The Miami Dolphins pulled this one off against division rivals New York Jets in what is known as the "Fake Spike Game." With only 22 seconds remaining in the clock, the Dolphins were out of timeouts and were down 21-24. The Dolphins reached the Jets 8-yard line. Quarterback Dan Marino seems to indicate that he was going to spike the ball to stop the clock and try a field goal that will tie the game and push for an overtime. Anticipating a spike, the Jets defense lined up haphazardly. Instead, Marino didn't spike the ball and with the Jets caught off guard, Marino passed the ball to wide-receiver Ingram for a touchdown. Because of the "fake spike," the Dolphins ended up winning the game 28-24 in regulation rather than pushing for overtime.
Most trick plays are a form of this. The play-action pass is a good example. Fake a hand-off to the runningback, expecting the defense to play as they should against the run: converge on the runningback. Meanwhile you, the quarterback, have all the time in the world to pass down the field. Like all true Batman Gambits, this can be ruined spectacularly. An all-out blitz can disrupt a play-action pass, as enough blitzers can get in the backfield to swallow BOTH the QB and RB, or the defenders see the hand-off as a fake and stick to their pass defending assignments.
The New England Patriots almost pull this off against the New York Giants in Super Bowl XLVI. Late in the fourth quarter, the New York Giants were running down the clock before scoring and taking the lead. With under a minute left in the game, Bill Belichick tells the defense to let the Giants score the touchdown, in order to give Tom Brady time to score a game winning touchdown. If not for his receivers dropping crucial passes, Tom Brady might have pulled it off.
The Green Bay Packers did the same thing in Super Bowl XXXII, letting the Denver Broncos score late to give Brett Favre a chance to come back and tie the score (betting that the Broncos were deep enough that they would have scored at least a field goal anyway, and holding them to that would have drained the clock too much). The gambit didn't work there either.
Oakland A's manager Dick Williams pulled off a masterful one in Game 3 of the 1972 World Series against Cincinnati Reds catcher Johnny Bench. Bench was red-hot at the plate and had worked the count against A's pitcher Rollie Fingers to 3 balls and 2 strikes. Williams called time and spoke to Fingers and catcher Gene Tenace. When Tenace returned to the plate, he held out his glove to indicate Fingers throw deliberately wide to walk Bench. Bench reacted just as Williams expected, relaxing at the plate and awaiting a free pass to first base. Tenace then dropped back into his crouch as Fingers fired a slider over the corner of the plate for strike 3.
Really, the vast majority of sports where one compete directly against their opponents are going to feature this as a matter of routine. In basketball, a pump fake is to get the defender to jump to block the shot he thinks is coming, allowing one to either dribble around him or draw a foul. In hockey, players will wind up as if they were going to take a slap shot to get goalies to react or defenders to hit the ice in an attempt to block it, only to hold back, and instead pass the puck to an open teammate with the defense too committed to blocking the threatened shot to react. Defenses in football will clearly show certain coverages and blitzes, only to run others, with the hopes of getting the offense to react a certain way, and Q Bs will pump fake hoping to get defensive backs to take a step in one direction, before throwing a real pass in another direction. About the only sport this doesn't happen in is golf, where the competitors are competing indirectly against each other by competing directly against the golf course, and naturally, it's difficult to play head games against dirt and grass.