Imagine a match between the two all-time best women players-Martina Navratilova and Chris Evert. Navratilova at the net has just volleyed a ball to Evert on the baseline, and Evert is about to attempt a passing shot. She can try to send the ball either down the line (DL; a hard, straight shot) or crosscourt (CC; a softer, diagonal shot). Navratilova must likewise prepare to cover one side or the other. Each player is aware that she must not give any indication of her planned action to her opponent, knowing that such information will be used against her. Navratilova would move to cover the side to which Evert is planning to hit or Evert would hit to the side that Navratilova is not planning to cover. Both must act in a fraction of a second, and both are equally good at concealing their intentions until the last possible moment; therefore their actions are effectively simultaneous, and we can analyze the point as a two-player simultaneous-move game.

Payoffs in this tennis-point game are given by the fraction of times a player wins the point in any particular combination of passing shot and covering play. Given that a down- the-line passing shot is stronger than a crosscourt shot and that Evert is more likely to win the point when Navratilova moves to cover the Wrong side of the court, we can work out a reasonable set of payoffs. Suppose Evert is successful with a down-the-line passing shot 80% of the time if Navratilova covers crosscourt; she is successful with the down-the-line shot only 50% of the time if Navratilova covers down the line. Similarly, Evert is successful with her crosscourt passing shot 90% of the time if Navratilova covers down the line. This success rate is higher than when Navratilova covers crosscourt, in which case Evert wins only 20% of the time. Clearly, the fraction of times that Navratilova wins this tennis point is just the difference between 100% and the fraction of time that Evert wins.

(a) Represent this game in a payoff-matrix form.

(b) Show that this game does not have a Nash Equilibrium in pure strategies.

(c) Find the mixed strategy Nash Equilibrium of this game.