Receiver Guess Guess Forehand Backhand Server to Forehand Serve To Backhand 90 45

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1 Homework 11, Math10170, Spring 2015 Question 1 Consider the following simplified model of a game of tennis, where the server must decide whether to serve to the receivers forehand(h) or backhand(b). The receiver who will not have enough time to react one they see the direction of the serve will anticipate a serve to their backhand (B) or a serve to their forehand (F). Suppose we know that the server wins the point 90% of the time when they serve to the receiver s backhand and the receiver guesses wrongly the server wins the point 45% of the time when they serve to the receiver s backhand and the receiver guesses correctly. The server wins the point 70% of the time when they serve to the receiver s forehand and the receiver guesses wrongly and the server wins the point 30% of the time when they serve to the receiver s forehand and the receiver guesses correctly. (a) Fill in the pay-off matrix for the server (row player) below. We give the payoff in percentages, each representing the probability that the row player (the server) will win the point in the given situation. This is a constant sum game with the payoff s adding to 100. Server Receiver Guess Guess Forehand Backhand Server to Forehand Serve To Backhand

2 Question 2 Endgame Basketball [Ruminski] Often in late game situations, a team may find themselves up by two points with the shot clock turned off. In this situation, the offensive team must decide whether to shoot for two points, hoping to tie the game and win in overtime, or to try for a three pointer and win the game without overtime. The defending team must decide whether to defend the inside or outside shot. we assume that the probability of winning in overtime is 50% for both teams. In this situation, the offensive team s coach will ask for a timeout in order to set up the play. Simultaneously, the defensive coach will decide how to set up the defense to ensure a win. Therefore we can consider this as a simultaneous move game with both coaches making their decisions without knowledge of the other s strategy. to calculate the probability of success for the offense, Ruminski uses League wide statistics on effective shooting percentages to determine probabilities of success for open and contested shots. He gets Shot Success rate open 2pt. 62.5% open 3pt. 50% Contested 2pt. 35.7% Contested 3pt. 22.8% Using this and the 50% probability of winning in overtime for each team, we can figure out the probability of winning for each team in all four scenarios using the following tree diagram: Start Off. Shoot 2 Off. Shoot 3 Def. Defend 2 Def. Defend 3 Def. Defend 2 Def. Defend Overtime Def. wins Overtime Def. wins Def. wins Off. wins Def. wins Off. wins Off. wins Def. wins Off. wins Def. wins (a) Use the above percentages to fill in the probabilities where appropriate on the tree diagram above. (b) Use those probabilities to fill in the probabilities of a win for the row player (offense) in the payoff matrix below. (Note the probability for a win for the defense team is 1 - prob. win for offense.) For each situation in the matrix below there is a single path in the above tree diagram leading to a win for the Offense, we multiply the probabilities along the path to get the probability that the offense will win in that situation. Offense Defending Team Defend 2 Defend 3 Shoot Shoot

3 Question 3: Dutta : Drug Testing Two swimmers, Rogers and Carter, are about to compete in a runoff. Each athlete has the option of using a performance enhancing drug (d) or not using it (n). Lets assume that both competitors have equal abilities and are the two top competitors with no serious competition for first and second place. In class, we saw that If no drug testing exists the payoff matrix for Rogers (R) looks like the one shown on the left below if we use the potability of a win for Rogers as the payoff ( This is a constant sum game where both payoffs add to 1). Rogers No Testing Carter d n d n Rogers Carter d n d 0 1 n -1 0 On the other hand, if we count a win for Rogers as a payoff of 1 for Rogers and a loss as a payoff of -1 for Rogers, we can use the expected payoff for Rogers in our payoff matrix as shown on the right above (This is a zero-sum game). How Drug Testing Changes The Game On the other hand lets consider the situation where the IOC tests (only) one of the swimmers and that both swimmers are equally likely to be tested. If a swimmer tests positive, then the race is awarded to the other swimmer and the swimmer who tested positive faces a further penalty. This penalty would have a large impact on the swimmer s career, we will denote its (payoff) value by b where b is a relatively large positive number. Using 1 as the payoff for being awarded first place in the race and (-1 for not being awarded first place ) fill out the probability distributions for each player s payoff for each scenario shown below and calculate his expected payoff for each situation. The tree diagrams should help to calculate probabilities in each case (Here the probability that R wins is the probability that they reach the finish line first; if either swimmer wins and then the tests come back positive, the race will be awarded to the other swimmer) : R(d), C(d) P(R wins) = 1/2 = P(C wins) 3

4 X = Y = P(X) = P(Y) Payoff R Payoff C Prob. XP(X) YP(Y) RT, RW -1-b (-1 - b)(0.25) 0.25 RT, CW -1-b (-1 - b)(0.25) 0.25 CT, RW 1-1-b (-1 - b)0.25 CT, CW 1-1-b (-1-b)0.25 E(x) = 2(-1-b)(0.25) E(Y) = = b(0.5) b/2 =-b/2 R(d), C(n) P(R wins) = 1, P(C wins) = 0 X = Y = P(X) = P(Y) XP(X) YP(Y) Payoff R Payoff C Prob. RT, RW -1-b (-1-b)(0.5) 0.5 RT, CW -1-b CT, RW CT, CW E(X) = -b/2 E(Y) = 0 R(n), C(d) P(R wins) = 0, P(C wins) = 1 X Y P(X) = P(Y) XP(X) YP(Y) Payoff R Payoff C Prob. RT, RW RT, CW CT, RW 1-1-b CT, CW 1-1-b (-1-b)(0.5) E(X) = 0 E(Y) = -b/2 4

5 R(n), C(n) P(R wins) = 1/2 = P(C wins) P(X) P(Y) P(X) = P(Y) XP(X) YP(Y) Payoff R Payoff C Prob. RT, RW RT, CW CT, RW CT, CW E(X) = 0 E(Y) = 0 (a) Calculate the expected payoffs for each player in each of the four possible scenarios (Note that this is not a constant sum game, so you need to include the payoff for both players for each scenario.) IOC Testing 5

6 Rogers Carter d n d (-b/2, -b/2) (-b/2, 0) n (0, -b/2) (0, 0 ) Question 4 The following graphs show the distance from the starting point x(t) at time t for two sprinters in a 100 meter race. Answer the following True/False questions by circling the correct answer in each case: (a) Sprinter B wins the race: T rue False (b) Sprinter A has a constant (instantaneous) velocity throughout the race : True F alse (c) Sprinter A s average velocity over the first ten seconds is greater than Sprinter B s average velocity over the first 10 seconds : True F alse (d) Sprinter A speeds up at the end of the race : True F alse (e) Sprinter B speeds up at the end of the race : T rue False (f) Sprinter A starts the race with a higher speed than Sprinter B : T rue False (g) Sprinter B accelerates at the beginning and end of the race and does not accelerate in the middle of the race : T rue False (h) Sprinter A has a positive acceleration throughout the race: True F alse 6