comparison, here in Finland we require that a voter has a valid ID, that is an ID card, a driver's licence with a photo in it, or a passport. Now what if you _don't_ have one of those IDs? Easy, you go to the police station and get yourself a temporary ID, which is by law given free of charge. You do need to get a passport photo taken for that, but those can usually be done in about 10 minutes for around 10 Euros.

And of course there are other things outside of the ID issue which make the elections not rigged against minorities. Simple things like having the actual election day on a Sunday, the polling stations being usually in walking distance in cities -- people living in the countryside most likely have a car -- and that said polling stations tend to be in schools, shops, and other such places where people would be going anyway. Oh, and you usually also get polling stations popping up before the actual election day so that people may vote in advance, in said shops, schools, and similar assorted places.

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The US could totally start doing ID mandates, but the problem is that in the US, their usage doesn't seem to be out of a genuine worry about election integrity or other similar concerns, but out of voter disenfranchisement, which again, hits minority populations disproportionately hard.