There are two cities A and B joined by two routes. There are 80 travelers who begin in city A and must travel to city B. There are two routes between A and B. Route I begins with a highway leaving city A, this highway takes one hour of travel time regardless of how many travelers use it and ends with a local street leading into city B. This local street near city B requires a travel time in minutes equal to 10 plus the number of travelers who use the street. Route II begins with a local street leaving city A, which requires a travel time in minutes equal to 10 plus the number of travelers who use this street and ends with a highway into city B which requires one hour of travel time regardless of the number of travelers who use this highway.

(a) Draw the network described above and label the edges with the travel time needed to move along the edge. Let x be the number of travelers who use Route I. The network should be a directed graph as all roads are one-way.

(b) Travelers simultaneously chose which route to use. Find the Nash equilibrium value of x

(c) Now the government builds a new (two-way) road connecting the nodes where local streets and highways meet. This adds two new routes. One new route consists of the new road and the local street into city B (on Route I). The second new route consists of the highway leaving city A (on Route I), the new road, and the highway leading into city B (on Route II). The new road is very short and takes no travel time. Find the new Nash equilibrium. (Hint: There is an equilibrium in which no one chooses to use the second new route described above.)

(d) What happens to total travel time as a result of the availability of the new road?

(e) If you can assign travelers to routes, then in fact it’s possible to reduce total travel time relative to what it was before the new road was built. That is, the total travel time of the population can be reduced (below that in the original Nash equilibrium from part (b)) by assigning travelers to routes. There are many assignments of routes that will accomplish this. Find one. Explain why your reassignment reduces total travel time. (Hint: Remember that travel on the new road can go in either direction. You do not need find the total travel time minimizing assignment of travelers. One approach to this question is to start with the Nash equilibrium from part (b) and look for a way to assign some travelers to different routes so as to reduce total travel time.)

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