Question

1. Jacob and William, 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 28 hours per week, so we measure the prize as 28 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 Jacob plans to train 15 hours per week, and that William plans to train 25 hours. What is the probability that William will be the county champ?

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

c. Suppose that he increases his training time to 30 hours per week. Does his payoff rise or fall? Explain.

d. Suppose that he reduces his training time to 15 hours per week. Does his payoff rise or fall (compared to part b)? Explain

. e. Is the allocation where Jacob trains 15 and William trains 25 a (Nash) equilibrium? Why or why not? f. Is the allocation where each athlete trains 7 hours per week as Nash equilibrium? (Hint: you can check to see if the payoff rises when, say, Jacob increases his training hours to 8 and then when he reduces them to

6. You don’t have to check for William’s incentives because the situation is symmetric.) Number of hours for Jacob Payoff for Jacob (assuming William trains 7 hours)

6 __________ 7 __________ 8 __________

Answer 1

a.

As, Jacob and William, both are equally talented athletes, probability that William will be the county champ will depend on number of hours William has done in comparison to Jacob.

Probability that William will be the county champ = number of train hours for William / Total number of hours for both

= 25 / (25 + 15) = 0.625

b.

William’s payoff = (prob x prize) – cost

= 0.625 * 28 - 25 = -7.5

c.

If the  training time increased to 30 hours per week,

Probability that William will be the county champ = 30 /(30 + 15) = 0.667

William’s payoff = (prob x prize) – cost

= 0.667 * 28 - 30 = -11.324

So, the payoff will fall.

d.

If the  training time decreased to 15 hours per week,

Probability that William will be the county champ = 15 /(15 + 15) = 0.5

William’s payoff = (prob x prize) – cost

= 0.5 * 28 - 15 = -1

So, the payoff will rise.

e.

If the training time for William decreased to 24 hours per week,

Probability that William will be the county champ = 24 /(24 + 15) = 0.6154

William’s payoff = (prob x prize) – cost

= 0.6154 * 28 - 24 = -6.7688

which is less than the payoff as in part (b). So, the William strategy would not be for 25 hour training. Hence the allocation where Jacob trains 15 and William trains 25 is not a (Nash) equilibrium.

f.

If the training time for Jacob is 6 hours per week,

Probability that Jacob will be the county champ = 6 /(6 + 7) = 0.4615

Jacob's payoff = (prob x prize) – cost

= 0.4615 * 28 - 6 = 6.922

If the training time for Jacob is 7 hours per week,

Probability that Jacob will be the county champ = 7 /(7 + 7) = 0.5

Jacob's payoff = (prob x prize) – cost

= 0.5 * 28 - 7 = 7

If the training time for Jacob is 8 hours per week,

Probability that Jacob will be the county champ = 8 /(8 + 7) = 0.53

Jacob's payoff = (prob x prize) – cost

= 0.53 * 28 - 8 = 6.84

Number of hours for Jacob Payoff for Jacob (assuming William trains 7 hours)

6 ____6.922______ 7 ____7______ 8 ____6.84______

So, the Jacob strategy would be for 7 hour training to maximize his expected payoff. As, the situation is symmetric, William strategy would also be for 7 hour training to maximize his expected payoff. Hence the allocation where where each athlete trains 7 hours per week is Nash equilibrium.