Bill Belichick should have used Signaling to win the Sequential Football Game

New England coach Bill Belichick has been known for his in-game coaching as a great decision maker; however, this moniker recently came into question after electing to go for the 1st down on a 4th and 2 yards to go up by 6 points. Upon not converting 4th down, the Patriots turned over the ball, allowing the Colts to make the game winning drive in their come from behind win, 34-35. Brian Burke of the New York Times writes in his article:

With 2:08 left and the Colts with only one timeout, a successful 4th-and-2 conversion wins the game for all practical purposes. A conversion on 4th-and-2 would be successful 60 percent of the time. Historically, in a situation with 2:00 left and needing a TD to either win or tie, teams get the TD 53 percent of the time from that field position. The total win probability for the 4th-down conversion attempt would therefore be:

(0.60 * 1) + (0.40 * (1-0.53)) = 0.79 WP (WP stands for win probability)

A punt from the 28 typically nets 38 yards, starting the Colts at their 34. Teams historically get the TD 30 percent of the time in that situation. So the punt gives the Pats about a 0.70 WP.

The math makes sense, but does the strategic move in this game make sense? Football is essentially a sequential game in that as the offense lines up for a play, the defense chooses their defensive strategy. The statistics of whether or not to go for it on 4th down is known by both coaches; therefore, along with the historical data of the Patriots and Bill Belichick, the Colts knew that the Patriots would attempt to get the 1st down. So what should Bill Belichick have done to give his team a better probability of winning the game? By signaling. Bill Belichick could have called a fake punt as a signal to the Colts of the offensive play the Patriots were calling. In this play, the Colts would have set their defense to defend a punt, but the Patriots would have enjoyed the same statistical probability of converting the 4th down while reducing the Colts probability of not allowing the 1st down. The statistics of 70% and 79% chance of winning are too close to make a significant decision based on pure statistics alone. This game situation needs to be changed by an improper signal to attempt to change the Colts response to the Patriots move.