edit: wait a minute, perhaps we have to bring the songs that were already chosen (the duplicates) down in our lists.
Generally speaking (meaning for the group overall, not one person): doing that is a double-edged sword.
The effect you seek will happen: if everyone does that, we'll get a lot of songs picked by a single person. But by the same token, songs that were selected by many people might miss the game!
For instance: the song selected by the most people is "Stargazer". If everyone moves that song to the bottom, it misses the game. This also means that "Stargazer" is the unofficial front-runner for the winning song... and the person that drafts it has a very good chance of winning Night Prowler's special secret prize.