Usually, the ten million letters win; after all, if ten million people believe something, there's probably a good reason for it. Probably, and let me stress that. After all, look at popular media, music, movies, television, books... just because millions of people like something doesn't mean its good. It just means its popular.
If this republic were meant to be run by popular opinion, you wouldn't need me. A trained monkey could weight letters and read polls and just do what everyone else thought was right. When people vote for me, they vote because they believe I can make the right decisions for my constituents. The right decision for my country.