Exercise 1: Part 1 and Part 2

2 Exercise 1. (20 points) (1) Consider the game F = (1, {5,}, {u,}) where I = {1,2,3}, 5'1 = {42, b}, 8'2 2 {L, R}, and S3 = {A, B}. The payoffs are given by the following matrices (in each cell, the rst entry is 1's utility, the second entry is 2's utility and the third entry is 3's utility): L R L R a-- 0,-- b 3,1,2 1,2,0 b 3,0,2 1,1,4 A B Find the pure Nash equilibria of this game (Hint: Player 1 chooses the row, player 2 chooses the column, and player 3 chooses the matrix). (2) Consider the road network shown in Figure 1, on which 2 drivers wish to travel from Start to End. Each arrow represents a road and the travel time in minutes is given by the number t. For example, it takes it = 45 minutes to drive from Start to point B and t = 20T minutes to drive from Start to point A (where T is the total number of drivers who use that road). Therefore, the total time needed to drive from Start to End (going through A) is 20T + 45. Suppose rst that the dashed road does not exist. Assume that a driver's utility function is strictly decreasing in the total time needed to drive from Start to End. a Derive a Nash equilibrium of this game. Is this the unique Nash equilibrium? How long does it take to go from Start to End in equilibrium? FIGURE 1. Road Network The State is considering to build a road that connects A to B (the dashed line). On that new road, the travel time would be extremely short: approximately 0 minutes. As a gametheory expert, you are hired to advise the State whether building the new road is a good decision, the objective being to reduce drivers' travel time (in equilibrium)