The Davis Removal Company and its main rival, Roland Ltd, have fleets of ten trucks each, which leave the companies' headquarters for Chicago each morning at 5 am for their daily assignments. At that early hour, these trucks are the only vehicles on the roads. Travel time along the road between the Davis Removal Company and Chicago is 20 + 2√x, where x is the number of cars on the road, and it is similarly 20 + 2√x, on the road connecting the headquarters of Roland Ltd with Chicago, where x is the number of cars on the road.
The Illinois Department of Transportation paves a new two-way road between the companies' headquarters, where travel time on this new road is 0.2, independent of the number of cars on the road. This situation is described in the following diagram.
Answer the following questions:
(a) Before the new road is constructed, what is the travel time of each truck between its headquarters and Chicago?
(b) Describe the situation after the construction of the new road as a two-player strategic-form game, in which the players are the managers of the removal companies and each player must determine the number of trucks to send on the road connecting his company with Chicago (with the rest traveling on the newly opened road and the road connecting the other company's headquarters and Chicago), with the goal of keeping to a minimum the total travel time to Chicago of all the trucks in its fleet. Note that if Davis, for example, instructs all its drivers to go on the road between company headquarters and Chicago, and Roland sends seven of its trucks directly to Chicago and three first to the Davis headquarters and then to Chicago, the total time racked up by the fleet of Roland Ltd is
7 × (20 + 2√7) + 3 × (0.2 + 20 + 2 √13).
(c) Is the strategy vector in which both Davis and Roland send their entire fleets directly to Chicago, ignoring the new road, a Nash equilibrium?
(d) Show that the strategy vector in which both Davis and Roland send six drivers directly to Chicago and four via the new road is an equilibrium. What is the total travel time of the trucks of the two companies in this equilibrium? Did the construction of a new road decrease or increase total travel time?
(e) Construct the payoff matrix of this game, with the aid of a spreadsheet program. Are there any additional equilibria in this game?