Can mixed strategies lead teams to behave in the opposite manner of what the game predicts?

After going over the mixed strategy game in class that involved the decision on whether to pass or run in football, I started thinking about what this could potentially imply.  To recap, after solving for the probabilities using the game, we found that the defense will prepare for pass about 66 percent of the time, while the offence will choose to pass about 33 percent of the time.  I found this interesting because, if both parties are aware of the thought processes going for each side, wouldn’t they react to this information?

I realize that the probability is a result of the associated payoffs; if the defense allows the offence to complete a pass, it is more likely to have a greater impact than allowing a run.  However, with this information, the defense will realize that the offence is expecting them to run a pass defense.  This realization is what will lead the defense to want to prepare for a run, because they know the offence is most likely to do so.  The knowledge of the game seems to hinder the decisions of each player in the game.

What if a player was to realize that this mixed strategy is what is determine the other player’s decisions?  Taking the side of the defense, even though it seems more beneficial to defend the pass most of the time, it would actually be in their favor to defend for a run.  Since the offence knows the defense is most likely to defend for a pass, they will be ready to run the football.

By now I am sure you can see the problem I am presenting.  With the mutual knowledge that the other player is going to react to the circumstances, it would be more likely for these players to go against the mixed strategy to achieve a better outcome.  It would seem counter intuitive to follow the strategy, knowing that the opponent has the same knowledge as you about the game.  It would almost seem more likely for the probabilities to represent their opposite.  For example, a strategy with a probability 1/3 would represent 2/3 because the player would have an incentive to the opposite, given the opponent has equal knowledge of his decision.

For purpose of this argument, I will pretend the game presented in class has the same possible payouts as in the NFL.  If this is the case, it would be likely that NFL teams would run more often than pass.  However, when taking the stats for 2013, every team, except for 2 teams, chose to pass more than 50 percent of the time.  I realize that in a real game of football there is more to making decisions than how we analyzed the game, but it is interesting to see that the real statistics yielded opposite results, just as my hypothesis suggested.