One of Andrew Feffer's jobs is getting Washington Nationals fans to come to the ballpark. But this year, the team's chief operating officer has another goal: keeping Philadelphia Phillies fans out.

Since the baseball franchise left Montreal for Washington, D.C., in 2005, the Nationals pastime has been losing games. They haven't yet had a winning season. Meanwhile, their rivals 125 miles up I-95 have been one of the best teams in the game.

With...