Yonggang Chen

Assignment Requirements

Economics of Sports
finish all of questions.

Problem Set #2

Ec 370                                                                                                                                    Summer, 2014

Michigan State University                                                                                            G. Wallsworth

 

1. Another popular sport on SASN is One-out, Two-base baseball. In this sport there are, obviously, two bases: home and 2^nd. The batter hits the ball and runs to 2^nd. 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 2^nd 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.

1. Draw the tree.

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 =   ________________

What is the probability that the team at bat with no one on base will win the game?

Probability =     ______________

1. 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

 

right                       center                   left                         pitcher’s

field                       field                       field                       mound

 

Bunt                      .300                        .400                        .350                        .250

 

Hit to right           .100                        .250                        .300                        .200

 

Team at bat

Hit to left             .350                        .250                        .100                        .220

 

Hit to center       .250                        .100                        .250                        .240

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?

______________

1. Write out the best responses for both teams.

 

Team at bat                                                                Team in field

Team in                Team at                                Team at                Team in

field                       bat                                         bat                         field

What are the Nash equilibrium strategies?

1. 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                             0.7                                                          0.2

Mike Caruso

Barrel roll                             0.1                                                          0.4

 

1. Is there a Nash equilibrium in pure strategies?       Explain briefly.

1. Determine the optimal mixing probability for Mike.

What is the probability that he makes the take-down.

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 data from similar grade-settling games with students.

 

Drive                     Jumper

 

Score                     80                           25

 

 

Fail                         30                           20

 

1. The Chi-square test for the difference in means gives a p-value of 0.0379. What would you recommend for Larry? (No facetious comments.) Explain.

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 your own 30 yard line?               _____________

 

b. If you have the ball on your own 30 and gain 18 yards, how many points have you gained in expected value?

______________

 

c. If you have the ball on the 30 and fumble, and your opponent recovers the ball at the same yard line, how many points have you lost in expected value?

 

_______________

 

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