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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 40 hours per week, and that
Shirley plans to train 20 hours. What is the probability that Shirley
will be the county champ?
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b. What is Laverne’s payoff? (i.e. prob x prize – cost) Note: the payoff is measured in hours, not money.
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c. Suppose that Laverne decreases her training time to 30 hours per week. Does her payoff rise or fall? Explain.
d. Suppose that Laverne increases her training time to 50 hours per week. Does her payoff rise or fall? Explain.
e. Is the allocation where Laverne trains 40 hours per week and Shirley trains 20 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.)