game theory

EXERCISE 34.3 (Choosing a route) Four people must drive from A to B at the same time. Each of them must choose a route. Two routes are available, one via X and one via Y. (Refer to the left panel of Figure 35.1.) The roads from A to X, and from Y to B are both short and narrow; in each case, one car takes 6 minutes, and each2.8 Best response functions 35 -val 6,9,12,15 Original network. 6,9,12,15 Network with new road from X to Y. Figure 35.1 Getting from A to B; the road networks in Exercise 34.3. The numbers beside each road are the travel times per car when 1, 2, 3, or 4 cars take that road. additional car increases the travel time per car by 3 minutes. (If two cars drive from A to X, for example, each car takes 9 minutes.) The roads from A to Y, and from X to B are long and wide; on A to Y one car takes 20 minutes, and each additional car increases the travel time per car by 1 minute; on X to B one car takes 20 minutes, and each additional car increases the travel time per car by 0.9 minutes. Formulate this situation as a strategic game and find the Nash equilibria. (If all four people take one of the routes, can any of them do better by taking the other route? What if three take one route and one takes the other route, or if two take each route?) Now suppose that a relatively short, wide road is built from X to Y, giving each person four options for travel from A to B: A-X-B, A-Y-B, A-X-Y-B, and A-Y- X-B. Assume that a person who takes A-X-Y-B travels the A-X portion at the same time as someone who takes A-X-B, and the Y-B portion at the same time as someone who takes A-Y-B. (Think of there being constant flows of traffic.) On the road between X and Y, one car takes 7 minutes and each additional car increases the travel time per car by 1 minute. Find the Nash equilibria in this new situation. Compare each person's travel time with her travel time in the equilibrium before the road from X to Y was built