Assignment
1. Another popular sport on SASN is One-out, Two-base baseball. In this sport there are, obviously, two bases: home and 2nd. The batter hits the ball and runs to 2nd. If he or she (it's a co-ed sport) gets a hit, then the batter is on second. If it's an out, then the inning is over and the other team gets to bat. Suppose that it's a hit. Then the next batter bats with a runner on 2nd base. If that batter makes an out, the inning is over. If that batter gets a hit, the run scores and his or her team wins. The probability of any batter getting a hit is 1/3.
a. There are four states in this game. List all four of them (win or lose are not counted as states).
b. Draw the tree. Why is it a "Recursive Dynamic Programming" problem?
c. Write an equation for the value of being the team at bat with no one on base, assuming that the value of winning is 10. (Hint: let V be the value of being at bat with no one on base. Then -V is the value of your opponent being at bat with no one on base.)
d. Solve the equation in part c. Value = ________________
e. What is the probability that the team at bat with no one on base will win the game?
Probability = ______________
2. More on One-out, Two-base baseball. The key strategic variable for the team in the field is where to position the rover fielder. The key strategic variable for the batting team is where to hit the ball. The following table gives the strategies and payoffs for the first play. The number in the table is the probability of a hit. The team at bat wants a large probability, the team in the field wants a small one. Experience has taught the rovers to wait at the mound and then move suddenly to the left, right or center field as the ball is hit. Thus, you can think of this as a simultaneous move situation.
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Team in the field
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Right field
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Center field
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Left field
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Pitcher's mound
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Team at bat
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Bunt
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.250
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.150
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.180
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.100
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Hit to right
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.240
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.280
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.300
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.230
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Hit to left
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.400
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.300
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.150
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.220
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Hit to center
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.350
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.200
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.350
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.210
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a. Does the team at bat have a dominant strategy? If so what is it?
b. Does the team at bat have a dominated strategy? If so what is it?
c. Write out the best responses, for each team, to each of the other team's moves.
Team at bat Team in field
Team in Team at Team at Team in
field bat bat field
d. What are the Nash equilibrium strategies? Show your reasoning
3. Mike Caruso, the three-time national wrestling champion from Lehigh University, had two great take-down moves: a quick single leg and his legendary barrel roll. (By the way, Mike wrestled in 1967 when Michigan State won the NCAA team championship.) His opponent could better defend the barrel roll with a more upright stance. The better option against the single leg was a more crouched stance. Of course, in wrestling everything happens so fast that you can think of it as a simultaneous game. The probability of a take-down for Mike is given in the table
Defender
Stand upright Crouch
Single leg 0.8 0.4
Mike Caruso
Barrel roll 0.2 0.6
a. Is there a Nash equilibrium in pure strategies? Explain briefly.
b. Determine the optimal mixing probability for Mike. Please show your calculation equation.
c. If Mike mixes as you suggest in part b., what is the probability that he makes the take-down?
4. Larry and Courtney are settling a grade dispute on the basketball court. Larry has a nice drive and scoop that sometimes works against younger players who don't know 1950s basketball. He also has a jumper patterned after the jumper of the former mayor of Detroit, Dave Bing. Here are his data from similar grade-settling games with students.
Drive Jumper
Score 80 25
Fail 30 20
a. The Chi-square test for the difference in means gives a p-value of 0.0379. What would you recommend for Larry? Explain.
b. Again analyzing his past data, Larry does a runs test. He finds the following results.
Expected Number of Runs: 64.9; sd: 5.1064
z-value= 2.96274;
His own data indicate 80 runs. What is he doing wrong? Explain.
5. The diagram below shows the point value of having the ball on each yard line. It is taken from Romer's "It's fourth and two."
a. What is the value of having the ball on the 50 yard line?
b. If you have the ball on your own 10 and gain 30 yards, how many points have you gained in expected value (Remember that my "own" side of the field is on the left hand side of the horizontal axis)? Please also write out the equation for your calculation.
c. If you have the ball on the 50 and fumble, and your opponent recovers the ball at the same yard line, how many points have you lost in expected value?