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Brian Burke who authors a blog, Advanced NFL Stats, produced detailed analysis, "Fourth Downs in the New Overtime: First Possession." In an NFL overtime game, the team that possesses the ball first can only win the game with a touchdown or safety. After that possession, any score will win the game.
The numbers show the team which has the ball first in overtime has a better chance of winning if, on fourth down deep in its own territory, it tries for a first down instead of punting. The deeper the team is in its own territory, the chances are better that that the team will make a first down rather than give the ball to the opposing team which should have great field position and is likely to score. Over the long run, this strategy may win more games on average but an NFL coach is looking at losing his job if it doesn't work just once.
Another finding is that long field goals should not be attempted on a team's first possession in overtime. Their projected success rate is overshadowed by the loss of field position if it is missed and even if it is good, the opposition still has a chance to win.