Let's think about the use of mixed strategies in tennis. Imagine that Venus

and Serena (Williams) are playing against each other. The match is almost

over (the tiebreak of the third set, say) and Venus is about to hit the ball. She

can hit either left (i.e., to Serena's left), denoted by L, or right (i.e., to Serena's

right), denoted by R. At that very moment Serena has to choose what side she

wants to go; either to her left (l) or her right (r). If Venus hits to Serena's

left and Serena goes to her right, then Venus wins the point with probability

.8 and Serena wins the point with probability .2 (this captures the possibility

that the ball goes outside the court). If, instead, Venus hits to Serena's right

and Serena goes to her left, then Venus wins the point with probability .9 and

Serena wins the point with probability .1 (this captures the idea that Venus

hits better to Serena's right). If Venus hits to Serena's left and Serena goes to

her left, then each win the point with probability .5. Finally, if Venus hits to

Serena's right and Serena goes to her right, then Serena wins the point with

probability .8 and Venus wins the point with probability .2 (this captures the

idea that, when she guesses correctly, Serena defends better her right side than

her left). Assume that the probabilities describe Venus and Serena's payoffs.

a. If Serena goes to her left with probability q > .6, where should Venus hit?

b. If Serena goes to her left with probability q < .6, where should Venus hit?

c. Find the Nash equilibrium in mixed strategies of this game.

Imagine now that Serena changes her coach and her new coach helps her improve her "left defense" quite a bit. So, next time she plays against Venus and Venus hits to her left, Venus wins the point with probability .3 and Serena wins the point with probability .7. All other probability (i.e.,payoffs) remain the same.

d. Find the Nash equilibrium in mixed strategies. How likely is it that Serena goes to her left now? Is it more or less likely than before? Why? Is it more or less likely that Venus hits to Serena's left? Why?