Reason 2

Playing a mixed strategy can keep your opponent off balance.  The worst case payoff of a mixed strategy may be better than the worst case payoff of a pure strategy.

The American women's soccer team played to a tie with with China in regulation play.  The consequence was a shots-on-goal shoot out.  Mia Hamm is on the line shooting on the Chinese goal.  She can shoot left or right.  The goalie can favor either left or right, in the sense of beginning to move in a split second before Hamm makes contact with the ball, in order to preempt and block the shot.

 China L R U.S. L .2 .8 R .7 .3

The numbers in the table are Mia's chances of making the shot. If Mia always kicks Left then the defender will know to play Left and Mia's chance of making the shot is just .20.  Mia could do better if she mixes up her shots 1/2 and 1/2.  That is,

 1/2(.2) + 1/2(.7) = .45 1/2(.8) +1/2(.3) = .55 When China plays Left When China plays Right

Even if China plays the same kind of mixed strategy Mia's chances of scoring are 1/2, much better than playing a pure strategy!

 China L R Mix U.S. L .2 .8 .5 R .7 .3 .5 Mix .45 .55 .5