They wouldn't! That's the downside. Or maybe not a downside -- there could be an even stronger incentive for politicians to get stuff done. Since voters would have to make their decisions based entirely actual results, and not an individual politician's voting record, the politicians might have a stronger incentive to build coalitions and influence other politicians' votes.
My half-baked point is that the harm of voters NOT having access to this information is less than the harm of lobbyists and major donors having access to it.
I should point out that places like Germany do vote for the party, not the person for positions like Parliament. In each region the parties submit slates of people they would put into the positions if they got all of the votes, and then once people vote they get a proportionate number of their slate into actual offices. There are lots of details here, but it is a system that works.
Any differential privacy experts care to weigh in? Is there a differential privacy method that can introduce a specific amount of noise in voting results in order to not fully hide nor fully reveal specific representative voting records?
What they say and how they behave.
I really don't believe there are a relevant number of people who can consistently say one thing and vote a completely different thing for years on end.
Maybe I'm naive, though. I really do think nearly everyone in congress does want to make the world a better place, they're just lost/confused about how to go about that, often getting caught up in the game of staying in congress rather than using their time there for good.
It's actually how congress worked before the "sunshine" act of 1971 which made voting records public.
The vast majority of people don't actually go and look up their senator's/congressperson's voting records, where as lobbyists do. So I think the voter disclosure argument is overblown; I don't think that voters actually need to know the exact voting records of their congressperson. I think voting should be selecting people based on character and rhetoric and then letting those people act as representatives.
It would be less democratic, but more functional (and more akin to what the founders imagined anyway).
My half-baked point is that the harm of voters NOT having access to this information is less than the harm of lobbyists and major donors having access to it.