Question

James and William, two equally talented athletes, expect to compete in the upcoming world championship in the 400-meter hurdles. Each of them is trying to decide how many hours to train each week for the first race. We will use the Tullock model to describe their behavior.

For each athlete, winning is worth 90 hours per week, so we measure the prize as 90 hours. The cost of an hour of effort is, of course, an hour.

Suppose that James plans to train 10 hours per week, and William plans to train 20 hours per week.

Question 1 (0.5 points)

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What is the probability that James wins the race?

Question 1 options:

	1/3
	2/3
	1
	None of the above

Question 2 (0.5 points)

What is the probability that William wins the race?

Question 2 options:

	1/3
	2/3
	1
	None of the above

Question 3 (0.5 points)

What is James's expected payoff?

Question 3 options:

	10
	20
	30
	40

Question 4 (0.5 points)

Suppose James increases his training time to 20 hours while William is still training for 20 hours. What is James's new expected payoff?

Question 4 options:

	10
	15
	25
	30

Question 5 (1 point)

Is the allocation where James trains 10 hours and William trains 20 hours a Nash equilibrium?

Question 5 options:

	No, because James's expected payoff can change by changing the number of hours he trains given that William is training 20 hours.
	No, because James's expected payoff can increase by changing the number of hours he trains given that William is training 20 hours .
	Yes, because James is maximizing his payoff given that William is training 20 hours.
	Yes, because both players are maximizing their payoffs given what the other player is doing.

Question 6 (1 point)

Is the allocation where James and William both train 25 hours a Nash equilibrium?

Question 6 options:

	No, because James's expected payoff can increase by changing the number of hours he trains, given that William is training 25 hours .
	No, because James is maximizing his payoff given that William is training 25 hours, but William can increase his payoff given that James is training 25 hours.
	Yes, because both players are maximizing their payoffs given what the other player is doing.
	The given information is not enough to determine the Nash equilibrium.

Question 7 (1.5 points)

Suppose that the prize increases from 90 hours to 120 hours. Which of the following is a Nash equilibrium with the new prize?

Question 7 options:

	Both James and William train 22.5 hours each.
	James trains 25 hours and William trains 30 hours.
	James trains 30 hours and William trains 25 hours.
	Both James and William train 30 hours each.

Answer 1

Q1) Let James' probability of winning the race be p

Then, Williams probability fo winning = 2p (he is training twice as hard)

=> p + 2p = 1

=> p = 1/3

Thus, the answer is (a) 1/3

Q2) Williams' probability fo winning = 2p = 2/3

Thus, the answer is (b) 2/3

Q3) James' expected payoff = pr(winning)*value from winning + pr(loss)*value from losing

= 1/3(80) + 2/3(-10)

= 60/3 = 20

Thus, the answer is (b) 20

Q4) now since both are training equally, James probability of winning = 1/2

James' expected payoff = pr(winning)*value from winning + pr(loss)*value from losing

= 1/2*(70) + 1/2(-20)

= 50/2 = 25

Thus, the answer is (c) 25