Game theory has many applications to sports, particularly when we consider mixed strategies. The normal-form game box below models a simple part of basketball. The players are a guard (who is on offense) and a defender. The guard can either drive to the basket or shoot a three-point shot. The defender can defend the drive, which leaves the shot open, or defend the shot, which leaves her susceptible to the drive. For simplicity, we'll assume that an undefended drive always goes into the basket, while an open three-point shot goes in with probability p. We'll also assume that a well-defended shot or drive always misses. The payoffs are in terms of points scored for the offense and given up by the defense.

Basic Basketball		Defender
		Defend drive	Defend shot
Guard	Drive	0, 0	2, -2
	Shoot	3p, -3p	0, 0

There is no pure strategy equilibrium to this game (check for yourself). Suppose that the guard will make a three-pointer with probability p=0.27p=0.27. Find the mixed strategy Nash equilibrium. In it, the guard will drive with probability ______ , and the defender will defend the drive with probability ______ . (Give your answers to two decimal points.) How many points is the guard expected to score per possession? ________

Suppose that the guard practices her three-point shot intensely and is able to improve the probability that she makes an open three-pointer to 0.32. Find the new optimal probability that she will drive in a mixed strategy equilibrium. _______ (Give your answer to two decimal points.)

The basketball team has hired a new coach who is emphasizing simplicity and specialization. He wants each player to focus on her strengths. The new coach recognizes that the guard from Parts 1 and 2 has worked very hard to improve her three-point shooting; she is now the best three-point shooter on the team. The coach, however, thinks she is not playing to her strength because she is not shooting as many three-pointers as she used to. He thus tells her to shoot on 80% of her possessions. Using the model above and assuming that the probability the guard makes a three-pointer is p=0.32p=0.32, how many points should the guard expect to score per possession if she shoots with probability g=0.8? _______ How many points should the guard expect to score per possession if she ignores her coach and shoots according to the mixed strategy probability from Part 2? ______

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