The $3 could be divided among all congresspersons whom vote in favor of their issues, and it could be pre-allocated for potential supporters so politicians know how much money is available if they support it.
I agree that people do not currently always vote in their own interest, and one of the reasons for this is that lobbyists are able to convince people that they don't want something they should. That's the power of propaganda. Lobbyists don't just lobby politicians -- they lobby the people too. One of the systems primary services will be to debunk lobbyist propaganda.
Also, $3 per month isn't a hard number -- $3 just shows how little money it would take to change the game -- compare that to how much extra we are paying in taxes to pay for unnecessary expenditures.