Problem Set #2
Michigan State University
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 the game and his or her team wins. The probability of getting a hit is 0.6.
a. Draw the tree.
b. Write an equation for the value of being the team at bat with no one on base.. (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.)
c. Solve the equation in part b.
Value = __.72______
d. What is the probability that the team at bat with no one on base will win the game?
Probability = ____.36______
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.
Team in the field
Hit to right
Team at bat
Hit to left
Hit to center
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 both teams.
Team at bat
Team in field
d. What are the Nash equilibrium strategies?
For the team in the field to play at the pitchers mound and the team at bat to bunt.
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
a. Is there a Nash equilibrium in pure strategies? Explain briefly.
No, there is no combination of strategies where each is playing the best response
b. Determine the optimal mixing probability for Mike.
c. What is the probability that he makes the take-down.
0.325 = 32.5%
4. Larry and Courtney are settling a grade dispute on the basketball court. It’s one point per basket. They play to 15. 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 mayor of Detroit (the current one, not Kwame). Here are his...
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