Question

Consider this interactive scenario: The 1000 employees of the EconoPlant all live in QuietTown miles away. There are two roads from QuietTown to the EconoPlant, Wide Road and Narrow Road. Every day each employee needs to decide which road to take for commuting to work. The two roads differ in their commute time such that:

• Wide Road has a low-speed limit, but is sufficiently wide so that traffic does not cause a slowdown. The commute on Wide Road takes 40 minutes.

• Narrow Road has a higher speed limit, but congestion can slow traffic. The commute on Narrow Road takes 20 + m/10 minutes, where m is the number of drivers that take Narrow Road.

Employees aim to minimize commute time. Assume that people are identical in that each person chooses Narrow Road with a probability of p at equilibrium. So this is a symmetric mixed strategy equilibrium. Answer the following questions for a specific employee Colin:

(a) (1 point) What is Colin’s commute time on Wide Road?

(b) (1 point) Describe the number of Colin’s colleagues on Narrow Road at equilibrium as a function of p.

(c) (1 point) Suppose Colin chooses Narrow Road. Describe the total number of people on Narrow Road at equilibrium as a function of p.

(d) (1 point) Describe Colin’s commute time on Narrow Road at equilibrium as a function of p.

(e) (2 points) Without solving for p, what is the numerical value of Colin’s equilibrium commute time on Narrow Road that you get in part (d)? Briefly explain how you get the numerical value.

(f) (2 points) What is the equilibrium strategy of each employee? (In other words, what is the value of p at equilibrium?)

Answer 1

Answer:- (a) What is Colin’s commute time on Wide Road?

40 minutes as it is the time taken on wider road.

Answer:- (b) Describe the number of Colin’s colleagues on Narrow Road at equilibrium as a function of p.

There are a total of 999 employees with Mike     . At equilibrium, every employee will select narrow road with probability p. Thus combined a fraction of p of these individual select narrow road and thus there are a total of 999p colleagues on narrow road.

Answer (c):- If Mike chooses to drive on Narrow Ride alongside the other 999p people, there will be 999p + 1 people on that r

Answer:- (d) : Now that there are m = 999p + 1 people in Narrow Road including Mike, his commute time on Narrow Road is

Mike’s commute time on Narrow Road = 20 + m/ 10 = 20 + (999p + 1)/10

Answer (e):- Since Mike is assumed to take Narrow Road with probability p, he is willing to randomize. So his commute time on Narrow Road should be the same as his commute time on Wide Road, which is 40 minutes

Answer (f):- Mike’s commute time on Narrow Road = Mike’s commute time on Wide Road

20 + (999p + 1)/ 10 = 40

P=199/999

P=0.199

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