EC Home work

Problem Set #3

Contests

Please provide explanations and calculations for your answers!

Ec 370 Summer, 2017

Michigan State University

1. Laverne and Shirley, two equally talented athletes, expect to compete for the county championship in the 400 meter hurdles in the up-coming season. Each plans to train hard, putting in several hours per week. We will use the Tullock model to describe their behavior.

For each athlete winning is worth 100 hours per week; so we measure the prize as 100 hours. The cost of an hour of effort is, of course, one hour. The probability is as described in the Tullock model.

a. Suppose that Laverne plans to train 48 hours per week, and that Shirley plans to train 16 hours. What is the probability that Shirley will be the county champ?

__________

b. What is Laverne’s payoff? (i.e. prob x prize – cost) Note: the payoff is measured in hours, not money.

___________

c. Suppose that Laverne decreases her training time to 32 hours per week. Does her payoff rise or fall? Explain.

d. Suppose that Laverne increases her training time to 64 hours per week. Does her payoff rise or fall? Explain.

e. Is the allocation where Laverne trains 48 hours per week and Shirley trains 16 hours per week an equilibrium? Why or why not?

f. Is the allocation where each athlete trains 25 hours per week as Nash equilibrium? (Hint: you can check to see if the payoff rises when, say, Laverne increases to 26 and then when she reduces to 24. You don’t have to check for Shirley’s incentives because the situation is symmetric.)

Number of hours for Laverne Payoff for Laverne

(assuming that Shirley trains 25 hours.)

24 __________

25 __________

26 __________

g. Assume that the prize falls to 60 hours. Show that the 25 hours each allocation is no longer a Nash equilibrium. Hint: you only have to check that the payoff is highest at 24 hours, 25 hours, or 26 hours for Laverne (again, holding Shirley at 25 hours.)

Number of hours for Laverne Payoff for Laverne

(assuming that Shirley trains 25 hours.)

24 __________

25 __________

26 __________

h. Suppose that with the payoff back at 100 hours, a third athlete, Edna, now enters the race. Edna has the exact same ability, and the exact same payoff for winning the race, as Laverne and Shirley. Is 25 hours training each still a Nash equilibrium? Hint: Do the same thing as before, that is hold both Shirley and Edna at 25 hours of effort.

Number of hours for Laverne Payoff for Laverne

(assuming that Shirley and Edna train 25 hours.)

24 __________

25 __________

26 __________

2. One of the predictions of contest theory is that effort is greater in symmetric contests, where ability is relatively equal as compared to asymmetric contests, where ability is unequal. In “the incentive effects of leveling the playing field,” Franke tests this idea with data on amateur golf tournaments. The paper is here.

https://hal.archives-ouvertes.fr/hal-00670763/document

He compares performance when the score uses unadjusted scores versus those that adjusts score by handicap. (Handicap scoring levels the playing field for the lower ability golfer.) Here is the distribution of performance difference in tournaments that use gross (unadjusted) vs net (adjusted). (Performance is measured using the Stabelford system. If there were no difference in performance, the distribution should be approximately binomial with a mean of zero.

a. Looking at the data, what is your guess about which type of tournament has higher (better) performance.

b. Is your answer to part a consistent with the predictions of contest theory? Why or why not?

The table shows regression coefficients (standard errors in parenthesis) for the independent variables. The dependent variable is score. (again, under the stabelford system higher is better).

c. Type is a dummy variable with 1 for net score tournaments and 0 for gross score tournaments. Explain the meaning of the coefficient.

d. Is it significant at the 5% level? Explain.

e. What is the t-statistic on TYPE? ________________

f. The variables PAR and CR are measures of difficulty of the course, and PHANDICAP is a measure of the golfer’s ability. Why are these variables included in the regression equation?

g. Which variable should be added to the regression if you want to test whether the effect of “TYPE” is different for male and female players? Why? (Hint: read slide 9 and slide 10 in lecture 16)