Question

Starting with the setup of questions 1-7, suppose that a third player, Mike, decides to compete in the race as well. Mike is equally talented as James and William. Suppose James and William are training 10 hours each and Mike is training for 20 hours. The prize for each of them is 90 hours.

Question 8 (1 point)

What is James's probability of winning?

Question 8 options:

	1/3
	2/3
	1/2
	1/4

Question 9 (1 point)

What is James's expected payoff?

Question 9 options:

	20
	30
	40
	None of the above

Question 10 (1.5 points)

Which of the following is a Nash equilibrium?

Question 10 options:

	Each player trains for 10 hours.
	Each player trains for 20 hours.
	James and William train for 10 hours each and Mike trains for 20 hours.
	James and William train for 20 hours each and Mike trains for 10 hours.

Answer 1

1> We know that the probability of winning a race is the ratio of the training hours for each of them

So, Jame's winning probability = James training hours/(James training hours+ Mike training hours+William training hours)

10/(10+20+10)=1/4

So, the correct option is D

2>

He loses with probability 3/4, so he gets a payoff of -10 from it.

He wins with probability 1/4, so he gets a payoff of 90-10=80 from it.

Thus expected payoff is 0.75*(-10)+0.25*80 = 12.5

Thus, the correct option is D

3> A,B

In a Nash equilibrium, everyone plays their best response, so we have a Nash equilibrium when each of them plays the same number of hours, then they do not have any incentive to play more or less.