5. City Planning (5 marks total) There are 1000 commuters that drive every weekday morning from their homes in Suburbia to their office jobs in Urbania. Each of these drivers has two choices for the route they take from home to work: the first goes through Allansville and the second through Borisberg. The highway joining Suburbia to Borisberg and the highway joining Allansville t \\J7,"\\ari wrhe 339 Assignment 2.paf ~ they always take 25 minutes, no matter how many drivers are on the highway. By contrast, the highways from Suburbia to Allansville and from Borisberg to Urbania are less reliable: the transit time depends on how many other commuters choose to travel along those routes. Specifically, if n commuters take the highway from Suburbia to Allansville in the morning, then the transit time for each commuter is minutes along this path. Similarly, if n commuters drive from Borisberg to Urbania, then the transit time along this road for each commuter is 7 minutes. We can model this situation as a simultaneous game with 1000 players - the commuters. Each morning, each commuter must decide on a route to take to get to work, either via Allansville (Suburbia Allansville Urbania) or via Borisberg (Suburbia Borisberg Urbania). We will set the payoff to each commuter to be the negative of their travel time (so that we can maximize payoffs by minimizing commuting time). (a) (2 marks) Find a pure strategy Nash equilibrium for this game. Be sure to justify why it is a Nash equilibrium. How long is travel time for the average commuter at this equilibrium? You should now be able to fairly easily find a mixed strategy Nash equilibrium in which all commuters are playing mixed strategies what is it and what are expected travel times at the mixed equilibrium? (b) (3 marks) In the hopes of easing congestion and shortening commutes, city planners are considering creating a tunnel that would directly link Allansville and Borisberg. With a tunnel, there would now be four options for every driver to get to work: e Suburbia Allansville Urbania e Suburbia Borisberg Urbania e Suburbia Allansville Borisberg Urbania e Suburbia Borisberg Allansville Urbania There are two proposals for the tunnel being considered. The first proposal is for a basic tunnel, and would allow for a travel time of 6 minutes in either direction between Allansville and Boris- berg no matter how many people use it. The second option is a higher quality tunnel that would reduce travel time to 3 minutes in either direction between the two, again regardless of how many people use it. The travel times among the other routes would follow the same rules as outlined in the original problem above. You have been hired to advise the city planners to help in their consideration of the two propos- als. We assume that all commuters are perfectly rational (i.e. they'll play according to a Nash equilibrium). Suppose that the six minute tunnel costs twenty million dollars and the 3 minute tunnel costs fifty million. We assume that each one-minute reduction in average commuting time between Suburbia and Urbania is worth some value v to the city as a whole. What would your advice to the planners be and how would this advice depend on this value v7